Detailed analysis of Rouse mode and dynamic scattering function of highly entangled polymer melts in equilibrium

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by  Hsiao-Ping Hsu

Detailed analysis of Rouse mode and dynamic scattering function of highly entangled
polymer melts in equilibrium
Hsiao-Ping Hsu∗ and Kurt Kremer†
Max-Planck-Institut fu¨r Polymerforschung, 55128 Mainz, Germany
Wepresentlarge-scalemoleculardynamicssimulationsforacoarse-grainedmodelofpolymermelts
in equilibrium. From detailed Rouse mode analysis we show that the time-dependentrelaxation of
the autocorrelation function (ACF) of modes p can be well described by the effective stretched
exponentialfunctionduetothecrossoverfromRousetoreptationregime. TheACFisindependent
of chain sizes N for N/p < N (N is the entanglement length), and there exists a minimum of
7 e e
the stretching exponent as N/p → N . As N/p increases, we verify the crossover scaling behavior
1 e
0 of the effective relaxation time τeff,p from the Rouse regime to the reptation regime. We have
2 also provided evidence that the incoherent dynamic scattering function follows the same crossover
scaling behavior of the mean square displacement of monomers at the corresponding characteristic
n
time scales. The decay of the coherent dynamic scattering function is slowed down and a plateau
a
developsaschainsizesincreaseattheintermediatetimeandwavelengthscales. Thetubediameter
J
extractedfromthecoherentdynamicscatteringfunctionisequivalenttothepreviousestimatefrom
0
themean squaredisplacement of monomers.
1

] I. INTRODUCTION fully equilibrated and highly entangled polymer melts
t
f were generated by a novel and very efficient method-
o
s The dynamics of polymer chains in a melt is a com- ology through a sequential backmapping of soft-sphere
t. plicated many-body problemwhere the motion ofchains coarse-grainedconfigurationsfromlowresolutiontohigh
a resolution, and finally the application of molecular dy-
depends not only on differentlength scales but also time
m
scales. Itiswellknownthatforshortunentangledchains namics (MD) simulations of the underlying bead-spring
- model [11–14]. We have also studied the dynamics
in a melt, excluded volume and hydrodynamic inter-
d
n actions are screened, and the viscoelastic properties of of fully equilibrated polymer melts, characterized by
the mean square displacement of monomers, and de-
o chains can be approximately described by the Rouse
c model [1–3]. If the polymer chains become long enough, termined the characteristic time scales: the character-
[ istic time τ , the entanglement time τ τ N2, the
the topological constraints dominate the dynamics of Rouse time0τ τ N2, and the disentean≈glem0enet time
v1 tchheainchsaainres.aAsstuminetdermtoedmiaotveetbimacekaannddlefonrgtthh(srceaplteast,iothne) τd τ0N2(N/RN≈e)1.40, according to the predictions given
≈
2 within a tube-like region created by surrounding entan- by the Rouse model and the reptation theory, where Ne
0 gled chains and depending on the entanglement length is the entanglement length and N is the chain size. For
7 N <N in the Rouse regime, there exist exact solutions
N . Thedynamicbehaviorwithinthistimeframeiswell e
2 deescribed by the reptation theory of de Gennes, Doi and of almost all physical observables. Therefore, based on
0
this work, we are interested in understanding to what
. Edwards [2–4].
1 extent the dynamics of single chains can be analyzed
Rouse mode analysis provides a straightforward way
0 through the Rouse mode analysis and check the scaling
of understanding the dynamics of single chains in a melt
7
predictions of the relaxation of the Rouse modes in the
1 bymappingthetrajectoriesofthechainsintoorthogonal
literature [1–9, 14–16] whenever it is possible. On the
: Rousemodes. Thechainsareassumedtobe Gaussianin
v other hand, Rouse mode decay should display a similar
this analysis. This has been applied widely to the analy-
i slowing down of the dynamic structure factor. The tube
X sisofexperimentaldataandsimulationdata. Besidesthe
diameter introduced in the reptation theory can be ex-
r original predictions of this model, in the literature [5–9]
a therealsoexistmodifiedtheoreticalpredictionsincluding tracted from the single chain dynamic structure factor
measured via neutron spin echo (NSE) experiments [17–
the excluded volume interaction, the intrinsic stiffness
19]. Therefore we also study the dynamic scattering
of chains, the intramolecular correlations in chains, and
function from single chains in a melt, and check the
the topological constraints for analyzing the relaxation
consistency of the tube diameter estimated from differ-
of Rouse modes.
ent physical quantities [3, 14, 15, 20, 21]. We use the
Inourrecentwork[10],wehaveinvestigatedthe chain
ESPResSo++ package [22] to perform the standard MD
conformations of fully equilibrated and highly entan-
simulations with Langevin thermostat at the tempera-
gled polymer melts and compared our simulation re-
ture T = 1ε/k where k is the Boltzmann factor to
sults to the relatedtheoreticalpredictions in detail. The B B
studyfully equilibratedpolymermelts consistingof1000
chains of sizes N = 500, 2000 for k = 1.5 (τ 2.89τ,
θ 0
≈
N 28[10,12])andofsizesN =1000,2000fork =0.0
e θ
≈
∗Electronicaddress: [email protected] (τ0 1.5τ, Ne 87 [12]) in the framework of the stan-
≈ ≈
†Electronicaddress: [email protected] dard bead-spring model [14] with a bond bending inter-
2
action parameter k at a volume fraction φ=0.85. and
θ
Theoutlineofthepaperisasfollows: Sec. IIdescribes
p −2 ζb2 p −2
the theoretical background of the Rouse model and the τ =τ = . (7)
Rouse mode analysis of highly entangled polymer melts. p 0 N 3kBTπ2 N
(cid:16) (cid:17) (cid:16) (cid:17)
Sec. III describes the scaling behavior of dynamic struc-
Forp=N, τ =τ is the shortestrelaxationtime ofthe
turefactorsandthecomparisonbetweentheoryandsim- N 0
Rouse model, where τ = ζb2/(3k Tπ2) is the charac-
ulation. Finally, our conclusions are summarized in Sec. 0 B
teristic relaxation time, while τ for p = 1 is the longest
IV. 1
relaxation time equal to the Rouse time, i.e. τ =τ .
1 R
However,astheexcludedvolumeinteractionandtopo-
logical constraint are taken into account, it has been
II. ROUSE MODE ANALYSIS
pointed out in the literature [5, 6, 24] that simulation
results of the normalized autocorrelation function are
In the Rouse model, neglecting inertia effects Rouse
well described by the stretched exponential Kohlrausch-
chains undergo Brownian motion and therefore the
Williams-Watts (KWW) function, i.e.
Langevinequationofmotionfortheithmonomeristhus
given by X~ (t)X~ (0)
h p p i =exp[ (t/τ∗)βp], (8)
d~r ∂U(~r ,...,~r ) X~ (0)X~ (0) − p
ζ i = 0 N−1 +f~R(t) (1) h p p i
dt ∂~r i
i where the KWW characteristic relaxation time τ∗ and
p
with the Rouse potential the stretching exponent βp depend on the mode p, and
both are measures of importance of excluded volume
3k T N−1 interactions and topological constraints. The effective
U(~r0,...,~rN−1)= 2Bb2 (~ri+1−~ri)2 (2) Rouse time of mode p is thus given by
i=1
X
∞ τ∗ 1
where b is the effective bond length, ~ri is the position τeff,p = dtexp[−(t/τp∗)βp]= βpΓ β , (9)
vector of the ith monomer, and f~R(t) is a random force. Z0 p (cid:18) p(cid:19)
i
The frictionζ andthe randomforcef~R(t)arerelatedby where Γ(x) is the gamma function.
i
the fluctuation dissipation theorem, i.e., Figure 1 shows the typical relaxation of the time-
dependent normalizedautocorrelationfunction ofmodes
hf~iR(t)·f~jR(t′)i=6kBTζδijδ(t−t′). (3) p, hXp(t)Xp(0)i/hXp2i, according to the definition of
Rouse modes X (t) given in Eq. (4). Data are for poly-
p
The Rouse modes X~p(t) are defined as the cosine trans- mermeltsofchainsizesN =500and2000withkθ =1.5.
forms of position vectors~ri at time t for i=1,...,N as The data sets shown in Fig. 1(a) from left to right cor-
given in Ref. [23], respond to N/p= 5,10,25,50,125,250,and500. We see
that for N/p < N (N 28 for k = 1.5) the relax-
e e θ
2 1/2 N pπ ation of X (t)X (0) / X≈2 is independent of chain size
X~ (t)= ~r (t)cos (i 1/2) , h p p i h pi
p N i N − N,namely,dataforN =500andN =2000areontopof
(cid:18) (cid:19) Xi=1 h i eachother. Similar results are alsoobservedfor polymer
p=0,...,N 1. (4)
melts of chain sizes N = 1000 and 2000 with k = 0.0
− θ
wheretheentanglementlengthN 87[12](notshown).
Here the p>0 modes describe the internal relaxationof e ≈
IntheregimewhereN/p>N thedeviationbetweentwo
a chain of N/p monomers while the 0th (p = 0) mode e
data sets corresponding to the same value of N/p be-
correspondstothemotionofthecenter-of-massofchain.
comes more prominent as N/p increases since the entan-
SinceeachRousemodeX~ (t)forp>0performsaBrow-
p glementeffectbetweenchainsegmentsbecomesmoreim-
nianmotioninaharmonicpotential,independentofeach portant. In Fig. 1(b), the plot of ln[ X (t)X (0) / X2 ]
other, the cross-correlations should vanish, the normal- h p p i h pi
versust/τ withτ =(N/p)2(seeEq.(5))showsthatthe
izedautocorrelationfunctionisexpectedtofollowanex- p p
exponential decay is only valid for small values of N/p
ponential decay,
at initial relaxation time t. As p becomes small and t
X~ (t)X~ (0) increases, one sees systematic deviations from the expo-
h p p i =exp( t/τ ) (5) nential decay due to the crossover from Rouse to repta-
X~p(0)X~p(0) − p tion behavior. In Fig. 1(c), the curves indicate the best
h i
fit of our data for N = 500 to the theoretical prediction
with
Eq.(8) withtwofittingparametersβ andτ∗. Polymer
{ } p p
pπ −1 chains containing N/p monomers are relaxedcompletely
hXp2i=hX~p(0)X~p(0)i = b2 4sin2 2N as hXp(t)Xp(0)i/hXp2i → 0 for t >> 1. Therefore, one
p/Nh≪1b2(cid:16)pπ (cid:17)−i2 , (6) claaxnveesrtyimloantegrcohuaginhslyinthaemreeqlutirthedrorueglahxtahtiisoncutrimveefittotirneg-
−→ N
(cid:16) (cid:17)
3
(a) (b)
1 0
>] kθ=1.5, N=500
2X >p 0.8 2< X p -0.5
< X(t)X(0) > / < pp 000...246 k2θ0=N010.5 n [< X(t)X(0) > / pp -1-.15 pppppp ====== 12334 500050000000 slope = -1
500 l -2
0 0 0.5 1 1.5
0.1 1 102 104 106 108 tp2 / N2 [τ]
t [τ]
(c)
1
N=500
2 >p 0.8
< X kθ=1.5
> / 0.6 N/p
0) 500
(t)X(pp 0.4 21552005
< X 0.2 2150
5
0
0.1 1 102 104 106 108
t [τ]
FIG. 1: (a) Semi-log plot of the normalized autocorrelation function of Rouse modes, hX (t)X (0)i/hX2i, versus time
p p p
t[τ] for polymer melts of size N = 500 and N = 2000. N/p = 5,10,25,50,125,250, and 500 from left to right. (b)
ln[hX (t)X (0)i/hX2i] versus tp2/N2[τ] for N = 500, and for several chosen values of p, as indicated. (c) Same data for
p p p
N =500asshownin(a),butincludingthecurveswhichpresentthebestfitofourdatausingEq.(8)forcomparison. Alldata
are for k =1.5.
θ
(a) (b)
1
00..89 kkkkθθθθ====1100....5500,,,, NNNN==== 212 500000000000 110068 kkkkθθθθ====1100....5500,,,, NNNN====2 21 050000000000
βp 0.7 τ []eff,p104 slope = 3.4
τ
0.6
102
0.5
slope = 2
N(kθ=1.5) N(kθ=0.0)
0.4 e e 1
1 10 100 1000 1 10 102 103
N / p N / p
FIG. 2: (a) Values of exponent β from fitting the stretched exponential function, Eq. (8), to the normalized autocorrelation
p
function of Rouse modes as shown in Fig. 1 plotted versus N/p. (b) Effective relaxation times τeff,p plotted as a function of
N/p. In(a)N/p≈Ne areindicatedbyarrowsforkθ =0.0and1.5. In(b)twoscalinglawsτeff,p ∼(N/p)2 andτeff,p ∼(N/p)3.4
predicted by Rouse model and reptation theory,respectively, are also shown for comparison.
procedure. Fitted values of the stretching exponent β straints on the chains [5, 6], and the recent work [16]
p
and the estimates of the effective relaxation time τ using the same simulation model but different cut-off
eff,p
from β and τ∗ Eq. (9) are shown in Fig. 2. We see for the LJ potential. At short length scales, the local
p p { }
that β reaches a minimum around N/p N for both constraints on the motion of monomers is related to the
p e
≈
cases k = 1.5 and 0.0 in Fig. 2a. For chains having the chain connectivity and the excluded volume interactions
θ
same intrinsic stiffness, β is independent of size N for betweenmonomerswhileatlonglengthscales,theentan-
p
N/p < N while β decreases with increasing size N for glementeffectsetsinthatthemotionofentangledchains
e p
N > N . Our results agree with the argument that the is strongly hindered by topological constraints. Results
e
development of a minimum in β is due to kinetic con- of the effective relaxation time τ show the crossover
p eff,p
4
(a) (b)
3.2
110066 kθ=0.0, N = 2000
kθ=0.0, N = 1000 >
kθ=1.5, N = 2000 2 Xp 2.4
2222X > / bX > / bpp 111100002424 kθ=1.5, N = 500 p/(2N)) < 1.6 kθ=1.5, N = 2000
< < 11 1/[(4πspi/nN2)(-p2π, /E(2qN.()6))] 2π4sin( 0.8 kkkθθθ===100...500,,, NNN === 521000000 00
Eq.(10) Eq.(11)
1100--22 0
1100--33 1100--22 1100--11 11 1 10 100 1000
pp // NN N / p
FIG. 3: Rescaled amplitude of the autocorrelation function of the Rouse modes, hX2i/b2 (a) and 4sin2(πp/(2N))hX2i (b),
p p
plotted versus p/N and N/p respectively. Two chain sizes N = 500, 2000 are chosen for k = 1.5, and N = 1000, 2000 for
θ
k = 0.0, as indicated. In (a), theoretical predictions given in Eqs. (6) and (10) are shown for comparison. b2 = 2.74σ2 for
θ
k =1.5 and b2 =1.74σ2 for k =0.0. In (b), Eq. (11) with C =0.23 and 0.32 for k =1.5 and k =0, respectively, are also
θ θ θ θ
shown for comparison.
behavior from the Rouse regime (τ (N/p)2) to the lation as p/N > (10−1) due to the lack of consider-
eff,p
reptation regime (τ (N/p)3.4) as N∼/p increases. ing the intrinsic stOiffness of the chains. Replacing ran-
eff,p
∼
In previous Monte Carlo simulations of polymer melts domwalk chains by freely rotatingchains with a specific
based on the bond fluctuation model [7], the authors bond angle θ, the analytical expression of the amplitude
showed that the Rouse model overestimates the corre- X (0)X (0) is as follows,
p p
h i
pπ −1 1 cosθ 2 pπ −1
X~ (0)X~ (0) =b2 4sin2 −|h i| +4sin2 (1+ (N−1) . (10)
p p
h i (h (cid:16)2N(cid:17)i −(cid:20) 4|hcosθi| (cid:16)2N(cid:17)(cid:21) O )
Figure 3a presents the rescaledamplitude of the auto- verified as shown in Fig. 3b. Since the entanglement ef-
correlation function of Rouse mode p, X (0)X (0) /b2, fect already sets in at N/p 28 for k = 1.5, we see
p p θ
plottedversusp/N. Hereb2 isdeterminhedbythebeistfit that for chains of size N = 2≈000, the data for k = 1.5
θ
ofourdatatoEq.(6)forsmallvaluesofp/N. b2 =2.74σ2 fluctuate more than that for k =0.0.
θ
fork =1.5andb2 =1.74σ2 fork =0.0. Ourfittedval-
θ θ
ues of b2 for both cases satisfy the relation b2 = ℓ2C
b ∞
predicted for freely rotating chains, where C is Flory’s III. DYNAMIC STRUCTURE FACTORS
∞
characteristic ratio (C = 2.88, 1.83 for k = 1.5,
∞ θ
0.0, respectively), and ℓb = 0.964σ is the mean bond Dynamic behavior of polymer chains in a melt can
length [10, 12]. Theoretical predictions given in Eqs. (6) also be described by the dynamic scattering from single
and (10) are also shown for comparison. The deviation chains. The coherent and incoherent dynamic structure
from the Rouse prediction for p/N > (10−1) is in- factors S (q,t) and S (q,t) for single chains are de-
O coh inc
deed seen as shown in Ref. [7]. Taking the estimates fined by
of cosθ from our simulations, our data are described
quihtewelilbyEq.(10)forp/N > (10−1). Forsmallp/N N N
1
(largeN/p),oneshouldexpectthOat4sin2(pπ/(2N))hXp2i Scoh(q,t)= Nh exp{i~q·[~ri(t)−~rj(0)]}i, (12)
reachesaplateauvalueb2. However,sinceatshortlength i=1j=1
XX
scale the local intramolecular correlations in the chains
and
(the correlation hole effect) are important, a correction
term [8, 9, 16] ((N/p)−1/2) is needed to be considered N
O 1
as follows, S (q,t)= exp i~q [~r (t) ~r (0)] . (13)
inc i i
Nh { · − }i
i=1
4sin2(pπ/(2N)) X2 =b2[(1 C(N/p)−1/2] (11) X
h pi −
The average denotes an average over all chains,
h···i
where C is a fitting parameter. The prediction is also many starting states (t=0), as well as over orientations
5
(a) (b)
103 q=0.05 kθ=1.5, N=500 ∼t1/2 103 q=0.05 kθ=1.5, N=2000
q= 0.2 q= 0.2
2σq,t)] [] 102 qqq=== 001...462 ∼t1/4 2σq,t)] [] 102 qqq=== 001...482 ∼t1/4
S(inc 10 q= 1.8 S(inc 10 q= 1.8
2 ln [ ∼t1/2 2 ln [ ∼t1/2
-q 1 -q 1
- 6 ∼t1 - 6 ∼t1
τ τ τ τ τ
10-1 0 e R,N=500 10-1 0 e
1 102 104 106 108 1 102 104 106 108
t [τ] t [τ]
FIG. 4: Log-log plot of −6q−2lnSinc(q,t) versus t for polymer melts of sizes N = 500 (a) and N = 2000 (b). Data are for
polymermeltswith k =1.5. Thetheoretical prediction of thescaling behaviorofthemean squaredisplacement ofmonomers
θ
is also shown for comparison since lnSinc(q,t)∝g1(t). Here the characteristic relaxation time τ0 ≈2.89[τ], the entanglement
time τe =τ0Ne2 ≈2266[τ], and the Rousetime τR,N=500 =τ0N2 ≈7.2×105[τ] for N =500 are taken from Ref. [10].
of the wave vector ~q having the same wave length. Note processes from the tube (creep motion) for longer times
that S (q) is the q-space representation of the Rouse and small values of q should be considered. Thus a pro-
coh
modes. nouncedplateauinS (q,t)ispredictedandcanloosely
coh
In the Rouse model, the displacement between be interpreted as a Debye-Waller factor for τ t τ ,
e d
≪ ≪
monomer positions is Gaussian distributed since the
S (q,t)
forcehas aGaussianprobabilitydistribution. Therefore, coh =1 q2d2/36. (19)
Eqs. (12) and (13) can be written as Scoh(q,0) −
N N Note that here the tube diameter is defined by d =
1 1
S (q,t)= exp q2 [~r (t) ~r (0)]2 , R (N )whereR (N )istheend-to-enddistanceofchains
coh i j e e e e
N {−6 h − i}
Xi=1Xj=1 of size Ne [3, 15]. In our previous work [10], the def-
(14) inition of tube diameter d is different by a factor of
T
and √3, i.e. our simulation estimate of tube diameter d =
T
(2 R2(N ) )1/2 =( R2(N ) /3)1/2 =d/√3. In the deep-
S (q,t)= 1 N exp 1q2 [~r (t) ~r (0)]2 . (15) rephtagtioneriegime, ahn aenaleytiic expression of the coherent
inc N {−6 h i − i i} dynamic structure used often in the neutron-spin-echo
i=1
X (NSE) measurements for the determination of the tube
where [~r (t) ~r (0)]2 g (t)issimplythe meansquare diameter is given by [3, 14, 15, 20, 21]
i i 1
h − i∼
displacement of monomers.
S (q,t) q2d2
For short chains (N < Ne), the scaling predictions coh = 1 exp f(q2(Wt)1/2)+
of Scoh(q,t) and Sinc(q,t) for t < τe (N < Ne) and Scoh(q,0) (cid:26)(cid:20) − (cid:18)− 36 (cid:19)(cid:21)
q >2π/Rg(N)(Rg(N)istheradiusofgyrationofchains q2d2 8 ∞ exp[ tn2/τ ]
d
containing N monomers) predicted by the Rouse model +exp − (20)
− 36 π2 p2
are as follows, (cid:20) (cid:21)(cid:27) n=1,odd
X
ln[Scoh(q,t)/Scoh(q,0)]= q2(Wt)1/2/6, (16) with f(u)=exp(u2/36)erfc(u/6).
−
Figure 4 shows the results of the incoherent dynamic
and structurefactorS (q,t)accordingtoEqs.(13)and(17)
inc
for polymer melts of sizes N = 500 and 2000 with
lnS (q,t)= q2(Wt)1/2/6 (17)
inc − kθ = 1.5. The characteristic time scales, τ0, τe, and
where the factor W = 12kBTb4 and R is the radius of τR,N=500 taken from Ref. [10] are indicated by arrows.
πζ g Since lnS (q,t) g (t), the scaling laws of g (t) show-
inc 1 1
gyration of the chain of size N. For t τ one expects ∼
≫ e ing the crossover behavior from the Rouse regime to
the standard diffusion behavior, i.e.
the reptation regime as t increases are also shown for
ln[S (q,t)/S (q,0)]=q2Dt (18) comparison. In the Rouse regime where q > 2π/dT
coh coh 1.25σ−1, we see that lnS (q,t) t for t < τ whi≈le
inc 0
duFeotrolotnhge tcohpaoinlosg(icNal >conNster)a,intthsebeenttwaenegnlemcheanintseffinecat lrnegSiimnce(qo,nt)e ∼shotu1/ld2 feoxrpeτc0t<thtat<ln∼τSei.nc(Iqn,tt)he retp1t/a4tifoonr
∼
meltbecomesimportant. Accordingtothereptationthe- τe <t<τR and2π/Rg(N)<q <2π/dT. Weseethatin-
ory, local reptation processes for short time and escape deedS t1/4 for0.4σ−1 <q <1.25σ−1 inFig.4aand
inc
∼
6
(a) (b)
0 0
0)] -1 0)] -1
q, q,
(oh -2 kθ=0.0 (oh -2 kθ=1.5
Sc N=2000 Sc N=2000
(q,t) / oh --43 qqq=== 000...468 (q,t) / oh --43 qqq === 000...684
Sc q= 1.2 Sc q = 1.2
n[ -5 q= 1.4 ∼ - t1/2 n[ -5 q = 1.4 ∼ - t1/2
l l
q= 1.6 q = 1.8
-6 -6
0 5 10 15 20 25 30 0 5 10 15 20 25 30
q2 t1/2 / 6 [σ-2τ1/2] q2 t1/2 / 6 [σ-2τ1/2]
FIG. 5: lnScoh(q,t)/Scoh(q,0) plotted versus q2t1/2/6[σ−2τ1/2] for polymer melts of size N = 2000 and for kθ = 0.0 (a) and
k =1.5 (b). The scaling law predicted by theRouse model, t1/2 is shown bya straight line.
θ
for 0.2σ−1 < q < 1.25σ−1 in Fig. 4b. Here for k = 1.5, The relaxation of time-dependent autocorrelation func-
θ
R √0.4839Nσ [10]. In Fig. 4a, we have also ob- tions of Rouse modes p is independent of chain size
g
≈
served that lnSinc(q) t1/2 for t > τR and q < 0.4σ−1. N for N/p < Ne (Fig. 1a) where the entanglement
∼
Therefore, our results are in perfect agreement with the lengthNeisobtainedthroughtheprimitivepathanalysis
theoretical predictions. (PPA) [10, 12, 25]. Estimates of the stretching exponent
Forcheckingthescalingbehaviorofthenormalizedco- βp alsoshowthattheminimumofβp occursinthevicin-
herent dynamic structure factor predicted by the Rouse ity of N/p Ne (Fig. 2a). The crossover behavior of
≈
model,weplotln[Scoh(q,t)/Scoh(q,0)]versusq2t1/2/6for effective relaxationtimes τeff,p fromthe Rouse regime to
polymer melts of size N = 2000 with k = 0.0 and 1.5 the reptation chain as N/p increases is verified as well.
θ
in Fig. 5. We see that in the intermediate time regime SincealltheseestimatedquantitiesforN/p<Ne behave
τ0 <t < τe, our data are also in perfect agreement with differently from that for N/p > Ne, we see that Ne can
the scaling law t1/2 for q > 2π/d N−1/2σ−1. For also be determined roughly via the Rouse mode analysis
T e
∝ oflargepolymermeltsystemsoftwodifferentchainsizes,
τ t τ , one would expect that S (q,t)/S (q,0)
d e coh coh
≫ ≫ and the value is consistent with that obtained through
reaches a plateau predicted by the reptation theory
PPA. Our results are also in perfect agreement with the
Eq. (19) . Therefore, in Fig. 6, we show the similar
{ } extendedtheoreticalpredictionsconsideringtheexcluded
dataasshowninFig.5butplotS (q,t)/S (q,0)ver-
coh coh
susq2t1/2/6. We seethatfirstdatatendto collapseonto interaction,topologicalconstraint,theintramolecularin-
teractions, and chain stiffness (Fig. 3).
a single master curve for τ < t < τ as q increases and
0 e
The scaling behavior of coherent and incoherent dy-
S (q,t)/S (q,0) is independent of chain size N. As
coh coh
namic structure factors strongly depends on the time t
t > τ , S (q,t)/S (q,0) for two different chain sizes
e coh coh
and wave length q. Therefore, it is a delicate matter to
N start to deviate from each other. For polymer chains
analyzethedynamicstructurefactors. However,wehave
of size N = 2000 in both cases (k = 0.0 and k = 1.5),
θ θ
provided evidence that the scaling behavior of S (q,t)
S (q,t)/S (q,0) slows down as t τ . It gives the inc
coh coh e
≫ (Fig.4)iscompatiblewiththemeansquaredisplacement
firstevidencefromsimulationsthatapronouncedplateau
ofmonomersg (t),andthecrossoverpointscharacterized
inS (q,t)shalloccuraschainsizeN increases. Finally, 1
coh
we determine the tube diameter d =d/√3 for k =1.5 by the characteristic time scales τ0, τe, and τR and the
T θ
corresponding wave length scales (the inverse of length
by fitting our simulation data of S (q,t)/S (q,0) to
coh coh
scales)areconsistentwitheachother. Theslowingdown
Eq. (20) (see Fig. 7). Our results show that d 7.10σ
T
≈ ofS (q,t)(Fig.6)givesthefirstevidencethatS (q,t)
for N = 500, and d 5.95σ for N = 2000 which are coh coh
T
≈ exhibits a plateau for τ t τ as the chain size N
compatible to our previous estimate of dT ≈5.02σ [10]. increases. The tube diame e≪ter d≪=dd/√3 extractedfrom
T
S (Fig. 7) is also in perfect agreement with d ob-
coh T
tained from g (t) in Ref. [10].
1
IV. CONCLUSION
We hope that the present work showing the detailed
analysis of the dynamic properties of highly entangled
By extensive molecular dynamics simulations and ac- chains covering the scaling regimes from Rouse to
companying theoretical predictions in the literature, we reptation will help for the further understanding of
have investigated the dynamic properties of polymer the dynamic behavior of the deformed polymer melts,
meltsinequilibriumbyanalyzingthechainRousemodes, polydispersepolymermelts,andtherelatedexperiments.
and the dynamic coherent and incoherent structure fac-
tors for chains of two different sizes and stiffnesses. Acknowledgments: This work was supported by Eu-
7
(a) (b)
1 1
q=0.1 q=0.1
q,0) 0.8 q=0.2 q,0) 0.8 q=0.2
(h (h
Sco 0.6 Sco 0.6 q=0.4
q,t) / 0.4 kθ=0.0 q,t) / 0.4 kθ=1.5
(h N=2000 q=0.4 (h N=2000 q=0.6
Sco 0.2 N=1000 Sco 0.2 N= 500
q=1.0 q=0.6 q=1.0
0 0
10-1 1 10 102 10-1 1 10 102 103
q2 t1/2 / 6 [σ-2τ1/2] q2 t1/2 / 6 [σ-2τ1/2]
FIG. 6: Semi-log plot of normalized coherent dynamic structure factor, Scoh(q,t)/Scoh(q,0), versus q2t1/2/6 for kθ = 0.0 (a)
and k =1.5 (b). Five values of q and two chain sizes N are chosen, as indicated.
θ
(a) (b)
1 1
0) 0.8 kθ=1.5, N=500 q=0.2, d=12.7 0) 0.8 kθ=1.5, N=2000 q=0.2, d=11.8
q, q,
(h (h
Sco 0.6 q=0.4, d=12.8 Sco 0.6 q=0.4, d=10.8
(q,t) / h 0.4 q=0.6, d=12.4 (q,t) / h 0.4 q=0.6, d=10.5
o o
Sc 0.2 q=0.8, d=12.3 Sc 0.2
q=0.8, d=10.3
q=1.0, d=12.3
0 0 q=1.0, d=10.3
0 0.4 0.8 1.2 1.6 2.0 0 0.2 0.4 0.6 0.8 1.0
t [105τ] t [106τ]
FIG.7: Normalizedcoherentdynamicstructurefactor,Scoh(q,t)/Scoh(q,0),plottedversustforpolymermeltsofsizesN =500
(a)and2000(b)withk =1.5. Fivevaluesofq arechosen,asindicated. Thecurvesshowthebestfitofourdatabyadjusting
θ
thefitting parameter d using Eq. (20).
ropean Research Council under the European Union’s Centre (JSC), and the Rechenzentrum Garching (RZG),
Seventh Framework Programme (FP7/2007-2013)/ERC the supercomputercenter ofthe Max PlanckSociety, for
Grant Agreement No. 340906-MOLPROCOMP. We are the use of their computers. This work is dedicated to
grateful to G. S. Grest for stimulating discussions, and Wolfhard Janke on the occasion of his 60th birthday.
A. C. Fogarty for a critical reading of the manuscript. H.-P. Hsu is thankful to Wolfhard for the invitations to
We are also grateful to the NIC Ju¨lich for a generous several scientific events and for the fruitful discussions
grant of computing time at the Ju¨lich Supercomputing during her annual visits to Leipzig.
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