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Neutron scattering investigations on

methyl group dynamics in polymers.

Juan Colmenero1,2,3, Angel J. Moreno1, Angel Alegŕıa2,3

1Donostia International Physics Center, Paseo Manuel de Lardizabal 4, 20018 San Sebastián, Spain.2Departamento de F́ısica de Materiales, Universidad del Páıs Vasco (UPV/EHU),

Apdo. 1072, 20080 San Sebastián, Spain.3Unidad de F́ısica de Materiales, Centro Mixto CSIC-UPV, Apdo. 1072, 20080 San Sebastián, Spain.

Abstract

Among the different dynamical processes that take place in polymers, methyl group rotationis perhaps the simplest one, since all the relevant interactions on the methyl group can becondensed in an effective mean-field one-dimensional potential. Recent experimental neutronscattering results have stimulated a new revival of the interest on methyl group dynamics inglasses and polymer systems. The existence of quantum rotational tunnelling of methyl groupsin polymers was expected for a long time but only very recently (1998), these processes havebeen directly observed by high-resolution neutron scattering techniques. This paper revisesand summarizes the work done on this topic over last ten years by means of neutron scattering.It is shown that the results obtained in many chemically and structurally different polymerscan be consistently described in the whole temperature range — from the quantum tunnellinglimit to the classical hopping regime — as well as in the librational spectrum, in terms of theRotation Rate Distribution Model (RRDM), which was first proposed in 1994. This modelintroduces a distribution of potential barriers for methyl group rotation, which is associated tothe disorder present in any structural glass. The molecular and structural origin of the barrierdistribution in polymers is discussed on the basis of a huge collection of investigations reportedin the literature, including recent fully atomistic molecular dynamics simulations.

Keywords: Methyl groups; Quantum tunnelling; Classical hopping; Neutron scattering; Polymers.

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http://arxiv.org/abs/cond-mat/0508668v1

Contents

I. Introduction

II. Neutron scattering techniques and instrumentationIIA. Neutron scattering techniquesIIB. Neutron scattering instrumentationIIC. Final experimental considerations

III. Methyl group dynamics in crystalline systems: Theory and scattering functionsIIIA. Rotational tunnellingIIIB. Crossover from rotational tunnelling to classical hoppingIIIC. Classical hoppingIIID. Additional remarks

IV. Methyl group dynamics in polymer systemsIVA. Rotation Rate Distribution Model (RRDM)IVB. Other approaches

V. Application of the RRDM to a showcase: poly(vinyl acetate)

VI. Summary of experimental results by neutron scattering on polymersVIA. Classical hoppingVIB. Rotational tunnelling and crossover to hoppingVIC. Librational levels

VII. Molecular origin of the barrier distribution for methyl group dynamicsVIIA. Influence of the chain conformationVIIB. Influence of mixing with other materialsVIIC. Molecular glasses and comparison with the crystalline stateVIID. Molecular dynamics simulations

VIII. Conclusions and outlook

Acknowledgements

References

Nomenclature

BS BackscatteringEISF Elastic incoherent structure factorFEW Fixed elastic windowHWHM Half-width at half-maximumINS Inelastic neutron scatteringMDS Molecular dynamics simulationsNMR Nuclear magnetic resonanceQENS Quasielastic neutron scatteringRRDM Rotation Rate Distribution ModelTOF Time of flightVDOS Vibrational density of states

A,Ea, Eb Sublevels of the librational ground stateA(Q) Elastic incoherent structure factorB Rotational constantEA Classical activation energy〈EA〉 Average barrier of f(EA)E01 First librational energyF (E01) Distribution of E01f(EA) Distribution of EAG(Tc) Distribution of Tcg(V3) Distribution of V3H(log Γ) Distribution of log Γh(h̄ωt) Distribution of ωtIinc(Q, t) Intermediate incoherent scattering function

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L(ω,Γ) Normalized Lorentzian functionlog Γ0 Average of H(log Γ)S(Q) Elastic coherent structure factorScoh(Q,ω) Coherent scattering functionSinc(Q,ω) Incoherent scattering functionSMGinc (Q,ω) Incoherent scattering function

for methyl group rotationTc Crossover temperatureV3 Threefold barrier height〈V3〉 Average barrier of g(V3)W c3 Coupling potential between methyl groupsZ(ω) Generalized VDOSh̄Q Neutron momentum transferh̄ω Neutron energy transferΓ Lorentzian HWHM for classical hoppingΓAE Inelastic crossover Lorentzian HWHMγAE Preexponential factor for ΓAEΓEaEb Quasielastic crossover Lorentzian HWHMγEaEb Preexponential factor for ΓEaEbγb RRDM preexponential factor for ΓAE and ΓEaEbγs RRDM preexponential factor for ∆ωtΓ∞ Preexponential factor for classical hopping∆ωt Shift of the tunneling frequencyσ Standard deviation of H(log Γ)σE Standard deviation of f(EA)σV Standard deviation of g(V3)σcoh Coherent cross-sectionσinc Incoherent cross-sectionτ Residence time for classical hoppingωt Tunneling frequencyPC PolycarbonatePDMS Poly(dimethyl siloxane)PEO Poly(ethylene oxide)PEP Poly(ethylene propylene)PH PhenoxyPI PolyisoprenePIB PolyisobutylenePMMA Poly(methyl methacrylate)PMPS Poly(methyl phenyl siloxane)PP Polypropylenehh-PP Head-to-head Poly(propylene)PPO Poly(propylene oxide)PS PolystyrenePSF PolysulfonePVAc Poly(vinyl acetate)PVME Poly(vinyl methyl ether)PαMS Poly(α-methylstyrene)SCPE Solution chlorinated polyethylene

I. INTRODUCTION

The different dynamic processes present in amor-phous polymers cover a extremely wide time scale,spanning from ca. 10−13 s to years (see, e.g., Refs.[1–4]). These processes include terminal relaxations,conformational rearrangements, segmental dynamics,localized Johari-Goldstein relaxations, rotations ofside groups, or fast vibrational dynamics. One ofthe simplest processes among the latters is methylgroup, -CH3, rotation. Many natural and syntheticmacromolecular chains contain this simple molecularunit as a side group or as a part of more compli-cated side groups. In the glassy state of polymericmaterials, at temperatures well below the glass transi-tion region, one can assume as a good approximationthat the main-chain dynamics are completely frozen-in and only small side units, as the methyl groups,can still remain mobile. In such conditions, the inter-action between the methyl group and its environmentis often well approximated by an effective mean-fieldor ”single-particle” rotational potential [5–7]. Themethyl group can be regarded as a rigid rotor becausethe strength of the C-H covalent bonds allows one toneglect the internal degrees of freedom in comparisonwith both translational and rotational motions of thegroup as a whole. Hence, the single-particle potentialonly depends on one characteristic angular coordinateφ, which is measured in the plane perpendicular tothe C3-symmetry axis of the methyl group. The lattercorresponds to the bond joining the carbon to the restof the molecule (see Fig. 1).In the simplest approximation (see Section III) the

rotational potential is threefold:

V (φ) =V32(1− cos 3φ). (1)

Note that, in this form, V3 corresponds to both themaximum and the amplitude of the potential V (φ).Figure 1 shows the quantized energy levels of a three-fold potential with barrier height V3 = 500 K (inthe following V3 will be given as V3/kB, with kB theBoltzmann constant). Such levels are obtained [5–7]by solving the corresponding stationary Schrödingerequation, which in the case of a threefold potentialtakes the form of the well-known Mathieu equation(see, e.g., Ref. [8]). The energy levels i = 0, 1, 2... ofthe individual potential wells are named torsional or

3

librational levels, and are split (with energy splitting∆i) due to the coupling between the single-well wave-functions. Three quantities characterizing the methylgroup rotational potential are experimentally acces-sible (see below): i) the energies E0i of the transi-tions between the librational levels, specially the tran-sition E01 between the ground and first excited level,ii) the energy splitting of the ground librational state,∆0 = h̄ωt, with ωt the quantum tunnelling frequency,iii) the activation energy EA for methyl group classicalhopping between adjacent wells, defined as the difer-ence between the top of the barrier and the groundstate (see Figure 1). The two former quantities can bedirectly accessed by inelastic spectroscopic techniques.The latter is indirectly derived from the temperaturedependence of the hopping rate, which can be deter-mined by quasielastic spectroscopy or by relaxationtechniques (see below).According to the scheme shown in Figure 1, methyl

group dynamics at very low temperature (typicallyT ∼ 80 K). Dueto the one-dimensional character of the rotational po-tential, methyl group dynamics is one of the simplestexamples of barrier penetration phenomena involvingthe interaction of a quantum system with lattice vi-brations. For that reason, it has attracted a greattheoretical interest as an ideal case for the understand-ing of the fundamental problem of the transition fromquantum tunnelling to classical hopping (see, e.g, Refs.[9–14]). Moreover, due to the high sensivity of the ro-tational potential to the chemical and geometrical na-ture of the environment, and to the relative simplic-ity of their dynamics, methyl groups can be used asbuilt-in probes for exploring structural and dynamicproperties of the host material.Dynamics, and in particular rotational tunnelling,

of methyl groups has been widely investigated inmolecular crystals by different techniques over lastdecades. (see, e.g., [5–7]). Measurements of tun-nelling frequencies of methyl groups in about 200 sys-tems of very different chemical nature have been com-piled in Ref. [5]. The used techniques include mostlyneutron scattering and nuclear magnetic resonance(NMR) methods (see, e.g., Ref. [15] for a review onNMR measurements), but also calorimetry, electron

nuclear double resonance, infrared or hole burning op-tical spectroscopy. However, no method yields such adirect model independent insight as inelastic neutronscattering (INS), where the tunnelling frequency is di-rectly observed as two resolution-width inelastic peaksat ±h̄ωt. Nevertheless, it must be stressed that thisobservation is only possible for moderate and weak po-tentials (V3

usually high temperature was considered as an unreli-able explanation of the observed features, and a pos-sible interpretation, by mantaining the picture of aunique rotational barrier, was discussed in terms ofsixfold symmetry contributions to the rotational po-tential [20,21]. Another explanation was introducedin terms of temperature dependent fractions of rotat-ing and non-rotating methyl groups [22,27]. Thougha reasonable description of the experimental data wasachieved, a physical justification of the latter hypoth-esis was lacking. Finally, it became clear that thebasic assumption of a unique barrier was wrong foramorphous polymers when experiments on a same sys-tem performed in different spectrometers, i.e., in dif-ferent time/energy windows, provided uncompatibleresults for the measured hopping rate [23,24]. In theseminal works of Refs. [25] and [26], QENS data formethyl group hopping in polymers were successfullyinterpreted by introducing a broad distribution of ro-tational barriers, resulting from the highly disorderednature of the glassy state. In last years, a noticeableamount of experiments in different systems have sup-ported this picture (see Sections V and VI).Another aspect of the problem of methyl group dy-

namics in polymers is the dynamic behavior at verylow temperature. From the early times, there was inthe literature a number of claims or predictions aboutlow temperature quantum effects. In particular, NMRmeasurements by Hoch et al. [28] showed methyl groupdynamics in poly(vinyl acetate), PVAc, to be activedown to very low temperature. Rotational tunnellingof methyl groups was also claimed to be present inpoly(methyl methacrylate), PMMA, which would ac-count for the reported non-Arrhenius behavior of themechanical relaxation time at very low temperature[29,30]. However, a quantitative and unambiguousinterpretation of the experimental data was not pro-vided. As mentioned above, the rotational tunnellingpicture has been evidenced in a vaste collection of crys-talline systems by different techniques, and speciallyby neutron scattering, where the tunnelling frequencyis directly observed as two resolution-width inelasticpeaks at ±h̄ωt. For the case of polymers, one mightexpect to observe broad peaks as a consequence of thedistribution effects. However, observation of peaks, orof any scattered intensity over the instrumental reso-lution at very low temperature was not reported for along time.

Finally, high resolution measurements in PVAc [31]and PMMA [32] at T ≈ 1 K showed, instead of well-defined peaks, a broad scattered intensity similar tothat commonly observed in the high temperature hop-ping regime. This result, which might have been inter-preted as a signature of hopping events at unusuallylow temperature, was indeed explained in terms of ro-tational tunnelling, namely as the result of the pres-ence of a strongly assymetric distribution of tunnellingfrequencies, with the maximum located well beneaththe instrumental resolution. The latter distributionjust followed from the distribution of potential barri-ers for methyl group rotation. This interpretation wassupported by exploting the expected isotope effect onthe tunnelling frequencies [32] (see Section VI). Theconsistency of the distribution picture, formalized inthe Rotation Rate Distribution Model (RRDM, seeSection IV) was supported by the fact that the samebarrier distribution was able to quantitatively accountfor the experimental spectra in all the temperaturerange [33,34], covering the crossover from rotationaltunnelling to classical hopping, as well as the libra-tional peaks observed by INS.The grounds of the RRDM can, in principle, be ex-

tended to non-polymeric disordered systems, as hasbeen successfully checked in the glassy state of toluene[35] and sodium acetate trihydrate [36] (see SectionVII). Contrary to the case of polymer systems, resultsfor methyl group dynamics in molecular glasses can bedirectly compared with those for the crystalline state,allowing one to get insight into the molecular origin ofthe barrier distribution.The goal of this review is to summarize the progress

in the topic of methyl group dynamics in polymersthat has been achieved over last 10 years by means ofneutron scattering techniques, together with the de-velopment of the RRDM. The molecular origin of thedistribution of potential barriers, which is an essentialingredient of the RRDM, is discussed on the basis ofthe experimental and computational investigations re-ported in the literature. A brief introduction to thetheoretical and experimental grounds of neutron scat-tering techniques is also given in next Section.

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II. NEUTRON SCATTERING TECHNIQUESAND INSTRUMENTATION

A. Neutron scattering techniques

Neutrons are adequate probes for the investigationof both structure and dynamics of condensed matter.There are three major features for that: i) their typ-ical wavelengths are of a few Å, which correspond tothe typical interatomic distance, ii) their typical ener-gies are of the order of some meV, which correspondto the energy scale of the typical excitations in con-densed matter, iii) neutrons being particles withoutelectrical charge, they can penetrate the sample andprovide information about bulk properties, in contrastto charged particles like electrons, which mainly probesurface properties.The range of the nucleus-neutron interaction is ∼

1.5 × 10−5 Å, much smaller than the neutron wave-length and the nuclear radius. Hence, scattering canbe approximated as isotropic, and the interaction ismodelled by the Fermi pseudopotential:

VF =2πh̄2

mbδ(r−R), (2)

where m and r are respectively the neutron mass andposition, andR is the nucleus position. The scatteringlength operator, b, is given by

b = bcoh +2binc

√

I(I + 1)S · I, (3)

where S and I are respectively the neutron and nuclearspins, and bcoh and binc are respectively the coherentand incoherent scattering lengths.As in any scattering technique, there are two ba-

sic quantities to be measured by neutron scattering:the energy transfer, h̄ω = h̄2(k2 − k20)/2m, which isthe difference between the final and the initial neutronenergy, and the momentum transfer, h̄Q = h̄k− h̄k0,where k and k0 are respectively the final and initialneutron wavevectors (see Fig. 2). The number of neu-trons scattered within a solid angle between Ω andΩ + dΩ, which have changed their energy in h̄ω andtheir momentum in h̄Q, is proportional to the double-differential cross-section [37–40]. The latter can besplit in two parts, the coherent and the incoherentdouble-differential cross-sections:

∂2σ

∂Ω∂ω=

(

∂2σ

∂Ω∂ω

)

coh

+

(

∂2σ

∂Ω∂ω

)

inc

, (4)

which can be expressed as a function of the scatteringfunctions:

(

∂2σ

∂Ω∂ω

)

coh

∝ kk0σcohScoh(Q,ω)

(

∂2σ

∂Ω∂ω

)

inc

∝ kk0σincSinc(Q,ω). (5)

In this equation σcoh and σinc are respectively the co-herent and incoherent cross-sections. Scoh(Q,ω) andSinc(Q,ω) are respectively the coherent and incoher-ent scattering functions. These functions give accountfor the amplitude transitions, mediated by the Fermipseudopotential, between the eigenstates of the sam-ple. While the coherent contribution results from in-terferences between the neutron waves dispersed bythe nuclei, the incoherent contribution is equivalentto the dispersion of the isolated nuclei. In the clas-sical limit, Scoh(Q,ω) and Sinc(Q,ω) are related viadouble Fourier transform, respectively, with the pair-correlation and self-correlation functions of the atomicpositions.Protium hydrogen (H) has a large incoherent scat-

tering cross-section as compared to its coherent partand to the total cross-section, σtot = σinc+σcoh, of theother elements typically present in polymer materials(see Table I). Hence, a neutron scattering experimenton a protonated polymer sample provides mainly in-formation on the self-correlation function of the pro-tium hydrogens. Moreover, due to the large differencebetween the incoherent cross-section of protium andthe total cross-section of deuterium (D), the intensityscattered by a subset of the protium hydrogens can bestrongly enhanced if selective deuteration of the otherhydrogens is possible by means of chemical methods.When neutron scattering is used for dynamic inves-

tigations, one finds three different types of scatteringprocesses: elastic, quasielastic and inelastic scatter-ing. Inelastic neutron scattering (INS) correspondsto the case where the neutron exchanges energy withthe sample excitations, and manifests in the experi-mental spectrum as two resolution-width lines of fi-nite energy (for neutron energy gain and loss). Onthe contrary, elastic scattering corresponds to the casewhere the neutron energy remains constant, yielding aresolution-width elastic line. Finally, quasielastic neu-

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tron scattering (QENS) is that associated with the en-ergy transfer resulting from the Doppler effect occur-ring when a neutron interacts elastically with a movingparticle, and manifests as a broad line around the elas-tic line. Of course, inelastic or quasielastic processesare no detectable when the involved energy changeis smaller than the instrumental energy resolution,yielding an additional contribution to the experimen-tal elastic line. Quasielastic and inelastic processestypically cover a range |h̄ω| < 2 meV and |h̄ω| > 2meV respectively, the latter corresponding to typi-cal vibrational excitations in condensed matter. Theterms QENS and INS are often used to refer to theseenergy ranges and the dynamic window of the corre-sponding spectrometers. However, it must be stressedthat excitacions in the µeV range can also be presentin the system, as the case of methyl group tunnelling.Hence, such excitations, which strictly correspond toINS, are accessible by means of instruments specifi-cally designed for QENS.In the liquid state, molecules are able to diffuse and

loss memory from their initial positions. Moleculardiffusion yields no elastic component in neutron scat-tering spectra. On the contrary, if the motion of agroup of particles is constrained into a localized region,as the case of methyl group rotation in a crystalline orglassy system, in addition to the eventual inelastic andquasielastic contributions, such a motion provides anelastic componet in the experimental spectrum. Theelastic peak is modulated by a Q-dependent quantitynamed the elastic-incoherent-structure-factor (EISF),A(Q). The EISF gives account, via Fourier transform,for the average positions of the individual particlesperforming the localized motion, i.e, it provides infor-mation about the geometry of the motion.Integration of the coherent scattering function pro-

vides the elastic coherent structure factor, S(Q).Fourier transformation of the latter gives account forthe probability distribution of the configurations of allthe ensemble of particles in the system, i.e., it providesstatic structural information.

B. Neutron scattering instrumentation

Neutron scattering investigations on methyl groupdynamics in polymers are carried out mainly by meansof time-of-flight (TOF) and high-resolution backscat-

tering (BS) techniques. In addition, in most of thecases it is convenient to evaluate the total coherentand incoherent contributions to the scattered intensity.For this purpose, diffraction experiments with spin po-larization analysis are neccessary. The showcase pre-sented here for PVAc (see Section V) corresponds toexperiments performed at the spectrometers IN5, IN6,IN16 and D7 of the Institute Laue Langevin, (ILL,Grenoble, France) and TOSCA at the source ISIS ofthe Rutherford Appleton Laboratory (RAL, Chilton,United Kingdom). Next we summarize the technicalcharacteristics of these spectrometers. A detailed de-scription can be found in Refs. [41–46].IN5 [41] and IN6 [42] are TOF spectrometers. By

means of a monochromator, different wavelengths, λ,can be selected in the ranges 1.8−16 Å for IN5 and 4−6Å for IN6. Most of the data presented here were ob-tained with λ = 6.5 Å at IN5 and λ = 5.1 Å at IN6. Inthis situation the elastic energy resolution (half-widthat half maximum, HWHM) is ≈ 25 µeV for IN5 and≈ 50 µeV for IN6. The explored energy window (inneutron energy gain) is -0.6 meV

resolution of IN16 makes it adequate for the investiga-tion of methyl group dynamics in, both, the classicalhopping regime at intermediate temperature, and thequantum tunnelling regime at very low temperature.By using a Si (111) monochromator the wavelengthof the incident neutrons is λ = 6.27 Å. The corre-sponding accessible Q-range is 0.2 − 1.9 Å−1. Whendeformed Si (111) crystals are used as analyzers a res-olution of nearly Gaussian shape with HWHM ≈ 0.45µeV is attained. In the standard operation mode themonochromator is subject to an oscillatory motion ata frequency of ≈ 14 Hz, which allows one to explore anenergy transfer range up to 15 µeV. As in backscatter-ing spectrometers the incoming beam energy is muchlarger than the maximum energy transfer, Eq. (6) canbe approximated by a direct relation between the posi-tion of the detector and the momentum transfer, givenby:

Q = (4π/λ) sin(Φ/2). (7)

BS spectrometers as IN16 can also be used in a differ-ent operation mode, known as the fixed-elastic-window(FEW) technique. In this technique the monochroma-tor is held fixed and the sample temperature is variedat a moderate rate. In this way one measures the elas-tic intensity as a function of the temperature, sincethe FEW technique only detects the fraction of neu-trons with an energy exchange within the instrumentalresolution. Hence, what is considered as elastic inten-sity depends on the resolution of the used spectrom-eter. For IN16, with a resolution function of HWHM≈ 0.45 µeV, the dynamics of scattering particles be-ing slower than a characteristic time τ ≈ 10 ns areobserved as elastic scattering. When increasing tem-perature, T , the fraction of particles slower than τdecreases, yielding a decrease of the elastic intensity.This decrease is exponential if dynamics in that timescale is purely vibrational, and it is given by the Q-dependent Debye-Waller factor e−2W (Q) [37–40], withW (Q) = Q2〈u2〉/3. In this expression 〈u2〉 is themean-squared displacement per particle. For vibra-tional motions 〈u2〉 ∝ T . The presence of other dy-namic processes is evidenced by deviations from ex-ponential behavior, yielding a step-like decay of theelastic intensity (see Section VI).The spectrometer D7 combines TOF methods with

spin polarization analysis capabilities [44,45]. Amongthe different types of experiments which are possible

with D7, that of particular interest for the investiga-tion of methyl group dynamics in polymers is the mea-surement of the coherent and incoherent contributionsto the total scattered intensity (without energy anal-ysis). These measurements are usually performed atlow temperature to ensure that no significant intensityis lost due to the finite range of energy transfer cov-ered by the instrument. By using a neutron incidentwavelength of 4.8 Å, the relevant accessible Q-range,

Q

for a strictly monoatomic sample Z(ω) is proportionalto the real VDOS. In the general case, though theexperimental peaks will correspond to the real excita-tions, their amplitudes will be controlled by the men-tioned factors, being larger for the modes involvingmotions of protium hydrogens.

C. Final experimental considerations

Instrumental resolution functions are usually de-termined from measurements on a vanadium sam-ple, which provides solely elastic scattering. Thesame measurements can be used to calibrate detec-tors. In order to obtain the actual experimental scat-tering function S(Q,ω), raw data must be correctedfor detector efficiency, sample container and absorp-tion. Due to its small scattering cross-section, alu-minium is the usual material for sample holders. Slabgeometry is usually preferred. The typical area of theneutron beam is of 10−15 cm2. The sample thicknessis selected to provide a transmission of 90− 95 %. Forfully protonated polymers this condition is attainedwith sample thickness of ∼ 0.1 mm. High transmis-sion allows one to neglect multiple scattering effects,provided that the very low Q-range is not consideredin the data analysis. This is the case for methyl groupdynamics, since the EISF at low Q is very close tounity (see Fig. 3), and correspondingly, the quasielas-tic intensity — which is weighted by 1 − A(Q), seeSection III— is very weak and difficult to analyze.

III. METHYL GROUP DYNAMICS INCRYSTALLINE SYSTEMS: THEORY AND

SCATTERING FUNCTIONS

A. Rotational tunnelling

As mentioned in the Introduction, methyl group mo-tion at very low temperature (T ∼ 1 K) can be mod-elled as that of a rigid rotor in an effective single-particle one-dimensional rotational potential V (φ).The corresponding Hamiltonian is given by [5–7]:

HR = −B∂2

∂φ2+ V (φ). (9)

In this equation B = h̄2/2I is the rotational constantof the rigid rotor, with I the moment of inertia of the

methyl group around its threefold symmetry axis. Fora fully protonated methyl group B = 0.655 meV. Fora fully deuterated one, B = 0.328 meV.The rotational potential V (φ) must be invariant un-

der 2nπ/3 rotations, with n = ±1,±2,±3..., becauseits symmetry cannot be lower than that of the rigidrotor, and moreover, both symmetries cannot be un-compatible. Hence, V (φ) can be expanded as a Fourierseries of 3n-fold terms of amplitude V3n:

V (φ) =

∞∑

n=1

V3n2

[1− cos(3nφ+ δ3n)], (10)

where δ3n are constant angular offsets. Indeed, it canbe formally demonstrated that the rotor-lattice inter-action can be expressed in this way provided that V (φ)is given by a sum of two-body additive potentials [51].As mentioned in the Introduction, in a first approx-imation only the threefold term of the expansion isretained:

HR = −h̄2

2I

∂2

∂φ2+V32(1− cos 3φ). (11)

If higher-order terms are neccesary to account forthe experimental results, they often contribute just assmall corrections to the main threefold term [5].As mentioned in the Introduction, the coupling of

the wavefunctions of the individual wells splits the li-brational levels. The resulting sub-levels are denotedas A and E, the latter consisting of the degeneratedoublet (Ea, Eb). These labels correspond to the ir-reducible representations of the symmetry group C3of the Hamiltonian HR. The eigenstate A remainsunchanged under a rotation 2π/3, while such an op-eration adds a phase ±2π/3 to the spatial part of thewave functions of Ea and Eb respectively. An eigen-state of HR is characterized by two discrete quantumnumbers, namely the librational, n, and symmetry, S,indexes: |nS〉. As shown in Fig. 1, the levels A and Ealternate between consecutive librational levels.At very low temperature, T ∼ 1 K, kBT ≪ E0i and

the occupation probability of the excited librationalstates i = 1, 2, 3... is negligible. Hence, the rotor os-cillates coherently between the three wells with an os-cillation frequency equal to the tunnelling frequencyωt. If the latter takes a value between the highest res-olution available in BS spectrometers, h̄ωt ≈ 0.3 µeV,and the free rotor limit, h̄ωt = B, it is detected as two

9

resolution-width inelastic peaks at ω = ±ωt (see Fig.4a).The incoherent scattering functions for rotational

tunnelling in different symmetries can be calculatedby using symmetry adapted scattering operators [6].Hence, the incoherent scattering operator

W =N∑

j=1

(S · Ij) exp(−iQ · rj) (12)

where rj , Ij are the hydrogen positions and spins, andS is the neutron spin, can be rewritten, for threefoldsymmetry, as:

W =WA +WEa +WEb . (13)

The eigenstates |A〉, |Ea〉 and |Eb〉 are unchanged byWA, and are permutated cyclically by WEa , and anti-cyclically by WEb . Hence, the decomposition (13) al-lows for a straightforward calculation of theW -matrixelements between the different eigenstates [6]. Afteraveraging over all the orientations of the scatteringvector Q, the following transition amplitudes are ob-tained:

i) fAE(Q) = [1− j0(Qr)]/3 for A↔ E (Ea, Eb) transi-tions. The corresponding contribution to the scatter-ing function is 2fAE(Q)δ(ω ± ωt), since there are twoinelastic transitions ω = +ωt and other two ω = −ωt.ii) fEaEb(Q) = [1 − j0(Qr)]/3 for Ea ↔ Eb tran-sitions. These two elastic transitions contribute as2fEaEb(Q)δ(ω).iii) fNSC(Q) = [1+2j0(Qr)]/3 for transitions withoutsymmetry change. These three elastic transitions con-tribute as 3fNSC(Q)δ(ω).

In these expressions, j0(Qr) = sin(Qr)/Qr, with r theH-H distance in the methyl group, (r = 1.78 Å). Af-ter normalization to one-hydrogen scattering, the cor-responding incoherent scattering function for methylgroup tunnelling is [5–7]:

SMGinc (Q,ω) =5 + 4j0(Qr)

9δ(ω) +

2[1− j0(Qr)]9

[δ(ω + ωt) + δ(ω − ωt)]. (14)

The EISF, A(Q) = [5+4j0(Qr)]/9, and [1−A(Q)]/2 =2[1− j0(Qr)]/9, which modulate respectively the elas-tic and inelastic contributions, are shown in Fig. 3.

B. Crossover from rotational tunnelling toclassical hopping

The usual temperature dependence of neutron scat-tering spectra for methyl group dynamics in crystallinesystems is illustrated in Fig. 4 for crystalline sodiumacetate trihydrate [36]. At T ∼ 1 K (Fig. 4a) thespectrum shows a central elastic peak and two inelas-tic peaks of resolution width, i.e., delta functions con-voluted with the instrumental resolution. Accordingto the picture exposed above, the inelastic peaks corre-spond to the two transitions A↔ E within the groundlibrational state. The elastic peak includes the groundstate transitions Ea ↔ Eb and the transitions withoutsymmetry change. It also includes the elastic inco-herent contribution from other atoms different frommethyl group hydrogens, and the total coherent con-tribution (see below).When increasing temperature [36,48,50,52–59], the

inelastic peaks progressively broaden and shift to theelastic peak. At the same time, a quasielastic com-ponent appears around the elastic peak, and broad-ens with increasing temperature (Figs. 4b,c). Finally,in a narrow temperature interval (∆T ∼ 7 K), thequasielastic and inelastic components merge in a sin-gle quasielastic line (Fig. 4d). In the following thewidth of this interval will be neglected and repre-sented as a unique temperature Tc, that will be de-noted as the crossover temperature. Above Tc, a clas-sical picture of thermally activated hopping over therotational barrier is commonly acepted [6,37]. Theobserved single quasielastic line is well described by aLorentzian function. The corresponding HWHM fol-lows an Arrhenius-like temperature dependence:

Γ = Γ∞ exp(−EA/kBT ). (15)

where the classical activation energy EA, as mentionedin the Introduction, is defined as the difference be-tween the top of the barrier and the ground state (seeFig. 1). In the following EA will be given in unitsof kB. The preexponential factor Γ∞ is temperatureindependent and typically takes values of ∼ 5 − 10meV.While rotational tunnelling and classical hopping

are well established pictures respectively in the lowand high temperature limits, no satisfactory theoryfor the crossover regime between both dymanic limitshas won general acceptance [9–14,60–65]. Every model

10

for the crossover regime must give account for the fol-lowing phenomenology observed in a large amount ofexperiments [36,48,50,52–59]:

i) Broadening of the quasielastic and inelasticcrossover components are well described by Lorentzianfunctions of the same intensity. The relative in-tensities to the elastic peak are the same thanin the purely rotational tunnelling regime, i.e.,2fAE(Q)/3fNSC(Q) for each of the inelastic lines and2fEaEb(Q)/3fNSC(Q) for the quasielastic line. Thetemperature dependence of the Lorentzian HWHMfollows Arrhenius-like behavior. The observed acti-vation energy corresponds approximately to the firstlibrational energy:

Γi = γi exp(−E01/kBT ). (16)

Γq = γq exp(−E01/kBT ). (17)

In these equations Γi and Γq are respectively theLorentzian HWHM for the inelastic and quasielasticcomponents. The preexponential factors γi and γq aretemperature independent, and typically take values inthe range 0.05-1 meV, much lower than the observedvalues for the classical Γ∞ (see above). It is usuallyfound that γq < γi, though both factors take values ofthe same order of magnitude (typically γq/γi ∼ 0.5).

ii) Shift of the tunnelling peaks also follows Arrhenius-like behavior:

h̄∆ωt = γs exp(−ES/kBT ), (18)

where ∆ωt is the shift (in absolute value) of the tun-nelling peak and γs a temperature independent preex-ponential factor, which takes values similar to γi andγq. The activation energy ES is generally smaller thanE01 though it takes a close value, typically ES/E01 ∼0.7. Fig. 5 shows all these features for crystallinesodium acetate trihydrate as a standard showcase [36].

All the theoretical models that aim to give accountfor these universal features state that its physical ori-gin is the coupling of the methyl group to the latticephonons, which progressively breaks the coherence ofthe wavefunction of the rotor, leading to incoherenttunnelling [9–14,51,60–84]. The standard approach,

and the only one that is able to give account quantita-tively for the observed behavior, at least below Tc, isto describe the system rotor-lattice by a Hamiltonian:

H = HR +HB +HC . (19)

In this equation HR is the rigid rotor Hamilto-nian (9), HB is a bath of harmonic oscillators[9,11–13,64,71–73,75–79,82,84]:

HB =

N∑

k=1

(

− h̄2

2mk

∂2

∂x2k+

1

2mkω

2kx

2k

)

, (20)

and HC is a coupling term between the rotor and thebath:

HC =

N∑

k=1

gkxk cos(3φ+ δk). (21)

In these equations xk, mk, ωk, gk and δk are respec-tively the displacements, masses and frequencies of theharmonic oscillators, the coupling constants and theangular offsets relative to the rotor.The total Hamiltonian (19) retains the threefold

symmetry of the interactions methyl group - envi-ronment. Hence, an argumentation on the basis oftransitions between different symmetry states is stillpossible. The eigenvectors of the Hamiltonian (19)are characterized by a large set of quantum numbers.However, since H is, as HR, unvariant under the sym-metry group C3, the symmetry indexes A, Ea and Ebare still good quantum numbers for H , and the sym-metry adapted formalism can still be used to calculatethe scattering function. In the limit N → ∞, H has acontinuous number of eigenstates. Hence, transitionswith symmetry change will take place not between twodiscrete eigenstates, but between two continuous setsof eigenstates. The delta functions for the low temper-ature limit [Eq. (14)], corresponding to transitions be-tween two eigenstates characterized by a symmetry in-dex, are now substituted by Lorentzians, correspond-ing to transitions between two large sets of eigenstates,each set characterized by a symmetry index, i.e.,

|ξS〉 → |ξ′S′〉, (22)

where S and S′ are the initial and final symmetrystates, and ξ and ξ′ a large set of quantum numberscharacterizing the initial and final state of the bath.

11

Within this picture, the inelastic Lorentzians involvetransitions A ↔ E, while the quasileastic Lorentzianinvolve transitions Ea ↔ Eb. Hence, Eqs. (16,17) canbe rewritten as:

ΓAE = γAE exp(−E01/kBT ) (23)

ΓEaEb = γEaEb exp(−E01/kBT ). (24)

It can be demonstrated that elastic transitions with-out symmetry change do not lead to broadening [13],yielding an elastic line as in the low temperature limit.Taking all this arguments into account, the incoherentscattering function for the crossover regime is given by[48,77]:

SMGinc (Q,ω) =1 + 2j0(Qr)

3δ(ω) +

2[1− j0(Qr)]9

×[L(ω; ΓEaEb) + L(ω + (ωt −∆ωt); ΓAE) +

L(ω − (ωt −∆ωt); ΓAE)], (25)

where L(ω−ν; η) is a Lorentzian function of area unity,centered at ω = ν and HWHM = η, i.e.:

L(ω − ν; η) = 1π

η

(ω − ν)2 + η2 . (26)

It must be stressed that the functional form (25) isa direct consequence of the threefold symmetry ofthe Hamiltonian (19). It is the description of thetemperature dependence of shift and broadening ofthe Lorentzian lines which depends on the particu-lar model for the coupling of the rotor to the bath[Eqs. (20, 21)]. In the different models based on thisapproach, Eqs. (23, 24) are obtained as effective ex-pressions from a sum over complicated functions con-taining Bose factors resonant on the librational fre-quencies, the density of states Z(ω) and the couplingconstants {gk}:

Γξξ′ =N∑

j=1

Gξξ′ [Z(E0j), {gk}, n(E0j)], (27)

with ξξ′ = AE or EaEb, and n(ω) the Bose occu-pation factor. In practice, for the temperature rangeT < Tc, where the crossover regime is experimentallyobserved, one has kBT ≪ E01. Hence, the sum (27)is dominated by the term in E01 and can be approx-imated as Γξξ′ = γξξ′ exp(−E01/kBT ), in agreementwith Eqs (23, 24).

For the case of the shift of the tunnelling frequency,the sum is done over all the phonon spectrum:

h̄∆ωt =

∫ ∞

0

dωGs[Z(ω), {gk}, n(ω)]. (28)

Again for kBT ≪ E01 the integral can be approx-imated as an effective Arrhenius equation as (18).Since the integration is done over all the phonon spec-trum, it is dominated by low-energy contributions,yielding an effective activation energy ES smaller butclose to E01.There is a strong controversy about the validity of

the model given by Eqs. (19,20,21) for arbitrarely hightemperatures. Some authors have claimed [64,77] thatit is a valid picture even above Tc, the temperatureabove which spectra are well described by the classi-cal hopping model. According to this statement, theobserved classical-like Arrhenius behavior (15) is alsoobtained as an effective expression of the sum (27),where the highest order contributions would lead, athigh temperature, to the observed apparent activationenergy EA, which is higher but close to the highest li-brational transition (see Fig 1).

C. Classical hopping

The incoherent scattering function above Tc can bederived in a classical hopping picture, disconnectedfrom the low temperature quantum features above de-scribed. A master equation [6,37] is assumed for theclassical probabilities (denoted as p1, p2, p3) of find-ing the rotor in each of the three equilibrium positions0,±2π/3:

ṗ1 =1

τ

[

−p1 +p22

+p32

]

ṗ2 =1

τ

[

−p2 +p32

+p12

]

(29)

ṗ3 =1

τ

[

−p3 +p12

+p22

]

.

The hopping time between two equilibrium positionsis assumed to be negligible in comparison with theresidence time, τ , between consecutive jumps. Thisassumption is confirmed (see e.g., [85–88]) by molec-ular dynamics simulations (MDS), which allow one toobserve the motion of each individual methyl group in

12

the system. Fig. 6 shows computed angular positionsfor a methyl group hindered by a moderate barrier. Ascan bee seen, sudden jumps occur between the threeequilibrium positions, with long intervals between con-secutive jumps. It is worth emphasizing that τ is anaverage value over all the time intervals between con-secutive jumps of a same methyl group. The fact thatthe latters take different values is a consequence of thestochastic nature of the hopping process. For a crys-talline system with a unique barrier, τ will be the samefor all the individual methyl groups. For a disorderedsystem, τ will be different for each individual methylgroup, as will be discussed in Section IV.After solving the master equation (29), averaging

over all the orientations of Q, and Fourier transform-ing to the frequency domain, the following expressionis obtained [6,37]:

SMGinc (Q,ω) =1 + 2j0(Qr)

3δ(ω) +

2[1− j0(Qr)]3

L(ω; Γ), (30)

where Γ = 3τ/2 is just assumed to follow the classi-cal Arrhenius-like behavior of Eq. (15). The EISF,A(Q) = [1 + 2j0(Qr)]/3, and 1 − A(Q) = 2[1 −j0(Qr)]/3, which modulate respectively the elastic andquasielastic contributions, are shown in Fig. 3.

D. Additional remarks

It must be noted that, as required by a consis-tent description of the temperature dependence of thespectra, the scattering function (25) for the crossoverregime is reduced to Eqs. (14) and (30) respectivelyin the low and high temperature limits. For very lowtemperature, shift and broadening in Eqs. (18,23,24)tend to zero, recovering Eq. (14). On the other hand,tunnelling is supressed above the crossover tempera-ture, i.e., ∆ωt = ωt, and the three Lorentzians mergeinto a single line if ΓAE = ΓEaEb = Γ. Hence, Eq.(30)is recovered.Finally, the total scattering function S(Q,ω) is ob-

tained by adding the incoherent contribution of theother atoms, and the total coherent contribution. If,except for methyl group rotation and lattice vibra-tions, no other dynamics are present in the considereddynamic window, the former contributions are elastic,and S(Q,ω) is given by:

S(Q,ω) = e−2W (Q)[σcohS(Q)δ(ω) +

(σinc − σMGinc )δ(ω) + σMGinc SMGinc (Q,ω)], (31)

with σcoh and σinc the total coherent and incoherentcross-sections and σMGinc the incoherent cross-section ofthe three hydrogens of the methyl group. σcohS(Q)is the coherent static intensity. As mentioned above,the Debye-Waller factor e−2W (Q) gives account for theintensity loss with increasing temperature due to vi-brations. Since the Debye-Waller factor only entersas a scaling factor in the total scattering function, itcan be removed in the analysis procedure of the ex-perimental spectra. Finally, it must be stressed thatthe function S̃(Q,ω) to be compared with the exper-imental spectrum is given by the convolution of thetotal scattering function (31) with the instrumentalresolution R(Q,ω):

S̃(Q,ω) =

∫

S(Q,ω)R(Q,ω − ω′)dω′ (32)

In the following, comparison between experimentalspectra and any of the theoretical scattering functionsgiven above, will be understood as done after convo-lution of the latter with the instrumental resolution.

IV. METHYL GROUP DYNAMICS INPOLYMER SYSTEMS

A. Rotation Rate Distribution Model (RRDM)

In this Section we summarize the grounds of theRRDM. Though the RRDM was initially developedfor polymer systems, it must be stressed that all itsbasic assumptions can, in principle, be also applied tonon-polymeric disordered systems (see Section VII).In the framework of the RRDM [25,31–36,88–94], it isassumed that the only effect of the structural disor-der on methyl group dynamics is to introduce a dis-tribution of rotational barriers g(V3), originated fromthe different local environments felt by the individualmethyl groups. A non-trivial assumption is that thedistribution of distances between neighbouring methylgroups does not introduce significant coupling forcesfor the smallest distances, which, in particular, wouldmodify the single-particle Hamiltonian (9) leading toa more complex energy level scheme [95–100].

13

According to this approach, the spectrum for thedisordered system is simply obtained as a superpo-sition of crystal-like spectra weighted by the distri-bution g(V3). Due to the normalization condition∫

g(V3)dV3 = 1, the elastic terms in Eq. (31) arenot affected by the introduction of the distribution.Hence, in the framework of the RRDM, S(Q,ω) forthe glassy system is formally equal to Eq. (31). How-ever, the introduction of g(V3) modifies the incoher-ent contribution for methyl group dynamics, which isgiven by:

SMGinc (Q,ω) =

∫ ∞

0

g(V3)SMGinc (Q,ω, V3)dV3, (33)

where each single crystal-like spectrum SMGinc (Q,ω, V3)evolves with temperature according to the descriptiongiven in Section III, summarized in Eqs. (14,15,18,23,24,25,30). The different parameters controlling thisevolution are therefore, for each individual methylgroup:

i) the tunnelling frequency ωt, first librational energyE01, and classical activation energy EA. These threequantities are direct functions of the potential barrierV3, and are obtained by calculating the eigenvalues ofthe Hamiltonian (9). Fig. 7 shows the correspond-ing results for the purely threefold case. Highly accu-rate numerical relations for the latters, in the rangeV3 < 3500 K, are given by:

EA(K) = 0.598V1.053 , (34)

E01(meV) = 0.470V0.5483 , (35)

h̄ωt(meV) =

0.655

(

1 +V32.67

)1.06

exp[− (V3/4)0.5]. (36)

In these equations V3 is given in units of K.ii) the preexponential factors Γ∞, γs, γAE andγEaEb for the different Arrhenius-like equations(15,18,23,24).iii) the crossover temperature Tc(V3), that determineswhich regime -crossover if T < Tc(V3) or classical hop-ping if T > Tc(V3)- governs the dynamics of a givenmethyl group of barrier height V3 at moderate tem-perature T .

Glassy systems show broad distributions g(V3). Forthat reason, shift, broadening, and merging of the in-dividual peaks with the quasielastic component, can-not be solved from the spectrum for the glass. On theother hand, it is expected that the temperature depen-dence of individual spectra is highly sensitive to thebarrier height. In principle, methyl groups with highvalues of V3 will reach the classical hopping regime athigher Tc than those with low values of V3. Moreover,there is a large set of parameters, namely the preexpo-nential factors for shift and broadening, which dependon the unknown constants {gk} describing the cou-pling between the methyl group and the phonon bath[Eqs. (27,28)]. Such preexponential factors cannotbe easily deduced for a glassy system, in contrast tothe crystalline case [36,48,50,52–59], where shift andbroadening of the single peaks can be directly observedand analyzed (see Figs. 4, 5). As mentioned above,the typical values of Γ∞ observed in crystalline sys-tems are in the same order of magnitude (a few meV),but γs, γAE and γEaEb can take a wide range of values.All these problems enormously complicate the anal-

ysis of the spectra in glassy systems. Next it is shownthat many of the unknown present paramaters, namelyγAE , γEaEb and γs, controlling the temperature depen-dence of the crystal-like spectrum for each individualmethyl group, can be removed without the need ofcarrying out complex calculations, as it would be re-quired by the introduction of a detailed model for thecoupling of the methyl group to the glass vibrations.Instead, a functional relation between the crossovertemperature and the barrier height, Tc = Tc(V3), isassumed [33–36]. It is found that a consistent de-scription of the spectra in all the temperature range ispossible in terms of only three parameters: the aver-age and standard deviation of the barrier distributiong(V3), and the preexponential factor Γ∞ for classicalhopping.In the approach introduced by the general version

of the RRDM [33–36], the crossover temperature Tcis operationally defined as the temperature where therate for coherent quantum tunnelling becomes equal tothe rate for incoherent classical hopping, i.e., Γ(Tc) =h̄ωt, or from Eq. (15):

kTc =EA

ln(Γ∞/h̄ωt). (37)

As mentioned above, the classical activation energy

14

EA and the tunnelling frequency ωt are direct func-tions of V3. It has been shown (see Refs. [25,35]for a detailed discussion) that Γ∞ can be taken asa barrier-independent quantity in a good approxima-tion. Therefore, Tc defined in this way depends onlyon the barrier height V3 and the preexponential fac-tor for classical hopping Γ∞. This latter material-dependent parameter can be interpreted as a measureof the strength of the coupling of the methyl groupsto the glass vibrations. For a given methyl groupof barrier V3, the larger Γ∞ —i.e., the stronger thecoupling—, the lower will be the Tc for the onset ofclassical hopping. Two technical approximations areintroduced:

i) For each individual methyl group, the preexponen-tial factors γAE and γEaEb are taken as equal (in thefollowing they will be denoted as γb). Therefore,

ΓAE = ΓEaEb = γb exp(−E01/kBT ). (38)

ii) The activation energy ES for the shift of the tun-nelling lines is assumed to be equal to E01:

h̄∆ωt = γs exp(−E01/kBT ). (39)

The general experimental evidence in crystalline sys-tems for the values of γAE , γEaEb and ES (see above)supports the reliability of these two approximations.In addition, two constraints are imposed to Eqs.(38,39) at the crossover temperature:

i) The tunnelling frequency must be shifted to zeroat the onset of the classical regime, ∆ωt(Tc) = ωt,as required by the disappearence of quantum effectsabove Tc. Hence, from Eq. (39):

γs = h̄ωt exp(E01/kTc) (40)

ii) Continuity condition for the HWHM of theLorentzians at the merging point Tc, i.e., ΓAE(Tc) =ΓEaEb(Tc) = Γ(Tc). Hence, from Eqs. (15,38):

γb = Γ∞ exp[(E01 − EA)/kTc]. (41)

A straightforward consequence of Eqs. (37,40,41) isthat γs = γb, which is also in agreement with the gen-erally close values found for both quantities in crys-talline systems.

All the quantities at the right sides of Eqs.(37,40,41) depend exclusively on V3 and on the barrier-independent factor Γ∞. In this way, the preexponen-tial factors for the crossover, γb and γs, can be deter-mined unambiguously for each value of V3 in the dis-tribution g(V3). Hence, the temperature dependenceof the spectra for the glass is modelled exclusively interms of Γ∞ and g(V3). Two-parameter (average bar-rier and standard deviation) simple distributions areused, as Gaussian [25,31–34,36,88–93] functions. In afew cases [35,94] two-parameter Gamma functions aremore adequate. In this way, the RRDM only intro-duces three independent parameters for the analysisof experimental spectra in all the temperature range.In the limit of low and high temperature, all the

individual methyl groups will perform respectively ro-tational tunnelling or classical hopping, and the cor-responding crystal-like contributions to the spectrumof the glass [Eq. (33)] will be governed respectively byEq. (14) for tunnelling and by Eqs. (15,30) for hop-ping. These limits are referred to as the tunnelling andhopping limits of the RRDM. At intermediate temper-ature the functional relation between Tc and V3 intro-duced in Eq. (37) allows one to select the correspond-ing dynamic regime for each individual methyl group—i.e., crossover, governed by Eqs. (25,38,39,40,41), orclassical hopping, governed by Eqs. (15,30).In a first approximation, a Gaussian distribution of

purely threefold rotational barriers is assumed:

g(V3) =1√

2πσVexp

[

− (V3 − 〈V3〉)2

2σ2V

]

, (42)

with 〈V3〉 the average barrier and σV the standard de-viation of the distribution. See Fig. 8a for 〈V3〉 =800 K, σV = 250 K, as an example of parameters forwhich tunnelling and hopping features are observableby neutron scattering (see below).The functional relations —e.g., Eqs. (34,35,36)—

between EA, E0i, h̄ωt, and V3, allow for a straight-forward transformation between g(V3) and the corre-sponding distributions of activation energies for classi-cal hopping, f(EA), rotational tunnelling frequencies,h(h̄ωt), and librational energies, F (E0i):

g(V3)dV3 = f(EA)dEA =

F (E0i)dE0i = −h(h̄ωt)d(h̄ωt). (43)

In a good approximation, the classical activationenergy EA depends linearly on V3 (see Fig. 7c).

15

Hence, the distribution f(EA) is also Gaussian (seeFig. 8b), with average energy 〈EA〉 and standard devi-ation σE . As mentioned above, the preexponential fac-tor for classical hopping Γ∞ is assumed to be barrier-independent. This approximation allows one, accord-ing to Eq (15), to transform the Gaussian distributionf(EA) into a log-Gaussian distribution of LorentzianHWHM for classical hopping:

H(log Γ) =1√2πσ

exp

[

− (log Γ− log Γ0)2

2σ2

]

, (44)

where

Γ0 = Γ∞ exp(−〈EA〉/kBT ) (45)

is the corresponding Lorentzian HWHM for the aver-age activation energy, and

σ = σE log(e)/kBT (46)

is the standard deviation for H(log Γ). Fig. 9 showsthe temperature dependence of this distribution forthe above parameters 〈V3〉 = 800 K, σV = 250 K, andfor a typical value of Γ∞ = 6 meV. As can be seen, athigh temperature the major part of the distribution isaccessible in the energy window of neutron scatteringspectrometers.The parameters Γ0 and σ at a given high temper-

ature can be obtained from a fitting procedure of theexperimental methyl group incoherent contributionSMGinc (Q,ω) to the classical limit of Eq. (33). From thetransformation g(V3)dV3 = −H(logΓ)d(log Γ), andfrom the incoherent scattering function (30) for a sin-gle barrier, Eq. (33) can be rewritten in the classicallimit as:

SMGinc (Q,ω) =1 + 2j0(Qr)

3+

2[1− j0(Qr)]3

∫ ∞

−∞

H(log Γ)L(ω; Γ)d(log Γ). (47)

The set of values Γ0(T ) and σ(T ) obtained for the dif-ferent temperatures is fitted to Eqs. (45,46) in orderto obtain Γ∞, 〈EA〉 and σE . Finally, the knowledgeof these two latter quantities allows, by transforma-tion, to determine 〈V3〉 and σV , i.e., the parametersof the barrier distribution g(V3). The latter, as ob-tained from the analysis of classical high temperaturespectra, can next be transformed into the distribu-tion of tunnelling frequencies h(h̄ωt) and librational

energies F (E01) (see Figs. 8c, 8d). The consistency ofthe RRDM requires that these two latter distributionsderived in this way, also reproduce, respectively, thecorresponding experimental tunnelling and librationalspectra.Due to the approximate exponential dependence of

h̄ωt on V3 (see Eq. (36) or Fig. 7a), h(h̄ωt) takesan extremely asymmetric shape (see Fig. 8d) withthe maximum shifted to low frequencies. From theincoherent scattering function for a single barrier (14),the tunnelling limit of Eq. (33) is given by:

SMGinc (Q,ω) =5 + 4j0(Qr)

9δ(ω) +

2[1− j0(Qr)]9

[h(ωt) + h(−ωt)]. (48)

For sufficiently high average barriers 〈V3〉 or broad dis-tributions g(V3), the maximum of h(±h̄ωt) is placedbeneath the instrumental resolution (see Fig. 8d).Hence, the tunnelling spectrum shows an apparentlyquasielastic contribution, which could be misinter-preted as a signature of hopping events at very lowtemperature. However, its actually inelastic originfinds a natural explanation, within the framework ofthe RRDM, in terms of a distribution of rotationaltunnelling lines.Once the tunnelling and hopping limits of the

RRDM are able to give account, with the same set ofparameters 〈V3〉, σV and Γ∞, respectively for the ex-perimental spectra at very low and high temperature,the consistency of the general version of the RRDMis tested, with such parameters, at intermediate tem-perature. It must be noted that this step does notinvolve any further fitting procedure. Instead, exper-imental spectra are directly compared with the the-oretical ones, which are constructed according to theprocedure exposed above for the general version of theRRDM, and by making use of the operational defini-tion of the crossover temperature, Eq. (37), in orderto select the dynamic regime (crossover or classical)for each individual methyl group.Eq. (37) also introduces a distribution of crossover

temperatures, G(Tc). Fig. 10 shows the latter, forthe same parameters, 〈V3〉 = 800 K, σV = 250 K andΓ∞ = 6 meV, of Figs. 8 and 9. As temperature Tincreases, the fraction of methyl groups that reach theclassical hopping regime, i.e., the area of G(Tc) forTc < T , will also increase and will be negligible above

16

some high temperature, where quantum effects will beunsolvable from the dominating hopping dynamics. Insuch conditions, the system will be well described bythe classical limit of the RRDM.

B. Other approaches

An alternative approach to the classical limit of theRRDM, also based on the idea of an underlying distri-bution of rotational barriers, has been introduced byArrighi et al. [23,24] for analyzing methyl group dy-namics in the high temperature hopping regime. ByFourier transforming into the time domain, the in-termediate incoherent scattering function for methylgroup classical hopping is given by:

IMGinc (Q, t) =1 + 2j0(Qr)

3+

2[1− j0(Qr)]3

F(t). (49)

Obviously for a crystalline system F (t) = exp(−t/τ),with τ the single residence time for hopping.For a disordered system, the corresponding function

F(t) in the framework of the RRDM is a log-Gaussiandistribution of time exponential functions:

F(t) =∫ ∞

−∞

H(log τ) exp(−t/τ)d(log τ), (50)

as trivially obtained by Fourier transformation of Eq.(47). In the approach introduced by Arrighi et al.,no functional form is assumed for the distributionH(log τ). On the contrary, F(t) is effectively rep-resented by a stretched exponential, or Kohlraush-Williams-Watts (KWW) function, exp[−(t/τ0)β ], withτ0 a characteristic rotational relaxation time. Thestretching exponent β takes values between 0 and 1.A KWW function can be effectively represented as

a log-distribution of time exponential functions:

exp[−(t/τ0)β ] =∫ ∞

−∞

K(log τ) exp(−t/τ)d(log τ). (51)

The distribution function K(log τ), which can be ob-tained by inverse Laplace transformation methods[101], is asymmetric. Therefore it is clear that theKWW function cannot be equivalent to the RRDMdistribution F(t) if a log-Gaussian form is selected forH(log τ) [89], (note that for the same reason τ0 6= τ).This fact is illustrated in Fig. 11 for typical values τ= 0.1 ns and σ = 1.

Both pictures (KWW and RRDM) achieved descrip-tions of similar quality for experimental spectra in alimited energy window, i.e., by using a single spec-trometer. The corresponding times τ and τ0 followedArrhenius behavior with close values of the respectiveactivation energies [23,24,89]. However, an analysis ina wide dynamic range [89], by combining several spec-trometers, showed that, while an excellent agreementof the experimental F(t) with the RRDM function (50)was obtained in all the time interval, a simultaneousfit to a KWW function was not satisfactory. More-over, independent fits in different experimental win-dows provided incompatible values of τ0 and β. Thisresult is illustrated in Fig. 12 for the same functionF(t) of Fig. 11, now analyzed in different narrow dy-namic windows.Another disadvantage of using a KWW functional

form for F(t) is that, in contrast to the unified pic-ture introduced by the RRDM, it does not provide adirect physical connection between the classical andquantum features observed respectively at high andlow temperature. It is worthy of remark that MDSsupport, by following separately the behavior of eachindividual methyl group in the system, the general va-lidity of the Gaussian approach for the distribution ofrotational barriers [85,86,88,102].

V. APPLICATION OF THE RRDM TO ASHOWCASE: POLY(VINYL ACETATE)

PVAc was the first polymer —and indeed the firsthighly disordered system— where a complete inves-tigation by neutron scattering was carried out onall the features of methyl group rotation (tunnelling,crossover, hopping and librations). It was also the firstsystem where a fully consistent description of the ex-perimental results was achieved in terms of the RRDM[31,33,89]. Figs. 13, 14 show respectively for a BS anda TOF spectrometer, experimental spectra (circles) forPVAc, at several temperatures in the tunnelling (T =2 K), crossover (T = 20 and 30 K) and classical hop-ping (T = 70, 120, 160 and 200 K) regimes. All thesetemperatures are far below the glass transition tem-perature (Tg ≈ 315 K), where motions different frommethyl group rotation just contribute as vibrationaldynamics via the Debye-Waller factor.An excellent agreement with the RRDM is observed

in all the temperature range (solid lines). As in the

17

usual analysis procedure, high temperature data ofPVAc were first analyzed in terms of the classicalRRDM [89]. Fig. 15 shows the temperature depen-dence of the parameters log Γ0 and σ of the distribu-tion H(log Γ) (taken as Gaussian) obtained from theanalysis in terms of the hopping limit of the RRDM.The use of several spectrometers with different energyresolutions and dynamic ranges reduced the uncertain-ties of the so determined parameters. The observed de-viations below T ≈ 70 K from the linear behavior pre-dicted by Eqs. (45,46) show the relevance of quantumeffects at low temperatures. The fact that the mag-nitude of such deviations depend on the experimentaldynamic window evidences the importance of usingseveral spectrometers in order to avoid, below sometemperature, an analysis biased by the used instru-mental resolution. The results of the fits for T > 70K to Eqs. (45,46) provide a preexponential factorΓ∞ = 9.1 meV, an average classical activation energy〈EA〉 = 450 K, and a standard deviation σE = 250 K.Transformation to g(V3) yields 〈V3〉 = 534 K and σV =274 K. The dashed lines in Fig. 15, correspondingto Γ = Γ∞ exp[−(〈EA〉 ± σE)/kBT ], have been intro-duced in order to stress the presence of a wide distri-bution of hopping times, which progressively spreadsover several orders of magnitude as temperature de-creases.The corresponding distribution of tunnelling fre-

quencies h(h̄ωt) is obtained by transformation of thebarrier distribution g(V3) independently derived fromthe high temperature classical analysis. Once h(h̄ωt)is known, the theoretical SMGinc (Q,ω) is constructed ac-cording to Eq. (48). An excellent agreement betweenthe tunnelling limit of the RRDM and the experimen-tal spectrum at 2 K is obtained (Fig. 13). The max-imum of h(h̄ωt) is at h̄ωt ≈ 0.03 µeV [31], i.e., wellbeneath the instrumental resolution, leading to the ap-parently quasielastic observed intensity .For the case of the librational distributions, a direct

comparison with experimental results is complicated,since librational energies are strongly superposed withthe phonon spectrum. This is particularly problem-atic in PVAc, which cannot be selectively deuteratedin order to attenuate the intensity scattered by nucleidifferent from methyl group protons. Fig. 16 shows acomparison of the generalized VDOS of PVAc with thedistribution of librational energies F (E01) obtained bytransformation of g(V3). A reasonable agreement be-

tween the maximum of the experimental peak andthe maximum of F (E01) is achieved, supporting thevalidity of the previuosly obtained distribution g(V3)and the consistency of the RRDM picture. It mustbe stressed that, since the generalized VDOS is notthe real density of states (see Section II), a rigorouscomparison between the widths of the experimentallibrational peaks and the theoretical librational distri-butions cannot be made.Once it has been checked that the distribution of

potential barriers g(V3) can give account, by transfor-mation to the other distributions in Eqs. (43, 44),for the spectra in the tunnelling and hopping dynamiclimits, as well as for the librational peak observed inthe generalized VDOS, the consistency of the generalRRDM is confirmed by also reproducing the experi-mental spectra at intermediate temperatures (Fig. 13for T = 20 and 30 K). An excellent agreement is againachieved. A more detailed comparison between exper-iment and theory can be obtained by investigating thetemperature dependence of the integrated intensitiesin different inelastic windows, as shown in Fig. 17.Since Eq. (48) is temperature-independent, and theDebye-Waller factor is a decreasing function of T , theinitial increase of the integrated intensities when heat-ing the system makes clear that the tunnelling limitof the RRDM is not appliable for T >∼ 2 K. It mustbe stressed that the observed double-peak structureis not an experimental artifact. Indeed, it can alsobe observed in a crystalline system when the inelas-tic window is properly selected, e.g., at lower energybut close to the single tunnelling line. As mentionedin Section III, when increasing temperature the tun-nelling line broadens and shifts to lower energies un-til it merges, when reaching the hopping regime, intoa single quasielastic line. As a consequence, the in-tensity measured in the selected inelastic window willpass through a first maximum. A second maximumwill be observed at higher temperature when broad-ening of the quasielastic line reaches the selected in-elastic window. In glassy PVAc, due to the broad dis-tribution of tunnelling frequencies in comparison withthe energy window accesible by the spectrometer, thedouble-peak structure is observed at any inelastic win-dow.As shown in Fig. 17, the double-peak structure

is nicely reproduced by the theoretical curves (solidlines) obtained by integration of the RRDM scattering

18

function in the corresponding inelastic windows. Thetheoretical curves quantitatively reproduce the exper-imental intensities in all the temperature range. Itmust be noted that the formers have been modulatedby a Debye-Waller factor e−2Q

2〈u2〉/3, with 〈u2〉 = θTand θ = 3 × 10−4 Å2K−1 [31]. The theoretical curvescorresponding to the classical hopping limit of theRRDM (dashed lines) have been extrapolated to verylow temperature in order to stress the relevance ofquantum effects below T ≈ 70 K, as was pointed outabove.

VI. SUMMARY OF EXPERIMENTALRESULTS BY NEUTRON SCATTERING ON

POLYMERS

A. Classical hopping

From early 80’s, neutron scattering has been widelyused to investigate methyl group dynamics in thequasielastic energy range |h̄ω|

The FEW technique requires much less acquisi-tion time than QENS measurements, but obviuoslyquasielastic spectra are preferred for an accurate de-termination of the parameters of the distribution ofhopping rates. Though for moderate average barri-ers, analysis of FEW data have provided reliable re-sults [90], this procedure is not recommended whenthe average rotational barrier is very high (V3 >∼ 2000K), since in such cases the two step-like decays of theFEW intensity overlap. Such a case is illustrated forpolysulfone (PSF) in Figure 18.First quasielastic investigations on methyl group

classical rotation in polymers were analyzed in termsof the usual approach for molecular crystals, i.e., by us-ing a single Lorentzian function (single hopping rate)to model the quasielastic component. As mentioned inthe Introduction, such a procedure fails when appliedto polymers (and in general to highly disordered sy-sems), providing inconsistent and unphysical results.In polymeric materials, and in general in disorderedsystems [35,36], the effect of structural disorder hasto be taken into account to describe properly methylgroup dynamics. In the framework of the RRDM, itis assumed that the only effect of the structural disor-der on methyl group dynamics is to introduce a dis-tribution of rotational barriers g(V3), originating fromthe different local environments felt by the individ-ual methyl groups. The initial version of the RRDM—the hopping limit— was introduced for analyzingquasielastic data from PVME [25]. Nearly at the sametime, a very close approach based in a Gaussian dis-tribution of activation energies was proposed by Frickand Fetters [26] to describe the temperature depen-dence of FEW data from PI. As mentioned in SectionIV, an alternative approach [23,24], first introducedto analyze time-domain data of PMMA and PVME,assumed a KWW stretched exponential function forthe intermediate scattering function I(Q, t). Althoughthis procedure was able to describe spectra obtainedin a single spectrometer, it failed to properly accountfor data collected in a wide dynamic range by usingseveral instruments [89].Classical hopping between three equivalent posi-

tions was assumed in all these approaches. It must benoted that classical hopping is just driven by the acti-vation energy EA, and that a purely threefold barrierand another one distorted by a small sixfold contribu-tion, V6(1− cos(6φ+ δ))/2, can provide the same EA

with an appropiate selection of the parameters V3, V6and δ. Hence, an analysis of high temperature classi-cal spectra provides a first approach to the functionalform of the rotational potential, but an unambiguousdetermination of the latter is only possible by means ofa complete analysis of hopping, tunnelling and libra-tional spectra. For most of the polymers or molecularglasses for which at least two of these three featureshave been investigated, it has been found that a dis-tribution of purely threefold barriers provides a gooddescription of experimental spectra. As shown below,small but significant corrections to the threefold termhave instead to be considered in order to consistentlyreproduce the different features of the ester-methylgroup dynamics in PMMA.Recent QENS investigations in PDMS [93], PVAc

[103] and in the blends hh-PP/PEP [111], andPEO/PMMA [116] have tackled the problem of methylgroup classical hopping in the very high temperaturerange where secondary relaxations or motions involvedin the glass transition process enter the dynamic win-dow of neutron scattering. In that situation the timescales of such processes and methyl group hopping su-perpose. A consistent description has been achievedby assuming that the mentioned motions and methylgroup rotation are statistically independent. This ap-proach was first introduced in an investigation, bymeans of molecular dynamics simulations, on methylgroups dynamics in PI above the glass transition tem-perature [129]. Within this approach, the RRDM pic-ture is maintained for methyl group rotation and thetotal scattering function is just obtained as a convo-lution of the RRDM scattering function (47) and thatcorresponding to the other relaxational modes [129].A unified description is achieved with the same values—or small variations— of the RRDM parameters de-rived at lower temperatures, where the dynamic con-tribution of the polymer matrix just enters in the vi-brational Debye-Waller factor. This result evidencesthat the barrier distribution for methyl group rotationin polymers is not substantially changed by the struc-tural rearrangements produced when approaching theglass transition [93,129].

20

B. Rotational tunnelling and crossover to hopping

As mentioned in the Introduction, the existence ofquantum rotational tunnelling for methyl group dy-namics in polymer systems was expected for a longtime but it was not directly observed by neutron scat-tering until late 90’s [31,32]. In contrast to crystallinesystems where, due to the unique value of the ro-tational barrier, resolution-width tunnelling lines areobserved at very low temperature, in amorphous sys-tems, a distribution of rotational barriers, and there-fore of tunnelling frequencies, is expected. Some ex-amples are misoriented methyl groups in crystallinelattices [130] or absorbed in water clathrates [131] orcrystalline zeolites [132], and other small molecularrotors in chemically or geometrically disordered en-vironments as non-stochiometric metal hexaammines[133], ammonia in metal alkali fullerides [134], dilutedsolutions of methane in noble gases [135–137], or ofammonium in metal alkali halides [138,139]. In thesesystems, disorder effects provide narrow distributionsof rotational barriers and of the corresponding tun-nelling frequencies, which just manifest as weak broad-ening of the tunnelling peaks in neutron scatteringspectra. However, in the case of methyl groups inpolymers [31,32,34], molecular glasses [35,36], or closeto pore walls in mesoporous silicates [94,140–142], dis-order leads to broad distributions of tunnelling fre-quencies. Due to the typical values of the rotationalbarriers in the latter systems, specially in polymers,most of the expected distribution of tunnelling linescorresponds to energies below ∼ 1 µeV, i.e., beyondthe dynamic limit explored by BS spectrometers. Thisfact prevented for a long time the observation of tun-nelling features in low temperature spectra.PVAc is the polymer having the lowest energy bar-

rier for methyl group rotation reported so far, and themost suitable one for the direct observation of rota-tional tunnelling transitions by means of neutron scat-tering [31,33]. An analogous investigation has beencarried out on rotational tunnelling of the ester-methylgroup of PMMA [32,34]. Though having a higher bar-rier, PMMA has the advantage of being selectivelydeuterable, in contrast to PVAc. In this way, the ob-served scattered intensity excess for PMMA-d5 (i.e.,with partial deuteration of the α-methyl group andthe main-chain hydrogens) over the elastic line at verylow temperature can be unambiguously assigned to the

ester-methyl group dynamics.A further support of the interpretation of the ap-

parently quasielastic low-temperature intensity as theresult of a distribution of rotational tunnelling linesis provided by exploting the well-known isotope ef-fect. Hence, deuteration of the methyl group yieldsa rotational constant B twice smaller than for theprotonated methyl group. The corresponding tun-nelling frequency is reduced by several orders of mag-nitude (dashed line in Fig. 7a), and consequently, thecorresponding distribution h(h̄ωt) is strongly shiftedto lower frequencies (see Fig.19 for the ester-methylgroup of PMMA). This effect predicts a strong su-pression of the scattered intensity excess in the exper-imental spectrum, as it is indeed observed [32] in afully deuterated sample (d8) of PMMA at the sametemperature (see Fig. 20). For a proper compari-son between the ester-methyl group dynamics in thed5- and d8-samples, the elastic contributions, corre-sponding to the total coherent cross-section and tothe incoherent cross-section of the other nuclei, aresubstracted from the spectra. This procedure intro-duces large uncertainties in the so obtained spectrumfor the d8-sample, due to the large relative weight ofsuch elastic contributions. However, the supression inthe d8-sample of the apparently quasielastic intensityis evident, and consistent with the result predicted onthe basis of the RRDM parameters derived for the d5-sample [32].Except for PVAc and for the ester-methyl group of

PMMA, for the rest of the polymers investigated sofar, the large values of the average barriers, 〈V3〉 >∼1000 K, determined from the analysis of high tempera-ture hopping data and/or of librational levels, preventthe observation of clear rotational tunnelling featuresat very low temperature. This situation is illustratedin Fig. 21 by a comparison between the expected dis-tributions of tunnelling frequencies for several poly-mers. As can be seen, within the neutron scatteringwindow, h(h̄ωt) for PVME and PI is orders of magni-tude smaller than for PVAc and for the ester-methylgroup of PMMA. Convolution of h(h̄ωt) with the in-strumental resolution provides a scattering intensityexcess unsolvable from the latter within the experi-mental noise, as indeed is experimentally confirmed.An analysis of high temperature data for PDMS interms of the hopping limit of the RRDM [93] provideda distribution of classical activation energies f(EA),

21

which, by transformation, would yield a distibution oftunnelling frequencies clearly observable by means ofhigh-resolution BS spectrometers. However, the mea-sured intensity at T = 2 K [143] is hardly distinguish-able from the instrumental resolution, as can be seenin Fig. 22. Moreover, the RRDM parameters givenin Ref. [93] provide, by transformation, a theoreticallibrational maximum around ≈ 17 meV, far from theexperimental value of 21.2 meV (see Fig. 23a). Onthe contrary, the RRDM parameters in Ref. [89] re-produce the librational peak and are compatible withthe experimental spectrum at T = 2 K (see Fig. 22).The reason for the mentioned uncompatibilities withparameters in Ref. [93] remains to be understood.Despite of the existence of very few polymer sys-

tems where methyl group rotational tunnelling can beobserved by neutron scattering, successful investiga-tions have been carried out in two molecular glasses—toluene [35] and sodium acetate trihydrate [36]—supporting the validity of the RRDM picture for rota-tional tunnelling.As mentioned in Section IV, at the crossover inter-

mediate temperature regime (typically 20 K

hopping regimes. It is noteworthy that such a con-sistent description has been achieved by fixing a ra-tio V6/V3 = 0.11 for all the individual methyl groups.Hence, as in the purely threefold case, the only inde-pendent parameter affected by the distribution effectsis the threefold term V3.Librational energies are also sensitive to coupling

between pairs of neighbouring methyl groups. In suchcases, the single-particle approach is modified by in-troducing a coupling Hamiltonian. In the simplest ap-proximation the coupling term is purely threefold andno angular offsets are included:

H = −B(

∂2

∂φ2+

∂2

∂ψ2

)

+U32(1 − cos 3φ)

+V32(1− cos 3ψ) + W

c3

2(1− cos 3(φ− ψ)). (54)

In a recent INS investigation on methyl group libra-tions in PSF [149], the double-peak structure in therange ≈ 34 − 44 meV (see Fig. 25) has been inter-preted as the result of a distribution of splitted firstlibrational energies, such a splitting resulting fromthe coupling between the two methyl groups in eachmonomeric unit. As in the single-particle approach,the terms U3 and V3 are taken as Gaussian distributed.Due to the highly disordered character of the poly-mer matrix and the symmetric chemical arrangementof both methyl groups in the monomeric unit, it isexpected that, on average, they feel the same typi-cal environment. Hence, the distributions g(U3) andg(V3) are assumed to be identical. Since the couplingterm W c3 is strictly due to the interaction between thetwo methyl groups of the same monomeric unit, it isexpected to be weakly affected by local packing con-ditions. Hence for simplicity, distribution effects areneglected for the coupling term.The classical hopping regime for PSF is not accesi-

ble by QENS at low and moderate temperature, andat high temperature superposes with secondary relax-ations which complicate the analysis. Instead, it canbe properly explored by NMR techniques. Hence, thetemperature dependence of the D-NMR line shape ofPSF has been reproduced in terms of a log-Gaussiandistribution of classical hopping frequencies as Eq.(44), with log Γ0 and σ respectively following Eqs.(45, 46). Analogously to the procedure exposed inSection IV, these results provide a Gaussian distri-bution of classical activation energies f(EA), which

can be transformed into a distribution of rotationalbarriers g(V3) and librational energies F (E01) via thefunctional relations EA = EA(V3), E01 = E01(V3).Such functional relations can be obtained by solvingthe eigenvalues of the Hamiltonian (54) for the selectedvalue ofW c3 and a large set of values U3 = V3 (see Ref.[149] for numerical relations).Fig. 25 shows the corresponding distributions

F (E01) for the single-particle approach, and for cou-pling of methyl groups, —with the conditions aboveexplained for the second case. While the single-particle approach only gives account for the lowest ex-perimental peak, the introduction of a coupling termreproduces the double-peak structure. Analogous re-sults are obtained for PC and PH [149]. It is note-worthy that the introduction of a coupling term doesnot substantially complicate the original RRDM, sincethe only model parameter affected by the distributioneffects is, as in the single-particle case, the rotationalbarrier height V3.

VII. MOLECULAR ORIGIN OF THEBARRIER DISTRIBUTION FOR METHYL

GROUP DYNAMICS

The results summarized in Table II show that theaverage barrier height 〈V3〉 for methyl group reorien-tation in amorphous polymers is highly sensitive tothe chemical structure of the monomeric unit. Thisfact suggests that the rotational potential is only par-tially determined by intermolecular interactions. Onthe other hand, there is not a clear correlation betweenthe width of the barrier distribution and the chemicalstructure of the monomeric unit. Next we summarizethe most relevant neutron scattering results relatingthe parameters of the barrier distribution with pack-ing conditions and intermolecular interactions.

A. Influence of the chain conformation

The relative orientation between identical sidegroups placed at different monomeric units can berandom or can adopt specific forms. Hence, in theisotactic (i) and sindiotactic (s) conformations, sidegroups at adjacent monomeric units are respectivelyparalell and anti-paralell. If both relative orientations

23

are randomly distributed, the conformation is hetero-tactic (h). When there is no dominating tacticity inthe ensemble of chains forming the system, the latteris atactic (a). The ester-methyl group dynamics inPMMA have been analyzed in terms of the RRDM fordifferent degrees of tacticity of the ester group. Theinfluence of local packing conditions on the barrier dis-tribution for samples chemically identical is evidencedby the small, but significant, differences between theobtained RRDM parameters [32,34,91,152] (see Ta-ble II). Hence, the average activation energy for apurely syndiotactic sample is 〈EA〉 = 710 K [34], whilefor an atactic sample with relation s:i:h = 50:10:40,〈EA〉 = 529 K [91].PMMA chains can be rearranged to form a stereo-

complex form, where syndiotactic chains wrap aroundisotactic ones to form a double stranded helix struc-ture. In Ref. [91] stereocomplexed samples with com-plementary deuteration of chains with different tac-ticities were investigated. In this way, the effect ofstereocomplexation on methyl group dynamics in dif-ferent types of chains could be discriminated. For theisotactic chains, stereocomplexation considerably re-duced the width of f(EA), yielding a standard devia-tion σE = 205 K, as compared to the value σE = 313K for the non-stereocomplexed sample [91].For the case of the α-methyl group in PMMA, INS

measurements showed a much higher librational peak[17], and consequently a much higher average rota-tional barrier, for the syndiotatic form as comparedwith the isotactic one. On the contrary, INS spectrafor PαMS [17] did not show significant differences be-tween a syndiotactic sample and an atactic one. FEWscans reported in Ref. [153] for different tacticities ofPP did not show apparent differences, within the ex-perimental noise, in the temperature range dominatedby methyl group dynamics. This result is supportedby the rather close values of the measured librationalpeaks for the isotactic [144] and atactic [145] confor-mations.There are very few systems where experimental data

are available for both, the usually investigated head-to-tail, and the head-to-head sequences of adjacentmonomeric units. For PαMS [17] and for PP [111]it has been found, repectively by means of INS andQENS measurements, that the average barrier in thehead-to-head conformation is much lower than in thehead-to-tail one. In the case of hh-PP, the distribution

of classical activation energies f(EA) has been calcu-lated from an analysis of high temperature data interms of the hopping limit of the RRDM [111]. Appartform the mentioned decrease of the average barrier,the analysis provides a much narrower distributionthan that obtained for head-to-tail PP [88,90]. It isworthy of remark that the maximum of the librationaldistribution F (E01) for hh-PP, obtained by transfor-mation from f(EA) in a purely threefold approach (seeSection IV), is close to the experimental value reportedin Ref. [145], though the agreement is not fully satis-factory [111]. A possible origin of such a differencemight be, similarly to the case of the ester-methylgroup in PMMA (see Section VI), the presence of smallhigher-order corrections to the main threefold term ofthe rotational potential, or small coupling effects be-tween neighbouring methyl groups, which could shiftthe librational energy to slightly higher values thanthe expected ones for a purely threefold single-particlepotential. This hypothesis is consistent with the prox-imity between methyl groups of adjacent units in thehead-to-head conformation of PP.

B. Influence of mixing with other materials

A few works report comparisons between methylgroup rotation in the neat polymer and blended withother polymers. QENS investigations in the blendsPVME/PS [114,115] and hh-PP/PEO [111] did notshow, for similar concentrations of the two compo-nents, changes within the error bar on the barrier dis-tribution as compared respectively to neat PVME andhh-PP. High dilution of PVME in PVME/PS provideda significantly broader distribution, but again did notaffect to the average barrier [115]. On the contrary,progressive dilution of PMMA in SCPE/PMMA in-creased the average barrier for the ester-methyl groupeven at moderate concentrat