Molecular Dynamics Analysis of Coupling between LibrationalMotions and Isomeric Jumps in Chain Molecules

Canan Baysal, Ali R. Atilgan, Burak Erman, and I4vet Bahar*

Polymer Research Center, School of Engineering, Bogazici University, and TUBITAKAdvanced Polymeric Materials Research Center, Bebek 80815, Istanbul, Turkey

Received June 20, 1995; Revised Manuscript Received December 12, 1995X

ABSTRACT: The coupling between librational motions and rotational isomeric jumps in polymers isinvestigated. Librations originate from bond stretching, bond angle distortion, and small-amplitudefluctuations in the bond dihedral angles. Molecular dynamics trajectories for a polymer chain of 100bonds in vacuo are used for assessing the contribution of librations to the dynamic behavior of chains.Each trajectory is resolved into two sets of components, low and high frequency, representing bondrotameric transitions and librations, respectively. Librational motions are observed to be stronglycorrelated with the rotational jumps of bonds between isomeric states. In fact, the passage from anisomeric state to another is accompanied by an enhancement in librational motions. Examination of thecontribution of librations to the local relaxation behavior of polymers indicates that librations affect themechanism of motion via two opposing effects. On the one hand, they provide an additional degree offreedom which causes a faster relaxation of the conformational correlation functions associated with thebond dihedral angles. On the other hand, the orientational correlation functions associated with thespatial reorientation of backbone bonds exhibit a slower time decay due to the compensating effect oflibrational motions. In fact, librational motions are not random but highly directed such that theycollectively preserve local chain direction and help in accommodating the local reorientations induced byisomeric jumps, which would otherwise result in larger displacements of chain segments. Finally, it isshown that replacing the librations of a trajectory with random, uncorrelated fluctuations leads to anoverestimation of the relaxation rate of configurational correlations.

Introduction

Two types of motions dominate local chain dynam-ics: (i) rotational transitions of backbone bonds fromone isomeric state to another and (ii) librational motionsincluding the torsional fluctuations within the energywell of rotameric states, the distortions of bond angles,and the stretching of bonds. The former lies in the lowerfrequency range of the spectrum, whereas the latter isresponsible for the higher frequency motions and con-stitutes a bath for the former. These two types ofmotions cooperate during the passage over the energybarriers of the conformational space1 and in the relax-ation to a new equilibrium state.One way of quantifying the cooperativity between

librational motions and isomeric jumps is to monitor theconformational and orientational motions of chain seg-ments. Conformational motions are those that may bedescribed without reference to an external coordinatesystem, such as the transitions between trans, gauche(+), and gauche (-) states and the fluctuations in bonddihedral angles. Orientational motions, on the otherhand, require a coordinate system for their specification.The angle between a bond at a given time and anotherbond at a later time is an example. Both conformationaland orientational motions are characterized by time-delayed correlation functions (CFs) and the associatedcorrelation times. These are not only predictable bycomputational methods but also measurable by experi-ments, such as time-resolved fluorescence or nuclearmagnetic resonance. However, the assumptions madein interpreting the experimental results can only berationalized by computer simulations.Although there have been several attempts to express

the experimental relaxation curves by analytical func-tions,2 most approaches have either neglected the

torsional librations3 or confined the motion to a fewbonds only.4 Some studies do not consider three-dimensional motions,5 and others associate high-fre-quency effects with additional fast anisotropic processes6incorporating Howarth’s treatment of librations.7 Deri-vation of correlation functions by analytical models israther complex in three dimensions. Cook and Helfand8developed a one-dimensional analytical model andshowed the occurrence of a single-exponential decay ratefor conformational correlations, representative of bothsingle and cooperative transitions. A similar model wasused by Moro where the coupling between librations andconformational jumps was analyzed on the basis of alinear chain of rotors.9 Therein it is demonstrated thatlibrational motions operating after a transition areresponsible for attaining the new equilibrium state.Recently Kim and Mattice10 took another path in thatthey investigated the effect of libration on the timeevolution of the conformational autocorrelation func-tions (CACFs) for dihedral angles. Interestingly, theyfound that the adoption of discrete conformational statesexcluding librations, as in the basic approach of thedynamic rotational isomeric states (DRIS) theory,11results in an overestimation of conformational correla-tion times.In all of the above treatments some questions remain

to be answered: How is the time decay of CFs affectedby the contribution of librational motions? Do thelibrations have a random character so that the overallchain dynamics can be adequately predicted by remov-ing them and adopting a DRIS-like treatment? Is thesuperposition of an additional single-exponential typefast relaxation mechanism on the relaxation spectrumsufficient to account for the effect of librations, asperformed in several studies, or do librations act in somecooperative manner and exert different effects on dif-ferent types of correlations? If some cooperativity exists,i.e., if librations are actively involved in the initiation

X Abstract published in Advance ACS Abstracts, February 15,1996.

2510 Macromolecules 1996, 29, 2510-2514

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or localization of rotameric jumps, for example, is itpossible to retain the effect imposed by librations onlarge-amplitude movements by simply eliminating thehigh-frequency components of the overall motion?In the present study, a molecular dynamics (MD)

trajectory is analyzed with the objective of testing thevalidity of various assumptions made in interpretingexperiments and clarifying the above raised issues.Time series obtained from MD simulations are filtered1to obtain smoother trajectories where only low-fre-quency modes remain. The fluctuations in bond lengthsand angles and/or dihedral angles are also turned offby using an operator formalism, and the consequencesof eliminating these degrees of freedom on the relaxationmechanism are explored. Alternatively, these degreesof freedom are assigned Gaussianly distributed randomvariables about their equilibrium values. CFs calcu-lated by using these modified trajectories are comparedto those determined from the original MD, in order toassess the effects of different levels of approximation.In the next section, the two types of correlation

functions considered in the present analysis are intro-duced as well as the methods adopted in interpretingthe results. Next, the time decay of correlation func-tions obtained after subjecting the original MD trajec-tory to various operations is presented, demonstratingthe strong coupling between low- and high-frequencymotions and the nontrivial dependence of the localrelaxation behavior on this coupling.

Method and Calculations

Correlation Functions. The effect of librations on thelocal relaxation phenomenon will be explored on the basis oftwo types of autocorrelation functions: (i) the orientationalcorrelation functions (OACFs) associated with the reorienta-tion of backbone bonds. This is expressed in terms of theinstantaneous unit vector li(τ) along the ith bond as

(ii) the second CACF describing the time evolution of therotational isomeric states of backbone bonds. This is givenby

where æi(t) is the dihedral angle of the ith bond at time t. Thebroken brackets in eqs 1 and 2 refer to the ensemble averageover internal bonds, and the overbar denotes the time averageover all conformational transitions starting at various t andoccurring within the time interval τ. Here, the section 30 e ie 70 will be considered for a chain of n ) 100 bonds.Molecular Dynamics Method. MD simulations are car-

ried out for a united atom model polyethylene (PE) chain of n) 100 CH2 units. Units are indexed from l to n. The backbonebond li connects the CH2 groups i - 1 and i. The time-evolvedcoordinates of atoms are obtained as an array of positionvectors ri(t) ) [xi(t) yi(t) zi(t)]T, 0 e i e n, where the superscriptT denotes the transpose. These are transformed into a set ofgeneralized coordinates, consisting of (i) bond dihedral anglesæi, 2 e i e n - 1, (ii) bond angles, θi, 1 e i e n - 1, (iii) bondlengths, li, 1 e i e n, of the particles at any instant in time,following the formulation previously presented.12 Here thetime arguments are omitted for brevity. For convenience, wewill denote as Q1 the transformation matrix or operator thatpermits the passage from the set of position vectors to that ofthe generalized coordinates, i.e.

The inverse operator, Q1-1, transforms from the generalized

coordinates [li, θi, æi]T to the set of position vectors, [xi, yi, zi]T.The set of generalized coordinates, obtained for all units in asequence of 40 bonds located in the middle of the chain andrecorded at 4 fs intervals, will be referred to as the originaltrajectory. The latter will be subjected to various operationsas will be presented below.The chain is chosen to be in vacuo so that only those

interactions generated by the polymer itself are operative. Theoverall potential is the sum of (i) a harmonic bond stretchingand (ii) a harmonic bond bending function, (iii) a quinticfunction that accounts for the intrinsic torsional potential ofdihedral angles and first-order interactions between chainatoms separated by three bonds, and (iv) a Lennard-Jonespotential, which operates on atom pairs that are separated byfour or more bonds and are within 12 Å distance of each other.Further details of the simulation method are provided in ourprevious work.1,12,13 Initially, a 25 ps equilibration time isallowed. The total simulation duration after equilibration is∼260 ps. The temperature is kept constant at 300 K.Filtering Technique. The contribution of modes with

particular frequency ranges νmin e ν e νmax may be extractedfrom the original MD trajectories by applying the followingfiltering method:14 (a) The original trajectory A(t) is Fouriertransformed into the frequency domain,

(b) A filter function F(ν) that extracts the frequency compo-nents within the range of interest is applied, as

(c) A′(ν) is inverse Fourier transformed to the time domain andthe frequency-filtered trajectory A′(νmin < ν < νmax, t) is obtainedas,

In a recent MD study with the same model chains, it wasshown that the modes responsible for isomeric transitions arethose having frequencies below 20 cm-1, approximately,whereas the overall relaxation spectrum contains frequenciesup to ∼4000 cm-1.1 Retaining these low-frequency modes istermed low-pass filtering below 20 cm-1. The cutoff value of20 cm-1 is not definitive, however, and lower or higher valuesmay be chosen depending on the precision required by thequantity of interest.Reconstruction of Trajectories Subject to Various

Operations. The original MD trajectories are subjected tothe following operations to obtain the so-called modifiedtrajectories: (a) The fluctuations in bond lengths and bondangles are removed from the trajectories by replacing thesevariables with their mean values. This may be expressed bythe operation

where the instantaneous bond lengths and bond angles areexpressed as a sum of their mean values and their fluctuations,as

M[l(τ)] ) 12

{3⟨[li(t)‚li(t + τ)]2⟩ - 1} (1)

M[æ(τ)] ) 12

{3⟨cos2[æi(t + τ) - æi(t)]⟩ - 1} (2)

Q1: [xiyizi ] f [ liθiæi

] (3)

A(ν) )∫-∞

∞A(t)e-jvt dt (4)

A′(ν) ) A(ν)F(ν)

F(ν) ) {1; νmin < ν < νmax0; ν < νmin, ν > νmax

(5)

A′(νmin < ν < νmax, t) )∫-∞

∞A′(ν)ejνt dν (6)

Q2: [ lhi + ∆liθhi + ∆θi

æi] f [ lhiθhi

æi] (7)

li ) lhi + ∆liθi ) θhi + ∆θi

(8)

Macromolecules, Vol. 29, No. 7, 1996 MDA of Coupling in Chain Molecules 2511

The mean bond lengths and bond angles are 1.53 Å and 109.5°,for all bonds. Comparison of the modified trajectory with theoriginal trajectories will provide information on the contribu-tion of the librations in bond lengths and bond angles on thelocal chain relaxation mechanism. Clearly the modified setof position vectors may be generated by the application of theinverse operatorQ1

-1 on the modified generalized coordinatesobtained in eq 7. In summary, a modified set of positionvectors is obtained by premultiplication of the original arrayby Q1

-1 Q2 Q1.(b) In addition to bond length and bond angles, torsional

librations about rotational isomeric minima may be eliminatedby replacing the original trajectories by a new one modifiedaccording to the operation

Here æj i represents the rotational isomeric state values as-sumed by the ith dihedral angle. This is given by a stepfunction of the form15

The values 0°, -120°, and +120° correspond to the respectiveisomeric states trans, gauche (-), and gauche (+).(c) Alternatively, normal fluctuations may be superimposed

on bond lengths and bond angles, which would provide insightsinto the differences between random librations and thoseoccurring in the original MD trajectories. Accordingly, theinstantaneous bond lengths and bond angles are expressed asGaussianly distributed random variables with the mean values1.53 Å and 109.5° and covariances 0.10 Å and 10°, respectively.This transformation may be written as

Here the superscripts n refer to the normal distribution, andthe dihedral angles are left unchanged.(d) A final transformation to be considered in the calcula-

tions is Q5, represented as

which is identical to Q4, except for the replacement of theoriginal torsional angles by their isomeric state values follow-ing eq 10. Clearly, in all cases a-d the modified set of positionvectors is constructed by application of the operation Q1

-1 Qm

Q1, with m ) 2-5, on the original position vectors.Table 1 provides a summary of the operations adopted in

obtaining modified trajectories subject to various approxima-tions, starting from the original MD trajectory. It should be

noted that the perturbations applied to the various degrees offreedom do not result from a change in the potential field usedin MD simulations. To the contrary, all trajectories originatefrom the same MD run with a given realistic potential field.In the earlier work of Helfand et al.16 the force constants inthe potentials were changed so as to constrain li and θi. Thisis a different approach which can provide an estimation of theeffect of constraining the librational motions on the relaxationmechanism. The present approach, on the other hand, pro-vides a direct measure of the perturbation of the MD trajec-tories by several factors, including the elimination of a classof operating modes via filtering technique, the removal of thelibrational motions, which may be directed, random, or both,and the imposition of a white noise for approximating theadditional mobility imparted by the high-frequency librationalmotions.

Results

Coupling between Low- and High-FrequencyModes. For illustrative purposes, a 250 ps portion ofthe trajectory for one of the dihedral angles is presentedin Figure 1a. The same trajectory, subjected to a low-pass filtering procedure truncating all modes withfrequencies above 20 cm-1, is shown in Figure 1b.Figure 1c shows the trajectory containing frequenciesabove 20 cm-1 only. Comparison of the curves inFigures 1a and b shows that motions associated withisomeric transitions are well represented by the 0-20cm-1 range of the frequency spectrum. It is interestingto note that the bond rotations in curve a or b are

Table 1. Characteristics of the Trajectories Utilizedfor Extracting OACFs and Their Orientational

Correlation Timesa

OACF operation τc(ps) τc/τco

i none 7.81 1.00ii filteredb 9.39 1.20iii Q1

-1 O2 Q1 5.62 0.72iv Q1

-1 Q3 Q1 4.09 0.52v Q1

-1 Q4 Q1 5.51 0.71vi Q1

-1 Q5 Q1 3.59 0.46a τc: orientational correlation times. τco: τc of original MD run.

b Using a low-pass filter of 20 cm-1.

Q3: [ lhi + ∆liθhi + ∆θiæj i + ∆æi

] f [ lhiθhiæj i ] (9)

æj i(τ) ) { 0°120°

-120°

-60° e æi(τ) e 60°æi(τ) g 60°

æi(τ) e -60°(10)

Q4: [ lhi + ∆liθhi + ∆θi

æi] f [ lhi + ∆li

n

θhi + ∆θin

æi] (11)

Q5: [ lhi + ∆liθhi + ∆θi

æi] f [ lhi + ∆li

n

θhi + ∆θin

æj i] (12)

Figure 1. Portion of a representative dihedral angle trajectoryobtained from MD simulations: (a) trajectory containing allfrequencies, (b) same trajectory containing frequencies below20 cm-1, and (c) same trajectory with frequencies above 20cm-1. The excited fluctuations in part c are synchronized withthe rotational isomeric transitions displayed in parts a and b,which demonstrate the coupling between low- and high-frequency modes.

2512 Baysal et al. Macromolecules, Vol. 29, No. 7, 1996

accompanied by an enhancement in the torsional fluc-tuations shown in part c. The occurrence of such excitedlibrations, synchronous with bond rotational transitions,is direct evidence of the correlations between fast andslow motions and, in particular, of their enhancedcoupling during rotational isomerization.Conformational Autocorrelations. The CACFs,

M[æ(τ)], evaluated using different approaches are pre-sented in Figure 2, up to τ ) 100 ps. Curve a representsthe result from the original MD trajectory. Curve b isobtained by applying a low-pass filtering below 20 cm-1.Curve c, on the other hand, is obtained by subjectingthe original generalized coordinates to the transforma-tion operator Q3. This eliminates all bond angle, bondlength, and torsional librations. However, eliminatingthe bond lengths and bond angles is inconsequentialsince the CACF is a quantity that depends purely ondihedral angles. In this approximation, the conforma-tional dynamics corresponds to that adopted in theDRIS model. This curve also compares to that givenby Kim and Mattice.10 It is evident from the figure thatboth approximations b and c predict a slower decaycompared to the original trajectory (a). The CACF fromthe filtered trajectory, curve b, indicates that thedeparture between the approximated and original be-havior arises primarily from a fast decay to ∼0.86 ofthe ordinate value in curve a within tenths of picosec-onds, which is absent in curve b. This initial decay incurve a is clearly induced by the high-frequency motionswhich have been eliminated in the filtered trajectory(b). The departure from the original CACF is evenstronger in curve c, where all torsional librations areextinguished following eqs 4-6.A comparison along the same lines was made by Kim

and Mattice for polybutadiene chains in the bulk stateand a similar result was obtained.10 Therein, it wasargued that the dynamics is governed by two relaxationphenomena at separate time scales: one is responsiblefor librations and the other for conformational transi-tions. Thus, neglect of librational motions leads toslower relaxation insofar as the CACF is concerned. Thisapproximation can be partly overcome by rescaling ofthe decay curves at short times. This explains the goodagreement between the CACFs extracted from Brown-ian dynamics simulations and those predicted by theDRIS formalism in which librational motions are notaccounted for.13

A mere analysis of the decays of CACF provides anincomplete picture of the relaxation mechanism. Amore thorough understanding of the role of high-frequency motions in local dynamics is gained byexamining the time-dependent orientation of chainsegments in space, in the presence and absence oflibrations.Orientational Autocorrelations. The OACFs of

bond vectors,M[l(τ)], obtained by subjecting a given MDtrajectory to different operations, are presented inFigure 3, up to τ ) 15 ps. Note that only the short-time range is given to emphasize the differences be-tween the initial decay of the curves. They all decay tozero eventually. Curve i represents the correlationfunction given by the original MD trajectory. Thecurves labeled ii-vi result from the trajectories modifiedfollowing the procedures summarized in Table 1. Mainly,curve ii refers to the low-pass (20 cm-1) filtered trajec-tory. Curves iii and iv result from the application ofthe operators Q1

-1 Q2 Q1 and Q1-1 Q3 Q1, respectively.

This allows the extraction of the contribution of libra-tional motions to the observed OACFs. Cases v and vifollow from the operations Q1

-1 Q4 Q1 and Q1-1 Q5 Q1

applied to the set of position vectors recorded in theoriginal MD trajectories. Here bond lengths and bondangles are assigned Gaussianly distributed randomfluctuations, which permit the visualization of thedifference between random librations and those coop-eratively operating in the original MD run. The orien-tational correlation times τc and their ratios τc/τco withrespect to that resulting from the original MD run, τco,are listed in the last two columns of Table 1 in order toprovide a measure of the effect of the various operationson the apparent OACFs.The low-pass filtered trajectory yields a slower decay

of the OACF than that of the original trajectory, as thecomparison of the curves i (solid) and ii reveals. Themore interesting feature observed in Figure 3 is thatall other curves, iii-vi, exhibit faster decays comparedto the original curve i. Even the supression of librationsin bond lengths and bond angles, which is representedby curve iii, enhances the decay rate of M[l(τ)] by aconsiderable amount. A decrease in orientational cor-relation time of 28% is induced by eliminating thevibrations in bond lengths and bond angles.In many experimental and computational studies it

is assumed that librations, especially those associatedwith stretching and bending, have a random charac-

Figure 2. Time decay of CACFs obtained from (a) the originaldihedral angle MD trajectories, (b) low-pass filtered trajectoriesbelow 20 cm-1 using eqs 4-6, and (c) regenerated trajectorieswhere only equilibrium values of the dihedral angles areallowed, conforming with eq 10. Logarithmic scale is used inthe ordinate for clarity.

Figure 3. Time decay of bond OACFs obtained with thevarious levels of approximation listed in Table 1. The solidcurve, labeled i, refers to the original trajectory. Curve ii isthe low-pass filtered trajectory. Curves iii-vi result fromdifferent constraints applied on bond bending, stretching, andtorsion degrees of freedom, as outlined in Table 1. Logarithmicscale is used in the ordinate for clarity.

Macromolecules, Vol. 29, No. 7, 1996 MDA of Coupling in Chain Molecules 2513

ter.6,7 Curves v and vi in Figure 3 result from theimposition of Gaussianly distributed random fluctua-tions on the equilibrium values of bond lengths andangles. Examination of these curves demonstrates thatthe replacement of the original directed librationalmotions by random fluctuations leads to a significantlyfaster loss of orientation. The relaxation time is de-creased by a factor of 0.71 and 0.46, respectively, incurves v and vi, compared to the original behavior.These results invite attention to the inadequacy ofinterpreting the stretching, bending, and torsional li-brational motions as a random accelerating relaxationmechanism.

Interpretation of Results and DiscussionOur recent molecular dynamics study of chain dy-

namics indicates that librations have components thatare strongly correlated with slower rotameric transi-tions.1 The basic feature of this coupling is that two ormore degrees of freedom may librate in a concertedfashion to localize a large-scale rotameric transition inspace which would otherwise result in excessive trans-lations of points along the chain. An inspection of thethree trajectories given in Figure 1 gives a good repre-sentation of the coupling between librations and rota-tional isomeric jumps. A definite coupling between bondrotational isomeric jumps and librational motions isevidenced by the enhanced activity of the high-frequencymodes (Figure 1c) during passage over rotational energybarriers. In addition to such coupled librations, thereare also librations in various degrees of freedom whichare random and not correlated in the manner stated.Removal of frequencies higher than 20 cm-1 from the

trajectory of Figure 1 eliminates most of the librationsthat are coupled to, or correlated with, rotameric transi-tions as well as those that are uncoupled. This resultsin the slowing down of the relaxation of conformationaland configurational transitions shown by curve b inFigure 2 and curve ii in Figure 3. This is essentiallydue to the elimination of some pathways of relaxationfrom the trajectory which otherwise would lead to fasterdecay of correlations.Operating on the trajectory with Q2 eliminates all

librations of bond lengths and angles and thereforeexcludes the possible localization effects that wouldresult from the concerted motions of these degrees offreedom. Consequently, any change in a torsional angleof a given bond has to move the remaining parts of thechain as a rigid body, which induces large displacementsand leads to a rapid decay of the OACF shown in Figure3, curve iii.Likewise, applying the operator Q3 on the trajectory

eliminates all localization effects resulting from libra-tions in torsion angles as well as from those in bondangles and bond lengths. In the absence of suchcooperative fluctuations, a jump in a torsion angle fromone isomeric minimum to another would induce large-scale displacements of the tails flanking the rotatingbond. Such large changes in orientations result in the

rapid decay of the OACF shown in curve iv. On theother hand, removal of librations by the application ofQ3 does not accelerate the loss of conformational cor-relations: A slowing down in the decay of the CACF(Figure 2c) results because of the removal of the rapidpathways of relaxation. It should be noted that CACFs,and in particular that of Hall and Helfand, have beencommonly used in literature for interpreting or fittingthe time decay of OACFs, without a rational justifica-tion. The present study clearly demonstrates that theresponses of OACFs and CACFs to librations differ fromeach other, and consequently, the use of the samefunctional form to express both correlation functionsmay not be adequate.It is to be noted that the replacement of the original

librations with uncorrelated ones results in OACFcurves (curves v and vi in Figure 3) that are significantlybelow the OACF curve of the original trajectory. In fact,the new OACF curves lie very close to the correspondingcurves iii and iv obtained in the absence of librations.It may thus be concluded that random librations arefar from representing the effects of the coordinatedlibrations and may not be adopted as appropriatemechanisms of relaxation in the real motion of macro-molecules.

Acknowledgment. Partial support by Bogazici Uni-versity Research Funds Project 95P0056 and by theTurkish Academy of Sciences is gratefully acknowledgedby B.E. and I.B.

References and Notes

(1) Baysal, C.; Atilgan, A. R.; Erman, B.; Bahar, I. J. Chem. Soc.,Faraday Trans. 1995, 91, 2483.

(2) Bahar, I.; Erman, B.; Monnerie, L. Adv. Polym. Sci. 1994,116, 145.

(3) Jones, A. A.; Stockmayer, W. H. J. Polym. Sci., Polym. Phys.Ed. 1977, 15, 847.

(4) Bendler, J. T.; Yaris, R. Macromolecules 1978, 11, 650.(5) Hall, C.; Helfand, E. J. Chem. Phys. 1982, 77, 3275.(6) Dejean de la Batie, R.; Laupretre, F.; Monnerie, L. Macro-

molecules 1989, 22, 122.(7) Howarth, O. W. J. Chem. Soc., Faraday Trans. 2 1978, 74,

1031.(8) Cook, R.; Helfand, E. J. Chem. Phys. 1985, 82, 1599.(9) Moro, G. J. Chem. Phys. Lett. 1990, 173, 903.(10) Kim, E.-G.; Mattice, W. L. J. Chem. Phys. 1994, 101, 6242.(11) Bahar, I.; Erman, B. Macromolecules 1987, 20, 1369.(12) Haliloglu, T.; Bahar, I.; Erman, B. J. Chem. Phys. 1992, 97,

4428.(13) Baysal, C.; Erman, B.; Bahar, J. Macromolecules 1994, 27,

3650.(14) Dauber-Osguthorpe, P.; Osguthorpe, D. J. J. Comput. Chem.

1993, 14, 1259.(15) For avoiding the redundant counting of isomeric transitions

and eliminating the rotational jumps which are back-restoredwithout stabilization of an isomeric state, the assignment ofdihedral angles were performed using the ranges |æi| < 30°for the trans and 90° < |æi| < 135° for gauche (() states. Therest of the dihedral angles are assigned their last definitestates.

(16) Helfand, E.; Wasserman, Z. R.; Weber, T. A. J. Chem. Phys.1981, 75, 4441.

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