Stress relaxation in a diatomic liquidR. C. Picu and J. H. Weiner Citation: The Journal of Chemical Physics 108, 4984 (1998); doi: 10.1063/1.475907 View online: http://dx.doi.org/10.1063/1.475907 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/108/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Relaxation of Voronoi shells in hydrated molecular ionic liquids J. Chem. Phys. 131, 174509 (2009); 10.1063/1.3256003 Infrared spectral profiles in liquids and atom-diatom interactions J. Chem. Phys. 121, 6353 (2004); 10.1063/1.1789937 Shear stress relaxation in liquids J. Chem. Phys. 120, 10188 (2004); 10.1063/1.1735628 Zero-shear stress relaxation and long time dynamics of a linear polyethylene melt: A test of Rouse theory J. Chem. Phys. 114, 8685 (2001); 10.1063/1.1368135 Structural changes during stress relaxation in simple liquids J. Chem. Phys. 107, 7214 (1997); 10.1063/1.474962

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Stress relaxation in a diatomic liquidR. C. Picu and J. H. WeinerDivision of Engineering, Brown University, Providence, Rhode Island 02912

~Received 20 October 1997; accepted 17 December 1997!

By the use of nonequilibrium molecular dynamics we have studied the stress relaxation followingimposition of a constant-volume elongation in thex1 direction on a model diatomic liquid. Threeconsecutive modes of relaxation of the stress differencet5t112

12(t221t33) are found, each

governed by exponentialse2a i t with a1.a2.a3 . Each mode is shown to correspond to the returnto isotropy of a different characteristic of the liquid structure that has been rendered anisotropic bythe deformation, namely~1! r (u), the mean distance from a generic atom of interacting atoms in theu direction, with the angleu measured from the stretch axisx1 ; ~2! N(u), the mean number densityof interacting atoms in theu direction; and~3! ^^P2(ub)&&, a measure of the orientationub ofmolecular axes with respect tox1 . The first two modes are identical in form to those studiedpreviously@R. C. Picu and J. H. Weiner, J. Chem. Phys.107, 7214~1997!# for stress relaxation ina monatomic liquid, but their rates of decay differ because of differences in packing and in atomicmobility. During the third mode of relaxation it is found thatt5C^^P2(ub)&&, whereC may beunderstood in terms of the intrinsic stress tensor, a tensor referred to a coordinate system fixed to themolecule@J. Gao and J. H. Weiner, J. Chem. Phys.90, 6749~1989!#. The relevance of these resultsto stress relaxation in polymer melts is discussed. ©1998 American Institute of Physics.@S0021-9606~98!50612-5#

I. INTRODUCTION

We are concerned in this paper with the structuralchanges that accompany stress relaxation in a diatomic liq-uid. This study is a sequel to a recent paper1 ~hereafter re-ferred to as I! that dealt with the same phenomena in simpleliquids.

The motivation for both these studies arises from thedesire for increased understanding of the process of stressrelaxation in polymer melts. Most treatments of viscoelastic-ity in polymeric systems are on the molecular level, withchains or portions of chains as the basic entities; these areregarded as entropic springs in tension and the anisotropic ordeviatoric stress in these systems is ascribed to these entropicforces. Interactions between nonbonded atoms are assumedto give rise only to isotropic or hydrostatic stresses and areneglected in such molecular treatments.

We have been engaged in a program to develop a physi-cal picture of this process in polymer melts on the atomiclevel, that is treating interactions between all pairs of atoms,bonded and nonbonded, on an equal footing.2 The principaltool in these investigations has been the computer simulationof model systems consisting of dense collections of long-chain molecules. In its most idealized form, the chain modelemployed has been of the ‘‘pearl necklace’’-type, with par-ticles representing atoms linked into chains by stiff linearsprings representing covalent bonds; nonbonded atoms inter-act with a short-range repulsive potential.3 It was found3,4

that at early times, the stress relaxation in this model poly-mer system was quantitatively very similar to that in thecorresponding simple liquid formed by eliminating all cova-lent bonds; it was this period of stress relaxation in melts thatmotivated our simple liquid study in I. Our simulations of

this model polymer system also showed that, following thissimple liquidlike period, the stress difference in the melt be-came proportional to the usual measure of bond orientation.5

It is the purpose of the present study of the diatomic liquid togain understanding on the atomic level of this result.

A description of the model and the NEMD algorithmemployed in the study are presented in Sec. II, stress relax-ation histories and accompanying structural changes resultsare detailed in Sec. III. The concept of intrinsic stress is usedin Sec. IV to clarify the role of bond orientation in the finalstage of stress relaxation; conclusions are summarized inSec. V.

II. MODEL DESCRIPTION AND SIMULATIONPROCEDURE

We employ the same model of a diatomic liquid as usedpreviously6 in simulation studies of equilibrium equations ofstate. The covalent bond between the pair of atoms of eachmolecule is represented by the potential

ub~r !5 12k~r 2b!2 ~1!

and all nonbonded atoms interact with a truncated Lennard-Jones potential,

unb~r !5H 4eLJS S sLJ

r D 12

2S sLJ

r D 6D r<Rc

unb~Rc! r .Rc

, ~2!

wherer denotes the distance between any pair of atoms andb is the bond length. In the present simulations,Rc

521/6sLJ so that only the repulsive part of the potential isconsidered. The units of length, energy and time of the prob-

JOURNAL OF CHEMICAL PHYSICS VOLUME 108, NUMBER 12 22 MARCH 1998

49840021-9606/98/108(12)/4984/8/$15.00 © 1998 American Institute of Physics

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lem are, respectively,sLJ , eLJ and sLJAm/eLJ, wherem isthe atomic mass. The units employed for stress areeLJ /sLJ

3 .The simulations are carried out using periodic boundary

conditions as customary in molecular dynamics simulations.The basic cell referred to a Cartesian system is, in the equi-librium state, a cube of dimensionsL. There areN atoms perunit cell which leads to a reduced density

r* 5NsLJ

3

L3 . ~3!

For all calculations reported here,N5632 atoms~316 mol-ecules! and the size of the unit cell is adjusted according tothe imposed density.

To maintain a constant temperature in the system duringstress relaxation, the velocity scaling algorithm due to Ber-endsenet al.7 is used. The position of a particle at time (t1Dt) is obtained from its position at timet and that at (t2Dt) by

x~ t1Dt !5~11j!x~ t !2jx~ t2Dt !1jf

mDt, ~4!

where the forcef acting on the atom in question is deter-mined from the atomic positions at timet. The scaling factorj is given by the equation

j5F11Dt

V S T0

T~ t2Dt/2!21D G1/2

, ~5!

whereT0 is the target temperature andT(t2Dt/2) is calcu-lated based on the velocities at time (t2Dt/2) obtained from

v~ t2Dt/2!5x~ t !2x~ t2Dt !

Dt. ~6!

The parameterV controls the speed of response of the algo-rithm to a perturbation in temperature. For our simulationsperformed under high strain rate–high energy input condi-tions, this parameter is taken asV510Dt. For this value itwas verified that the temperature at the onset of the relax-ation is the target temperature and remains so through theremaining relaxation. The time step of integrationDt is keptconstant for the whole loading-relaxation history and equalto 0.001. It was verified that further decrease inDt did notaffect the results.

The stress in the system is calculated using the virialstress formula8

vt i j 52NkTd i j 1 (aPb

^r a21ub8~r a!r a i r a j&

1 (aPnb

^r a21unb8 ~r a!r a i r a j&, ~7!

wherev5L3 is the volume of the unit cell,t i j are the com-ponents of the stress tensor referred to the fixed Cartesiancoordinate system of the cell,r a is the length of the vectorra

between a paira of particles and has componentsr a i , andu85du/dr. The sums are taken over all pairs of atoms~bonded,aPb, or nonbonded,aPnb! with at least oneatom in the basic cell. The brackets indicate an ensembleaverage taken over simulations with independent initial con-

ditions as discussed further in the following. The first term inEq. ~7! represents the kinetic contribution to the stress whichis assumed to contribute only to the hydrostatic stress. Thisassumption is based on the observation that the thermal ve-locities are much higher than that corresponding to the de-formation of the liquid and therefore the velocity anisotropyinduced by deformation can be neglected. The last two termsrepresent the potential contributions due to bonded and non-bonded interactions, respectively.

During the loading period we impose a volume-preserving elongational deformation of the unit cell duringwhich the periodic boundary conditions remain unchanged.For such a deformation, with the stretch directionx1 , the cellsize is modified according to

L15L~11 et !,

L25L/~11 et !1/2, ~8!

L35L/~11 et !1/2,

where e is the deformation strain rate. All simulations re-ported here are performed with a strain ratee50.1 and thetotal deformation of the cell in the stretch direction is 10%.Therefore, for a time step of 0.001, the deformation of Eq.~8! is imposed on the cell for 1000 steps after which thestrain rate drops sharply to zero and stress relaxation begins.We are concerned here primarily with the deviatoric stress orequivalently, with the stress differencet, where

t5t11212~ t221t33!. ~9!

Substitutingt i j from the virial formula, Eq.~7!, the differ-ence stress becomes

vt5v~tb1tnb!5 (aPb

^r aub8~r a!P2~uab !&

1 (aPnb

^r aunb8 ~r a!P2~ua!&, ~10!

whereP2(ua)5 12(3 cos2 ua21) is the second Legendre poly-

nomial, ua is the angle made by the position vectorra be-tween a pair of nonbonded atoms with the stretch axisx1 andr a5urau. For aPb, we use the notationP2(ua

b), whereuab

is the angle between the molecule bonda and axisx1 .Therefore, the fluid deformation is imposed simply by

the deformation of the unit cell and is transmitted inside thecell through the periodic boundary conditions imposed onthe system. Further details are presented in I.

Due to the small size of the system, for a random initialconfiguration, the stresst is nonzero in the unloaded state.Therefore, the calculation has to be repeatedNc number oftimes using independent initial conditions and the resultshave to be averaged over all runs. The initial conditions~po-sitions and velocities! are obtained by storing these data atthe end of the equilibration of the previous run. For the firstrun, we start from an FCC-type distribution of atoms in thecell and with a random velocity field and distribution of bondorientations. Moreover, for symmetry purposes, two addi-tional independent configurations are obtained for each runby simply permuting the axesxi and therefore replacing thestretch directionx1 with x2 andx3 , respectively. We refer to

4985J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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the time-history of atomic positions for a given set of initialpositions and velocities as a trajectory. The results reportedhere are obtained by averaging overNc52700 trajectories.The high number of iterations was necessary to reduce thenoise in the measured quantities.

III. STRESS RELAXATION HISTORY

For most of the discussions we confine attention to theunloading period, that is beginning att51 with the drop ofthe strain rate to zero. The parameter values for the simula-tion reported are summarized in Table I. The choice of di-atomic molecule withb5sLJ is motivated by the use ofthese values in the ‘‘pearl necklace’’ chain model frequentlyemployed in model polymer system simulations.

The stress relaxation results are shown in two differentrepresentations in Figs. 1 and 2. The first representation~solid line in Fig. 1! follows the typical method of plottingstress relaxation in polymer melts in a log–log plot. Thesecond representation, Fig. 2, shows the variation of log(t)with time t and shows clearly three successive modes ofexponential decaye2a i t, with a1512.5, a250.63, anda3

50.49. As we will discuss in greater detail in the following,the first two modes are accompanied by structural changes of

the same type as studied in I for a simple monatomic liquid,while the third mode is related to the bond orientation decay.

A. Simple liquid modes

The existence of an anisotropic or nonzero deviatoricstress at the end of the loading period must be accompaniedby an anisotropic radial distribution functiong(r ) and therelaxation of this stress is accompanied by the relaxation ofg(r ) back to isotropy. During the first two modes of stressrelaxation we find that the change ing(r ) is well describedby the use of the same formalism and anisotropy measuresemployed in I for the monatomic liquid. That is, if we con-fine ourselves for the diatomic liquid to the distribution ofnonbonded atoms in the vicinity of a generic atom, the func-tion gnb(r ) has the same appearance and qualitative behavioras observed in I for the monatomic liquid, when described inthe fixed laboratory frame. In this frame, the bonded atomoccurs in various orientations relative to the generic atomand its effect, therefore, is not apparent. However, as weshall see in what follows, the bonded atom has a large effectupongnb(r ) when the latter is described in a coordinate sys-tem fixed in the molecule.

We use the cylindrical symmetry of the extensional de-formation about x1 , the stretch axis, and writegnb(r )5gnb(r ,u), wherer 5ur u and u is the angle betweenr andx1 . We refer to the distributiongnb(r ,u), 0,r ,Rc , as the‘‘cage’’ surrounding any atom and characterizegnb by thetwo anisotropy measuresr (u) and N(u) defined in I. Thereit is seen thatr (u) is the mean radius of nonbonded atoms inthe cage, andN(u) is proportional to the number density ofnonbonded atoms in the cage, both at angleu. Figure 3shows the variation of these two measures with time foru50, in the stretch direction, and foru5p/2, in a directionperpendicular to the stretch direction. As discussed at lengthin I, during loading, the equilibrium spherical shape of thecage becomes, as described by the functionr (u), ellipsoidalwith the long semiaxis in the stretch direction. Similarly,

TABLE I. Parameter values for the diatomic liquid simulation.

Density r* 50.9Temperature kT/eLJ51.0Potential cut-off radius Rc521/651.13Bond length b5sLJ

Bond stiffness kb2/kT5100Strain rate e50.1Total strain e510%Time step Dt50.001

FIG. 1. Stress relaxation history in diatomic liquid in log–log representation~solid curve!. Dashed curve shows stress relaxation in simple liquid of samereduced density,r* 50.9, with simple liquid scaled to agree with diatomicliquid at t51. As discussed in text and shown in Fig. 2, Modes I, II, and IIIrefer to periods in which different structural characteristics of the diatomicliquid return to isotropy, namely I:r (u), II: N(u) and III: ^^P2(ub)&&.

FIG. 2. Stress relaxation history in diatomic liquid in semilog representation~solid curve!. Dashed lines show exponential fits to curve applicable toModes I, II, and III. Noise in stress history seen in Fig. 1 has been smoothedby use of a time-domain filter~Ref. 9! with a window of 0.3.

4986 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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under loading, the number densityN(u) in the stretch direc-tion becomes smaller than the normalized~as in I! equilib-rium value of 1, with the opposite trend in theu5p/2 direc-tion. During relaxation, here, as in the simple liquid case, itis possible to identify the first stress relaxation mode with theisotropization ofr (u); both are very fast processes whichoccur over a time intervalDt1;0.1 @Figs. 2 and 3~a!#. Thesecond stress relaxation mode is associated with the return ofN(u) to isotropy through a slower, diffusion-dominated pro-cess. This is evidenced by the good agreement between theexponential fit to the relaxation part ofN(u) with an expo-nent of 0.67, and the exponent characterizing the second re-laxation mode of the stress in Fig. 2,a250.63.

Further comparison with the simple liquid case is madein Fig. 1 where the broken line corresponds to the stressrelaxation in a simple fluid of same density and at the sametemperature as the diatomic liquid under consideration. Thecurve was scaled such that the stress at timet51 in themonatomic and diatomic systems coincide. The first mode of

stress relaxation of the simple liquid occurs at a slightlyslower rate than in the diatomic liquid or conversely, thediatomic liquid behaves in the early relaxation times as alower density simple fluid. Since, as we showed in I, thestress relaxation in the first mode is due to a fast local rear-rangement of atoms in the cage, the increased rate in thediatomic case is apparently related to the uneven packing ofatoms in it and consequent larger open regions found there.For the second relaxation mode a different situation exists,with the simple liquid relaxing faster than the diatomic liq-uid. This is due to the reduced mobility of atoms in thediatomic case which implies slower diffusion and a reducedrelaxation rate during this mode.

It is also of interest to consider the individual contribu-tions to t made by the bonded and nonbonded interactions.These are shown in Fig. 4. Note that the bonded contribution,tb , is negative. This is also found to be the case in polymermelt simulations,10 and is a consequence of the fact that thebonds are in compression in both cases.

B. Bond orientation decay

Even after the values ofr (u) andN(u) have decayed bythe time t;3.5 to values corresponding to isotropy of thecage, it is clear that neithergb(r ) andgnb(r ) is in fact per-fectly isotropic sincet(t), tb(t) and tnb(t) are all not yetzero ~Fig. 4!. The fact that these stresses are still nonzeroafter the simple liquid characteristicsr (u) and N(u) havedecayed to isotropy is due to the persistence of anisotropy inthe bond distribution. As a measure for bond orientation, wecompute the quantity^P2(ub)&&, whereub is the angle be-tween the molecule bond and stretch axisx1 , and where thedouble brackets denote averaging over the ensemble of inde-pendent trajectories and over all the molecules. The time-dependence of^P2(ub)&& during stress relaxation is shown

FIG. 3. ~a! Behavior ofr (u), mean radius of nonbonded atoms interactingwith generic atom, in sectors of widthDu5p/18 centered atu50 andp/2.~b! Behavior ofN(u), normalized measure of the number density of non-bonded atoms interacting with generic atom, in same sectors as defined in~a!.

FIG. 4. Decomposition oft5tb1tnb wheretnb is due nonbonded interac-tions andtb is due to bonded interactions.

4987J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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in Fig. 5 and it is seen that it is well characterized by anexponential decaye2bt, with b50.51, i.e.,b'a3 , so thatin this simulation one can write the equation

t5C^^P2~ub!&& ~11!

with C52.0, relating stress and bond orientation with goodaccuracy throughout the final mode of stress relaxation.

IV. INTRINSIC STRESSES

In order to understand the physical basis of Eq.~11!, weturn to the concept of intrinsic stress, introduced initially totreat dense polymeric systems11 and adapted here to the caseof a diatomic liquid. For this purpose, we change the nota-tion of Eq. ~7! slightly in that we provide an index for eachatom of an interacting pair and condense the bonded and thenonbonded contributions in one term, writing

vt i j 52NkTd i j 11

2 (a51

N

(b~a!

^r ab21uab8 ~r ab!r ab i r ab j&,

~12!

where the sum overb~a! is over allb atoms interacting withatoma anduab denotes the respective potential,ub or unb .For each atoma, for a particular timet and trajectory, wedefine

s i j ~a!51

2 (b~a!

r ab21uab8 ~r ab!r ab i r ab j ~13!

so thats i j (a) is the contribution that atoma makes tovt i j attime t. Therefore, Eq.~12! becomes

t i j 52rkTd i j 11

v (a51

N

^s i j ~a!&, ~14!

wherer5N/v is the atom number density, or

t i j 52rkTd i j 1r^^s i j &&, ~15!

where, as before, the double brackets indicate an averageover all atoms as well as over all trajectories.

The above discussion has all been with respect to thefixed laboratory coordinate systemxi , wherex1 is the stretch

direction. We next introduce an intrinsic coordinate systemxr for each atom, with the intrinsic systems for the twobonded atoms of a molecule related as shown in Fig. 6. Letei

be the unit base vectors of thexi system, andar(a) thecorresponding vectors of thexr system. We can now expressthe tensors rs(a) with respect to the intrinsic systemxr interms ofs i j (a) as

s rs~a!5s i j ~a!ari ~a!as j~a!, ~16!

whereari (a)5ar(a)–ei and with the usual summation con-vention on repeated indices,i ,r 51,2,3 employed. It followsfrom Eqs.~13! and ~16! that we can also write

s rs~a!51

2 (b~a!

r ab21uab8 ~r ab! r abr~a! r abs~a! ~17!

where r abr(a) are the components ofrab with respect toxr(a).

The quantities ^s rs&& have been calculated both for theliquid in equilibrium and when it undergoes loading andstress relaxation. It is found that^^s rs&& is a cylindrical ten-sor, that is the only nonzero components are^^s11&& and^^s22&&5^^s33&&. The results are shown in Fig. 7 where it isseen that, except for a small variation during the loadingperiod, the components of^^s rs&& are constant during non-equilibrium and equal to the values obtained from an equi-librium simulation. We refer therefore, to s rs&& as the in-trinsic stress tensor associated with each molecule. Theseparate contributions ofub andunb to this tensor are listedin Table II. We note, parenthetically, that~a! the averagevalue of the force in the bonds corresponds to compression,~b! the components ofs rs and s i j satisfy the relations rr (a)5s j j (a) identically, and ~c! the relation f b1 f nb

52kT/b, where, for a particular atom,f b is the time averageof the bond force,f nb is the time average of the projection ofthe nonbonded forces acting on that atom in the bond direc-tion, is found to be satisfied.6

A. Steric hindrance

In order to understand further the existence of the intrin-sic stress tensor s rs&&, we consider the distribution of in-teracting nonbonded atoms about a given molecule whenviewed from the intrinsic coordinate system associated with

FIG. 5. Time-dependent behavior of^^P2(ub)&&, measure of orientation ofmolecular axes~solid curve!. Dashed line corresponding toe2bt is nearlycoincident with the solid line.

FIG. 6. Intrinsic coordinate system,xr , for individual molecules.

4988 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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that molecule. For comparison, Fig. 8 shows this distributionnormalized by the equivalent distribution determined in asimilar equilibrium simulation of a simple liquid. The non-uniform character of the distribution is seen to be a conse-quence of steric hindrance and gives rise to the intrinsicstress tensor, whose components are constant when referredto the intrinsic coordinate system as shown above.

B. Inversion problem

We turn next to the important problem of the expressionof t i j , the stress tensor referred to the laboratory frame, interms of^^s rs&&. For this purpose it is convenient to intro-duceDti j , the deviatoric part oft i j , defined as

Dti j 5t i j 213d i j tkk ~18!

with analogous definitions forDs i j andDs rs . Then,Dti j 5r^^Ds i j &&5r^^Ds rsari as j&&. ~19!

We are primarily interested inDt11 which equals23t, wheret

is the stress difference defined in Eq.~9!. By use of theorthonormality relationsari ar j 5d i j and ar j as j5d rs , it canbe shown that

Dt115r@^^Ds1112~3a11

2 21!&&1^^~ s222s33!12~a21

2 2a312 !&&

12^^s12a11a211s13a11a311s23a21a31&&#

5I1II1III. ~20!

If Ds rs for every atom were independent ofub, the bondorientation with respect tox1 of the molecule to which itbelongs, then the quantitiesDs rs andari would be indepen-dent and we could write

^^Ds rsari as j&&5^^Ds rs&&^^ari as j&&. ~21!

It would follow from the fact that ^Ds rs&& is cylindrical,that the second and third terms of Eq.~20! ~terms II and III!vanish and that

Dt115r@^^Ds1112~3a11

2 21!&&#5r^^Ds11&&^^P2~ub!&&.~22!

Then, a simple relation would be established between theproportionality constantC of Eq. ~11! and ^^Ds11&&.

It is necessary at this point to consider separately thecontributions ofub andunb to s rs and write

s rs5s rsb 1s rs

nb . ~23!

It is clear thats rsb (a) is a cylindrical tensor withs 11

b (a) theonly nonzero component and that furthermore its value isindependent ofua

b . We then conclude from Eq.~22! that

tb5 32Dt11

b 5 32r^^Ds 11

b &&^^P2~ub!&&. ~24!

However, as we shall show, it is not the case thats rsnb(a) is

independent ofuab or, equivalently, ofari (a). We have

tested the independence of the quantitiesDs rsnb and ari by

direct calculation of all three terms in Eq.~20! during thestress relaxation simulation. The results are shown in Fig. 9.The nonvanishing of terms II and III is a direct proof of the

FIG. 7. Intrinsic stress components^^s rr && ~no sum onr ! during loading,unloading and in equilibrium.

TABLE II. Values of the blonded and nonbonded contributions to the in-trinsic stress tensor.

^^s 11b &&522.0 ^^s 11

nb&&524.75^^s 22

b &&5^^s 33b &&50. ^^s 22

nb&&5^^s 33nb&&527.25

^^Ds 11b &&521.33 ^^Ds 11

nb&&51.66

FIG. 8. Normalized number density distribution,N( u)/Nsl( u), of non-bonded atoms about a representative atom in intrinsic coordinates. The nor-malization is made with the similar distribution computed for the simpleliquid system of same density and temperature.~a! Cartesian representation.~b! Polar representation. Light curves representN( u)50 as well as ‘‘hardspheres’’ of atoms. Distribution is cylindrically symmetric aboutx1 axis.

4989J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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nonindependence of these quantities upon bond orientation.In turn, this result implies that the interaction of a moleculewith its neighborhood, even when viewed in the frameworkof a coordinate system intrinsic to that molecule, depends tosome extent on the orientation of the molecule with regard tothe stretch axis. It is possible to gain some understanding ofthis dependence by consideration of the distribution of thebond orientation in the deformed liquid, Fig. 10. We see, forexample, that a molecule oriented close to the stretch direc-tion will have more like-oriented molecules to interact withthan one oriented perpendicular to that direction.

Although terms II and III of Eq.~20! are not zero fors rs

nb , their time dependence of decay, as observed in thesimulation, is proportional to that of^^P2(ub)&&. Because ofthese two additional terms, we find that

tnb5 32Dt11

nb5 32Dr^^Ds 11

nb&&^^P2~ub!&& ~25!

with the factorD51.7 determined empirically from the non-equilibrium simulation. Apart from this factor, the otherquantities^^Ds 11

b && and ^^Ds 11nb&& may be determined from

equilibrium simulations, Table II. By use of these factors, wefind from Eqs.~24! and ~25! that

C5 32r@^^Ds 11

b &&1D^^Ds 11nb&&#52.0 ~26!

in good agreement with the nonequilibrium results found inthe third mode of relaxation.

V. CONCLUSIONS

Our simulations of a diatomic liquid show that at earlytimes the manner of stress relaxation and the accompanyingstructural changes parallel completely those observed1 insimulations of a simple monatomic liquid, although the timescales at which these processes occur are slightly different inthe two systems. A new phenomenon then appears in thediatomic liquid. After the simple liquid-type relaxation, fol-lows a period in which the dominant structural change is thedecay of the quantity^P2(ub)&&, a measure of the anisot-ropy of the molecular bond orientation. In this last period,the difference stress obeys the relationt5C^^P2(ub)&&, andthe molecular mechanism underlying this relation may beunderstood in terms of the intrinsic stress tensor. We maythen write

t5tsl1tor , ~27!

where the decay to zero oftsl(t) accompanies the return toisotropy ofgnb(r ) as observed in the fixed laboratory frameby the same mechanisms as in the simple liquid, whiletor

5C^^P2(ub)&& is the portion of the stress due to the orien-tation of the bonds induced by the imposed deformation.Since the relaxation oftsl is more rapid thantor , this givesrise to the proportionality relation between the total stressand ^^P2(ub)&& that is observed at later times. The closerelation between the stress autocorrelation function and theorientation autocorrelation function has been observed andstudied by Luo and Hoheisel13 and by Hounkonnouet al.14

in their simulations of diatomic liquids. The present workshows that the intrinsic stress concept provides a useful toolfor understanding this relation.

The present simulation of a diatomic liquid has beenundertaken in order to increase our understanding of thestress relaxation in polymer melts. Stress relaxation simula-tions of a melt employing a ‘‘pearl necklace’’ chain modelwith chains havingN5200 bonds and withN5300 bondshave been performed.5,12 Two versions of the ‘‘pearl neck-lace’’ model must be distinguished here. The first representsa coarse-grained version of the real chain with the bondsbetween beads regarded as soft entropic springs exerting atemperature dependent attractive force.15 The second is anidealized atomic-level model of a chain with the beads rep-resenting single monomers connected by covalent bonds, thelatter modeled by stiff linear springs, using the potentialub(r ) of Eq. ~1! as in the present diatomic model. In ourstudies,5,12 we used the second, idealized atomic-levelmodel, since it permits a systematic analysis of different de-grees of coarse graining with results that, we hope, provide

FIG. 9. Time-dependent behavior of deviatoric stress componentDt11nb and

of the individual terms I, II, and III of the sum in Eq.~20!. Points shown onthe curve forDt11

nb correspond toD3Term I, with D51.7.

FIG. 10. Distribution of bond orientation att54. Distribution normalized sothat equilibrium distribution corresponds tof (ub)[1.

4990 J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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insight into the coarse graining of more realistic atomic-levelchain models. The melt simulations show proportionality be-tweent and^^P2(ub)&& at later times, behavior that is quitesimilar to that found in the diatomic case, although the timescales are quite different. The long-time limiting value forthe ratio ^^P2(ub)&&/t for the melt was found to beCpol

;4, while in the diatomic case we obtainedC52.0. How-ever, whereas the limiting behavior for the diatomic liquidwas reached att;3.5, in the melt it was not reached untilt;400. The melt result is particularly surprising since it im-plies that after the simple liquid-type of behavior relaxes, thedifference stress in the melt becomes proportional to the ori-entation of individual bonds not, as in the usual view, to theorientation of the end-to-end vectors of chain segmentswhere each segment has a sufficient number of bonds to beregarded as a Gaussian entropic spring. Adding to thecounter-intuitive character of the result is the previouslynoted fact that, at the simulated liquidlike densities, the co-valent bonds of our atomic-level chain model~and of thediatomic molecules! are in compression.16 The present treat-ment of the diatomic liquid in terms of the concept of intrin-sic stresses serves to make the melt behavior more under-standable.

ACKNOWLEDGMENTS

This work was supported by the Gas Research Institute~Contract No. 5091-260-2237!

1R. C. Picu and J. H. Weiner, J. Chem. Phys.107, 7214~1997!.2J. Gao and J. H. Weiner, Science266, 748 ~1994!.3J. Gao and J. H. Weiner, Macromolecules25, 1348~1992!.4M. Kroger, C. Luap and R. Muller, Macromolecules30, 526 ~1997!.5G. Loriot and J. H. Weiner, J. Polym. Sci.: Phys.36, 143 ~1998!.6J. Gao and J. H. Weiner, Mol. Phys.70, 299 ~1990!.7H. Berendsen, J. Postma, and W. van Gunsteren, J. Chem. Phys.81, 3684~1984!.

8R. J. Swenson, Am. J. Phys.55, 746 ~1987!.9W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery,Numerical Recipes in Fortran~Cambridge University Press, Cambridge,1992!, p. 644.

10J. Gao and J. H. Weiner, Macromolecules24, 1519~1991!.11J. Gao and J. H. Weiner, J. Chem. Phys.90, 6749~1989!.12J. Gao and J. H. Weiner, Macromolecules29, 6048~1996!.13H. Luo and C. Hoheisel, J. Chem. Phys.94, 8378~1991!.14M. N. Hounkonnou, C. Pierleoni, and J. P. Ryckaert, J. Chem. Phys.97,

9335 ~1992!.15See, for example, M. Doi and S. F. Edwards,The Theory of Polymer

Dynamics~Clarendon, Oxford, 1986!, p. 15, Eq.~2.40!.16The covalent bonds in melts employing more realistic atomic-level chain

models are also found to be in compression; see J. Gao and J. H. Weiner,J. Chem. Phys.98, 8256~1993!.

4991J. Chem. Phys., Vol. 108, No. 12, 22 March 1998 R. C. Picu and J. H. Weiner

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