Dynamics and Rheology of a Supercooled Polymer Melt in Shear Flow

a r X i v :c o n d -m a t /0203207v 1 [c o n d -m a t .s o f t ] 9 M a r 2002

Dynamics and Rheology of a Supercooled Polymer Melt in Shear Flow

Ryoichi Yamamoto and Akira Onuki

Department of Physics,Kyoto University,Kyoto 606-8502,Japan

(February 1,2008)

Using molecular dynamics simulations,we study dynamics of a model polymer melt composed of short chains with bead number N =10in supercooled states.In quiescent conditions,the stress relaxation function G (t )is calculated,which exhibits a stretched exponential relaxation on the time scale of the αrelaxation time ταand ultimately follows the Rouse dynamics characterized by the time τR ～N 2τα.After application of shear ˙γ,transient stress growth σxy (t )/˙γ?rst obeys the linear

growth t 0dt ′G (t ′

)for strain less than 0.1but saturates into a non-Newtonian viscosity for larger strain.In steady states,shear-thinning and elongation of chains into ellipsoidal shapes take place

for shear ˙γlarger than τ?1

R .In such strong shear,we ?nd that the chains undergo random tumbling motion taking stretched and compact shapes alternatively.We examine the validity of the stress-optical relation between the anisotropic parts of the stress tensor and the dielectric tensor,which are violated in transient states due to the presence of a large glassy component of the stress.We furthermore introduce time-correlation functions in shear to calculate the shear-dependent relaxation times,τα(T,˙γ)and τR (T,˙γ),which decrease nonlinearly as functions of ˙γin the shear-thinning regime.

PACS numbers:83.10.Nn,83.20.Jp,83.50.By,64.70.Pf

I.INTRODUCTION

The dynamics and rheology of glassy polymers are

known to be very complicated and are still not well un-derstood.We summarize salient features of such systems below.

First,in the linear response regime,thermal relax-ations of the chain conformations occur from microscopic to macroscopic time scales,as revealed in measurements of stress and dielectric responses.1,2In a relatively early stage,the stress relaxation function G (t ),which describes linear response to small shear deformations,can be ?tted to the Kohlrausch-Williams-Watts (KWW)form,

G G (t )=G 0exp[?(t/τs )c ],

(1.1)

after a microscopic transient time t tra .The time τs (?τtra )is of the order of the structural αrelaxation time τα(to be de?ned in (3.8)below),which grows dramati-cally as the temperature T is lowered towards the glass transition temperature T g .When the time t consider-ably exceeds τs ,the relaxation of the chain conforma-tions is relevant and is well described by the Rouse or reptation dynamics,depending on N N e ,respectively.3Here N is the polymerization index and N e is that between entanglements on a chain.For short chain systems with N

G (t )=G G (t )+G R (t ).

(1.2)

The G R (t )is the stress time-correlation function in the Rouse model whose terminal relaxation time τR is of or-der N 2τα.For entangled chain systems with N >N e ,the KWW function in (1.1)is followed by the power-law decay,

G (t )～=e ?1G 0(t/τs )?ν

(1.3)

with ν～0.5until the rubbery plateau G (t )?G (0)

N is

reached,1,2where G (0)

N assumes the modulus nk B T/N e of entangled polymers with n being the bead number density.Ultimately,G (t )follows the reptation relax-ation G rep (t )on the time scale of a very long rep-tation time τrep .These hierarchical relaxations arise from rearrangements of jammed atomic con?gurations and subsequent evolution of chain conformations.They also give rise to the corresponding characteristic behav-iors in the frequency-dependent shear modulus G ?(ω)≡iω ∞

0dte ?iωt G (t ),depending on the frequency ωrelative to the inverse characteristic times introduced so far.1–3Second,in the nonlinear response regime,glassy ?u-ids generally exhibit highly viscous non-Newtonian ?ow close to (but above)T g even if they are low-molecular-weight ?uids.4In such ?uids,if ˙γ>τ?1

α,atomic rear-rangements are induced not by thermal agitations but by externally applied shear.5–7In chain systems with-out entanglements,on one hand,shear thinning occurs at su?ciently high (but sometimes unrealistically large)shear rates due to chain elongation.8–12In entangled polymers,on the other hand,shear thinning occurs at

very small shear larger than τ?1

rep ,where disentanglements

are induced by shear.3

Thus supercooled chain systems are most easily driven into a nonlinear response regime even by extremely small shear,though the crossover shear stress from linear to nonlinear regimes may not be very small.Furthermore,in glassy ?uids below T g ,plastic de-formations are often induced in the form of large-scale shear bands above a yield stress (corresponding to a few %strain).13,14It is of great importance to understand

how these nonlinear e?ects occur depending on˙γ,T,and N.

Third,in rheological experiments on polymers,use has been made of the stress-optical relation between the devi-atoric(anisotropic)parts of the dielectric tensor?αβ(at optical frequencies)and the average stress tensorσαβ.3,15 In shear?ow with mean velocity˙γy in the x direction,it is expressed as

?xy=C0σxy,?αα??ββ=C0(σαα?σββ),(1.4) where C0is called the stress-optical coe?cient.For melts this relation excellently holds at relatively high T(>T g) for general time-dependent nonlinear shear deformations. If measurements are made in steady states,it holds even close to T g.16For its validity,we need to require that the form contribution to?αβis negligible as compared to the intrinsic contribution15and that the glassy part of the stress is negligible as compared to the usual entropic part.Thus it is violated when the form part is relevant such as in polymer solutions close to the demixing critical point or when measurements are made in transient states close to T g.In the latter case,as is evident from the en-hancement ofτs in(1.1),the glassy part of the stress is dominant for relatively rapid deformations.16–19

While the predictive power of analytic theories in poly-mer science is still poor,computer simulations20–22can provide us a useful tool to investigate the microscopic origins of experimentally observed macroscopic phenom-ena.In quiescent states,di?usive motions in supercooled melts have been extensively studied using molecular dy-namics(MD)23–27and Monte Carlo28–30simulations.In another application,nonequilibrium molecular dynam-ics(NEMD)simulations have been useful to investigate chain deformations and rheology in?ow.8–12,17,31In par-ticular,Kr¨o ger et al.17studied the molecular mechanisms of the violations of stress-optical behavior for a melt con-sisting M=260chains with bead number N=30after application of elongational?ow.

In this paper,we will present results of very long MD simulations to study linear and nonlinear dynamics of a supercooled polymer melt in the absence and presence of shear?ow.Long simulation times are needed to calculate the terminal relaxation of G(t),which has not yet been undertaken in the literature.As a new?nding,we will show that each chain in our melt system is changing its orientation(tumbling)randomly in shear?https://www.360docs.net/doc/cd7725727.html,e will be made of techniques and concepts introduced in our previous papers on supercooled binary mixtures under shear?ow.5–7Some of our results were already published elsewhere.7,32

II.MODEL AND SIMULATION METHOD Our system is composed of M=100chains with N=10beads con?ned in a cubic box with length L=10σand volume V=L3=103σ3.The number density is?xed at n=NM/V=1/σ3,which results in severely jammed con?gurations at low T.All the bead particles interact with a truncated Lennard-Jones poten-tial de?ned by20

U LJ(r)=4? σr 6 +?(r<21/6σ).(2.1)

The right hand side is minimum at r=21/6σand the potential is truncated for larger r(U LJ(r)=0for r>21/6σ).By using the repulsive part of the Lennard-Jones potential only,we may prevent spatial overlap of the particles.20Consecutive beads on each chain are con-nected by an anharmonic spring of the form,

U F(r)=?

1

m

M

k=1N j=1p k jαp k jβ? all pairs U′LJ(ξ)ξαξβ

ξ

,(2.3)

where m is the mass of a bead,U′LJ(ξ)=dU LJ(ξ)/dξ, and U′F(ξ)=dU F(ξ)/dξ.Hereξ=(ξx,ξy,ξz)in the right hand side represents the relative vector R k j?R k′j′between the two beads,R k j and R k′j′,in the second term and the relative vector R k j?R k j+1between the two consecutive beads,R k j and R k j+1,of the same chain in the third term. To avoid cumbersome notation,we will write the bead positions simply as R j(j=1,···,N)suppressing the in-dex k.When they will appear in the statistical averages ··· ,the average over all the chains M k=1(···)/M will be implied even if not written explicitly.Furthermore, it is convenient here to introduce the usual notationσαβfor the stress tensor by

1

We will hereafter measure space and time in units of σandτ0≡(mσ2/?)1/2.The temperature T will be mea-sured in units of?/k B.The original units will also be used when confusion may occur.Our simulations cover nor-mal(T=1.0)and supercooled(T=0.2)states with and without shear?ow(˙γ=0,10?4,10?3,10?2,and10?1). Simulation data were taken after very long equilibration periods(～102τR?5×106at T=0.2)so that no ap-preciable aging(slow equilibration)e?ect was detected in the course of taking data in various quantities such as the pressure or the density time-correlation functions.(i)In quiescent cases,we impose the micro-canonical condition and integrate the Newton’s equations of motion,

d

m p j,

d

dt R j=

1

dt p j=f j?˙γp y

j

e x??ζp j,(2.6)

where e x is the unit vectors in the x(?ow)direction, R j=(X j,Y j,Z j),and p j=(p xj,p yj,p zj).The friction coe?cient?ζwas set equal to

?ζ= j(f j·p j?˙γp xj p yj) j p2j.(2.7) The temperature T(≡ j p2j/mNM)could then be kept at a desired value.The time increment was?t=0.0025. After an equilibration run in a quiescent state for t<0, we gave all the particles the average?ow velocity˙γY j e x at t=0and then imposed the Lee-Edwards boundary condition33,34to maintain the shear?ow.Steady sheared states were realized after transient relaxations.

III.DYNAMICS IN QUIESCENT STATES

Although it is highly nontrivial,it has been con?rmed by computer simulations20,23–25,28–30,35that the single-chain near-equilibrium dynamics in unentangled melts can be reasonably well described by(or mapped onto) the simple Rouse model.In the Rouse dynamics,the re-laxation time of the p-th mode of a chain is expressed in terms of a friction coe?cientζand a segment length b as36

τp=ζb2/[12k B T sin2(πp/2N)],(3.1) where1≤p≤N?1.The Rouse relaxation timeτR is the slowest relaxation time,

τR≡τ1～=N2ζb2/(3π2k B T).(3.2) The segment length b in the corresponding Rouse model may be related to the variance of the end-to-end vector of a chain P≡R N?R1in our microscopic model by

|P|2 =b2(N?1).(3.3) As a result,b is dependent on T but its dependence turns out to be weak as b=1.17,1.18,1.19for T=1.0,0.4, 0.2,respectively.Note that b is larger than the mini-mum distance b min～=0.96of the bond-potential.Let us consider the time-correlation function of P(t),

C(t)= P(t+t0)·P(t0) / |P|2 ,(3.4) which is normalized such that C(0)= 1.Here C(t) should be independent of the initial time t0in steady states in the limit of large system size.However,our system is not very large,so we took the average over the initial time t0.This statistical averaging will not be mentioned hereafter in showing our MD results of time-correlation functions.In the Rouse dynamics C(t)is cal-culated as

C R(t)=

2

2N e?t/τp,(3.5)

where the summation is over odd p but the?rst term (p=1)is dominant in the whole time region(so we may determineτR by C(τR)=e?1).As shown in Fig.1,our MD data of C(t)can be excellently?tted to C R(t).The τR thus determined increases drastically with lowering T asτR=250,1800,and6×104for T=1.0,0.4,and0.2, respectively.In the previous simulations on nonentangled polymer melts,24,25,28–30,35numerical results were con-sistent with the Rouse dynamics for small p(large-scale motions),but deviations are enhanced for large p(small-scale motions)in supercooled states.Furthermore,we here give the expression for the stress relaxation function in the Rouse model,

G R(t)=

nk B T

1

F q(t)=

M chain1b0b jβ

independent of T both in the linear (Q xy 0.05)and nonlinear (Q xy 0.05)regimes.

We next consider

anisotropy of chain conformations

in

shear ?ow.In Fig.7(a),we plot the x -y cross-section (z =0)of the steady state bead distribution function,

g s (r )=

1

N 2

N

i,j =1

exp[i q ·(R i ?R j ] ,(4.4)

which is proportional to the scattering intensity from la-beled chains in shear.37In these ?gures θis the relative

angle of the ellipses with respect to the y (shear gradient)direction.These ?gures demonstrate high elongation of

the chains for ˙γ>τ?1

R .As will be shown in Fig.8below,they almost saturate into the shapes shown in Fig.7once

˙γexceeds τ?1

R .

Let us de?ne the tensor,

I αβ=

1

N

N j =1

(R jα?R G α)(R jβ?R G β) .(4.5)

For small q =(q x ,q y ,0),S (q )is expanded as S (q )=1?

1

2

a 21(q

·e 1)2

?

1

˙γσxy (t )～=

t

dt ′G (t ′).

(5.1)

In the nonlinear regime,σxy (t )/˙γtends to the non-Newtonian viscosity η(˙γ).As a guide,we also display the

linear growth function t

0dt ′G R (t ′)in the Rouse model.In the very early time region 1?t τα,the growth (～=G 0t )is much larger than the Rouse initial growth (～=k B T t ).We can see a rounded peak for ˙γ

=0.1be-fore approach to the steady state in Fig.5.More pro-nounced overshoot was already reported at high shear in MD simulation of much longer unentangled alkane chains (C 100H 202).12

In experiments,16–19the stress-optical relation is tran-siently violated at low T after application of elongational ?ow due to the enhancement of the glassy component of the stress.For shear ?ow,Fig.10displays our MD results at T =0.2after application of shear at t =0in a stress-optical diagram,where the solid lines are the averages over ten independent runs.As time goes on,the system traces the curve of a given ˙γ,passes across the dashed curve representing the steady-state universal relation in Fig.6,and ?nally comes back to the dashed curve.The deviation from the steady-state curve be-comes larger with increasing ˙γ,as in the experiments of elongational ?ow.

VI.TIME-CORRELATION FUNCTIONS AND

TUMBLING IN SHEAR FLOW

In Fig.11,we show the end-to-end vector correlation functions C (t )= P (t )·P (0) / |P |2 with and with-out shear ?ow for various ˙γat T =0.2.For ˙γ=0,it rapidly decreases,negatively overshoots,and ?nally approaches zero with dumped oscillation superimposed.This oscillatory behavior arises from random rotation of chains in shear ?ow,which is well known in dilute poly-mer solutions 38–40but has not been reported in polymer melts.This is more evidently seen in Fig.12,where we show time development of the x component of the end-to-end vector P j =R N ?R 1of one chain for ˙γ=10?3(a),10?2(b),and 10?1(c)at T =0.2.The correspond-ing Weisenberg number (4.1)is given by W i =60,600,and 6000,respectively.In Fig.12(d),we show chain con-tours projected onto the x -y plane at the points 1～8indicated in Fig.12(c).When the chains change their ori-entation,their shapes are contracted as in the case of a

single chain in solution.39The average period of tumbling is about35/˙γin our case.

We may introduce the van Hove time-correlation func-tion(2.14)even in shear?ow if the particle displacement vector is rede?ned as6

?R j(t)=R j(t+t0)?R j(t0)

?˙γ t0dt′Y G(t0+t′)e x,(6.1)

where Y G is the y-component of R G=N?1 N j=1R j. From the net displacement,the?rst term,we have sub-tracted the?ow-induced displacement,the second term. Figure13shows F q(t)with q=2πfor various˙γat T=https://www.360docs.net/doc/cd7725727.html,parison of this?gure with Fig.2suggests that applying shear is analogous to raising the tempera-ture.This tendency was already reported for the case of supercooled binary mixtures.6,41

We introduce the shear-dependent Rouse timeτR=τR(T,˙γ)and theαrelaxation timeτα=τα(T,˙γ)by

C(τR)=e?1,F q(τα)=e?1.(6.2) We may then examine how shear can accelerate the mo-tions of chains and individual beads in shear?ow.Fig-ure14showsτR andταas functions of˙γat T=0.2. In our short chain system,bothτR andταdecrease for ˙γ τR(T,0)?1?τα(T,0)?1.Our data at T=0.2are consistent with

τR(T,˙γ)?1=τR(T,0)?1[1+A R˙γ],(6.3)

τα(T,˙γ)?1=τα(T,0)?1[1+(Aα˙γ)μ],(6.4) where A R～=104～τR(T,0),Aα～=6000?20τα(T,0), andμ～=0.77.The average tumbling period in Fig.12is about4τR(T,˙γ).For simple supercooled liquids we al-ready introduced the van Hove time-correlation function in shear?ow6and obtainedτα(T,˙γ)?1=τα(T,0)?1[1+ Aα˙γ]with Aα～τα(T,0).

The sensitive shear-dependence ofτα(T,˙γ)predicted by(6.4)suggests potential importance of dielectric mea-surements in shear?ow.6As a?rst experiment,Mat-suyama et al.measured the dielectric loss function ?′′(ω,˙γ)in steady shear˙γin oligostyrene and polyiso-prene melts.19In the former melt at T=42?C,?′′(ω,˙γ) decreased nonlinearly as a function of˙γat low frequencies (ω 105s?1)in the shear-thinning regime(˙γ 15s?1). Their?nding indicates thatταdecreases as a function of ˙γin the non-Newtonian regime,consistently with(6.4), More systematic dielectric measurements in supercooled systems under shear?ow are very informative.

VII.SUMMARY

We have performed very long MD simulations of a su-percooled polymer melt composed of M=100short chains with bead number N=10in quiescent and sheared conditions.We here summarize our main sim-ulation results together with remarks.

(i)The stress relaxation function G(t)is shown to fol-low a stretched exponential decay(1.1)on the scale of theαrelaxation timeταand then the Rouse relaxation (3.6)on the scale ofτR.

(ii)The nonlinear shear regime sets in at extremely small shear rate of orderτ?1

R

in supercooled states,where marked shear-thinning and shape changes of chains are found.Scattering and birefringence experiments from weakly sheared melts near T g seem to be very promising. (iii)In the nonlinear shear regime,each chain under-goes random tumbling in our melt as in the case of iso-lated polymer chains in shear?ow.It is of great interest how this e?ect is universal in solutions and melts and how it in?uences macroscopic rheological properties.For ex-ample,we are interested in whether or not such tumbling occurs in sheared entangled polymers.

(iv)Transient stress divided by˙γafter application of shear?ow obeys the linear growth t0dt′G(t′)?G0t for strain less than0.1and then saturates into a non-Newtonian steady-state viscosity.This initial growth is much steeper than that predicted by the Rouse model. As a result,the stress-optical relation does not hold tran-siently under deformation even in the linear(zero-shear) limit.Its violation is more enhanced for larger shear rates.These are consistent with the experiments.

(v)The time-correlation functions in shear?ow are cal-culated for the end-to-end vector and the modi?ed par-ticle displacement in(6.1).The former represents the relaxation of chain conformations,while the latter the monomeric relaxation on the spatial scale of the particle distance.We can then determine the shear-dependent relaxation times,τR(T,˙γ)andτα(T,˙γ).They decrease nonlinearly and behave di?erently as functions of˙γin the nonlinear shear regime as in(6.3)and(6.4).It is then of great interest how these times behave in strong shear for much larger N.We conjecture that if N is su?ciently large,shear should?rst in?uence the overall chain conformations,while it does not much a?ect the monomeric relaxations.We propose dielectric measure-ments in shear?ow,which should give information of τα(T,˙γ).19In addition,as demonstrated in Fig.13,the e?ect of shear on the van Hove self-correlation function is analogous to raising the temperature above T g as in supercooled binary mixtures.6,41

ACKNOWLEDGMENTS

We would like to thank T.Inoue for valuable discus-sions on the stress-optical relation.This work is sup-ported by Grants in Aid for Scienti?c Research from the Ministry of Education,Science,Sports and Culture of https://www.360docs.net/doc/cd7725727.html,putations have been performed at the Hu-man Genome Center,Institute of Medical Science,Uni-

versity of Tokyo.

1010

t

C (t ), C R (t )

FIG.1.Normalized end-to-end vector time-correlation function C (t )in (3.4)for T =1.0,0.4,and 0.2.The dot-ted lines are the results of the Rouse model (3.5).The Rouse time τR in (3.2)is indicated by arrows.

t

F q (t )

FIG.2.The van Hove self-correlation function F q (t )at q =2πfor T =1.0,0.4,and 0.2on a semi-logarithmic scale.The dotted line represents the stretched exponential decay ∝exp[?(t/τα)0.64].

1010

10

t

G (t ), G R (t )

FIG.3.The stress relaxation function G (t )in (3.10)(thin-solid lines)at T =0.2in a supercooled state and T =1in a normal liquid state.For T =0.2,it can be ?tted to the stretched exponential form,exp[?(t/τs )0.5]with τs =90,(dotted line)for 1 t 103and tends to the Rouse relax-ation function G R (t )(bold-dashed lines)at later times.

10

-2

10

-1

10

10

1

102

10

3

10

4

10

5

10-3

10

-2

10

-1

10

10

1

10

2

T = 0.2

T = 1.0

t

G b (t ), G R (t )

https://www.360docs.net/doc/cd7725727.html,parison of the time-correlation function G b (t )in (3.14)(thin-solid lines)and the Rouse relaxation function G R (t )in (3.6)(bold-dashed lines).

γ

.

η(γ )

.

FIG.5.The steady-state viscosity η(˙γ)vs shear ˙γfor T =0.2,0.4,and 1.A line of slope ?0.7is a view

guide.

1010

10

Q xy (γ )

.

σx y (γ ) / T

.

FIG.6.Universal stress-optical relation σxy /T vs Q xy in steady states under shear ?ow for T =0.2,0.4,and 1.

FIG.7.(a)Isointensity curves of g s (r )in (4.3)in the x -y plane (?3.75

1010

1010γ τR

.

FIG.8.tan θ,1?a 1/a 2,and Q xy vs ˙γτR in steady states.

1010

t / τ R

σx y (t ) / γ

.

FIG.9.Shear stress divided by shear rate σxy (t )/˙γvs t/τR

for ˙γ=10?1,10?2,10?3,10?4

(thin-solid lines)at T =0.2where τR =6×104.The curves follow the linear viscosity growth function (bold-solid line)for ˙γt 0.1,but depart from it for ˙γt 0.1.The linear growth function in the Rouse model is also plotted(bold-dashed line).The arrows indicate onset of the nonlinear behavior.

0.02

0.04

0.060.080.10.12

0123

4

56Q xy (t )

σx y (t ) / T

γ = 10

-2

γ = 10

-3

.

.

γ = 10

-4

γ = 10

-1

.

.

FIG.10.Parametric plots of σxy (t )/T vs Q xy (t )after ap-plication of shear at t =0for T =0.2.The curves initially de-viate from the universal steady-state curve obtained in Fig.6(dashed line)but approach it ultimately.The deviations in-crease with increasing ˙γ.

10-1

10010110210310410510610

7

-0.20

0.20.40.60.81t

C (t )

γ = 10

-1

γ = 0

.

.

FIG.11.Normalized time-correlation function of the end-to-end vector C (t )in (3.4)at T =0.2for ˙γ=0,10?4,10?3,10?2,and 10?1from right to left on a semi-logarithmic scale.The negative overshoot for ˙γ>0arises from rotational motions of chains.

–10010 t

.–100

10

.(b)–10010.(a)P x ( t )

γ = 10

-3γ = 10-2γ123456

78

γ = 10

-1.(c)12345678

(d)

FIG.12.Time-evolution of the x component of the end-to-end vector P x (t )=X N (t )?X 1(t .Here T =0.2

and ˙γ=10?3(a),10?2(b),and 10?1

(c)from above.Typical tumbling motions at the points 1～8indicated in (c)are shown in (d),where the chain conformations are projected on the x -y plane.

10-2

10-110010110210310410510

6

00.2

0.4

0.6

0.8

1

t

F s (q ,t )

γ = 0

γ = 0.1

.

.

T = 0.2q = 2π

FIG.13.The van Hove self correlation function (3.7)with (6.1)at T =0.2for ˙γ=0,10?4,10?3,10?2,and 10?1from right to left on a semi-logarithmic

scale.

1010

1010101010γ

.

FIG.14.Two relaxation times τR (˙γ)and τα(˙γ)as func-tions of shear ˙γat T =0.2determined from (6.2).Both these times decrease for ˙γ τR (0)?1～N ?2τα(0)?1in our short chain system.The solid and dashed lines represent (6.3)and (6.4),respectively.The slopes of the curves at high shear are ?1for τR (˙γ)and ?0.77for τα(˙γ).