Dynamics of entangled polymer melts: A computer simulationhp0242/P_ABBinder_JCP81.pdf · there is no hope to bridge, in a computer simulation, the many decades of time from the microscopic

Dynamics of entangled polymer melts: A computer simulation

A. BaumgEirtner and K. Binder

I nstitut fiir Festkorperforschung, Kernforschungsanlage Jiilich, Postfach 1913, D-5170 Jiilich, Federal Republic of Germany (Received 23 February 1981; accepted 15 May 1981)

As a model for dense polymer solutions or melts we consider an ensemble of n freely jointed chains consisting of N rigid links in a box. A Lennard-Jones interaction is taken to model repulsive and attractive interactions between the units of the chains. Dynamics is introduced into the model by allowing for stochastic rotations of the beads of the chains, where the transition probability satisfies detailed balance and only such rotations are allowed which do not lead to any intersection of chains, in order to take entanglement restrictions into account. Although the chains treated by the simulation are very short (N = 16), with increasing density and/or decreasing temperature we do find a transition from a regime of essentially Rouse-like dynamics (with decreased mobilities) to a glass-like state, where the "monomer" displacements are very small and follow different power laws of time than in the Rouse model. We also treat the case where all chains apart from the mobile one are strictly frozen in, so that the mobile chain moves through fixed obstacles. We clearly confirm the de Gennes reptation mechanism for this case, while it does not become effective in the other cases above where either "tube rearrangement" is not negligible or all motions are more or less frozen. The experimental relevance of these results is briefly discussed.

I. INTRODUCTION

In a dense system of long flexible polymers, interchain entanglements are expected to modify the dynamics of the chains substantially. Various concepts to describe the dynamics of such a system have been proposed by suitably modifying the Rouse model1 which would adequately describe the Brownian motion of an isolated chain in the limit where hydrodynamic backflow effects2 can be neglected. A simple extension would be to increase the local friction constants depending on the degree of entanglement. 3,4 Other concepts involve approximate treatments of "transient entanglements,,5 or "transient networks,,6-9 (where one introduces a finite lifetime of crosslinks into the theory of rubber elasticity, etc. ). These treatments, as well as attempts of normal mode analyses for entangled networks, 10 have recently been criticized because they do not take into account correctly the "topological" character of the entanglement constraints. 11 In fact, if a chain moves in the presence of fixed obstacles (e. g., a chain inside a gel), a "reptation" type of movement where the chain diffuses inside a randomly shaped "tube" formed by the obstacles, is predicted. 12 While originally12 it was considered to be very doubtful whether this mechanism is relevant for polymer melts (where the "obstacles" would themselves be mobile), in later work11 ,13-16 the dynamics of all dense polymer systems was described on this basis. However, since fluctuations of the "tube" and "tube renewal" are hardly included explicitly, it is not completely clear under which conditions these treatments are self-consistent, and how the dynamic behavior "crosses over" into the less dense regime where the chains are much more mobile and some of the treatments mentioned above might be preferred. In addition, the experimental evidence for reptation either stems from very indirect analyses17,18 or from studying the diffusion of rather short chains in the environment of (much less mobile!) longer chains. 19 Thus, in this situation it is no surprise that theoretical reviews20,21 state

that "the dynamics of entangled chains is still poorly understood. ,,21

In this Situation, the present study wishes to shed some light on these questions by extending our previous simulations of the statics and dynamics of single polymer chains22- 25 to dense polymer systems. Of course, there is no hope to bridge, in a computer simulation, the many decades of time from the microscopic time scale of, local conformational changes (_1O-10 sec) to the macroscopic scales studied in the creep compliance function of dense polymers (_102 sec, e.g., Ref. 18). In fact, the simulations of the model which are presented below are restricted to rather short time scales, on which the dynamic behavior of the polymer melt is essentially not very different from the glassy state. 20 However, this restriction is not in all respects a drawback, since there is also considerable experimental i-nterest in this regime of short times: it is the only regime which is accessible to inelastic neutron scattering, and very interesting results from this technique have in fact been obtained. 26 It has already been recognized12

that due to the slowness of the reptation mechanism one does not expect to see it by neutron scattering-but what other dynamic degrees of freedom is one going to probe with this technique then? A priori, there is little reason to believe that the scattering from such dense systems would be similar to the scattering from isolated chains in dilute solution. 27

A more important caveat concerns the smallness of the number of links per chain in the computer simulation: although the assumption that successive links of the freely jointed chain can form arbitrary angles means that a link may correspond phYSically to a group of about ten successive monomers, the degree of polymerization of the model chains is still very small. It has been shown that nevertheless one can see the asymptotic behavior of "swollen" chains (end-to-end distance (R2) IX N 1• 18 , etc.), 22,23 as well as e-point behavior and col-

2994 J. Chern. Phys. 75(6), 15 Sept. 1981 0021-9606/81/182994-12$01.00 © 1981 American I nstitute of Physics

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A. Baumgartner and K. Binder: Dynamics of entangled polymer melts 2995

lapse transition23 as well as the "glass transition" of a single polymer chain, 24 in spite of its shortness. Since the predictions of the Rouse model for single-chain dynamics could easily be seen in the simulations, 23,24 we expect also a study of an ensemble of short chains to be useful. In fact, it is a challenge for a complete theory of polymer dynamics to make reasonable predictions also for this regime.

Section II describes the model and defines the quantities of interest for our investigation. For the sake of comparison, Sec. III briefly presents the results for the dynamics of isolated chains of otherwise precisely the same model. Section IV then shows our main results for several densities and temperatures of the system. Section V gives our results for the case of one mobile chain in a frozen environment. We interpret these results in terms of a scaling analysis in Sec. VI. Finally, in Sec. VII a qualitative interpretation of out findings is attempted, and the possible relevance of our results for experiments is discussed.

II. THE MODEL SYSTEM AND THE SIMULATION TECHNIQUE

We consider an ensemble of n chains, each consisting of N + 1 point-like beads joined together by N rigid links of length 1 (Fig. 1). As an interaction between the beads, we choose a Lennard-Jones form as in our previous work on single chains, 23,24 rl j = r l - r j'

(1)

where E: sets the scale of temperature, and we-arbitrarily-choose a= 0.41. For high densities of chains in the considered volume V, the interactions are expected to have little effect on the conformations of individual chains. 21 Therefore their average end-to-end distance should have the ideal value

(2)

As volume V we choose a rectangular box V = L 3 with periodic boundary conditions. For computational reasons, we wish to choose the size of this box as small as possible. The minimum possible size, for which the smallness of the box should not affect the chain conformations too much, is hence given by choosing

L~ 1N1 . (3)

Thus for N = 16 we choose L = 41 in the following.

We now assume that microscopic degrees of freedom not explicitly taken into account in this model act like a heat bath inducing stochastic changes in the conformation of our model chains. The simplest dynamics of a local character28 consists in trying rotations of two links around the axis joining their end points by an angle qJ

chosen randomly from the interval [ - 6.qJ, + 6.qJ] (Fig. 1). The parameter 6. qJ is arbitrary, and it is convenient to choose it so that about one half of the attempted rotations are successful. Thus one starts with a suitable initial configuration of all the chains, and chooses then randomly one of the chains as well as a bead i on this chain for which a rotation shall be tried. The transition

FIG. 1. Local motion of the freely jointed chain. A point i of the chain is moving on a circle through an angle ({J from its old position r, to a new one ri, while the positions of all other points on this chain (as well as of other chains) are held fixed.

probability WI (rl - r;) for this rotation is calculated, and compared to a random number 1'/ with 0 < 1'/ < 1. If WI> 1'/, the rotation is actually performed and the new configuration (in which bead i is on the site r;) is accepted; otherwise this rotation is rejected, and the old configuration is counted once more for the averaging.

As is well known, this standard Monte Carlo procedure29 ,30 is equivalent to a numerical realization of the master equation for the probability P({r j }, t) that the n polymers are in a state described by the coordinates {r j} at time t,30

:tp({rj}, t) = - ~wl(rl- r;)P({rj~, r l}, t)

+ Lw,(r: - r,)P({rj~I' r;}, t) (4) I

Here the transition probability per unit time, WI (rl - r;) has to be constructed so that it satisfies detailed balance with the equilibrium distribution Po({r j }) ,

wl(rl - r;)Po({rj~" r l}) =wl(r; - rl)Po({rj~, raJ, (5)

with [k B = Boltzmann's constant and T = temperature]

Po = (l/Z) exp( -J<'/kBT), Z =Tr exp(-JC/kBT) , (6)

the Hamiltonian JC being just the total potential energy, as there is no kinetic energy in our stochastic model

JC({rj})=LU(r,j) . (7) '~j

The sums in Eqs. (4) and (7) extend both over the N + 1 beads of each chain and over the n chains. A choice of the transition probability consistent with Eq. (5) is in our case [<'iJ<'=J<'({rj~, r,}) -JC({rj~, raJ]:

=Ie~xp[ <'iJC/k B T ]" if <'iJC< 01 (8a)

no intersection, if <'iJC2': 0 (8b).

, if the rotation would require link intersection. (8c)

Here 'Ts denotes the relaxation time of a Single free bead interacting with the heat bath, which is an arbitrary pa-

J. Chern. Phys., Vol. 75, No.6, 15 September 1981


2996 A. Baumgartner and K. Binder: Dynamics of entangled polymer melts

FIG. 2. A polymer chain in the basic cell I (encircled) and its images generated by the periodic boundary condition. Each cell has a volume V = L3.

rameter just setting the time scale for the process (as there is no intrinsic time scale defined by the Monte Carlo averaging, of course30). Since we want to simulate the effect of entanglements between the chains on dynamic correlation functions, we reject all rotations (Fig. 1) where moving the two links from their old positions (r/ - r/ -b r/ +1 - r/) to their new ones (r: - r/ -1> r/ <1 - r;) would intersect any other links [Eq. (8c)]. Of course, if we were interested in static thermal equilibrium averages only, this restriction would be unnecessary (and, in fact, it has not been applied in previous work21 - 24) .

In computing W/ (r/ - r~), care is necessary for taking into account the periodic boundary condition. Unlike the conditions for point-like atoms, any polymer chain usually will not be restricted to just the basic L 3 -cell, but will typically extend over two (or more) neighboring cells (Fig. 2). In order to avoid any ambiguity of labeling, we concentrate on that "image" of the chain for which r/ lies within the basic cell. We then consider the interaction between r/ and all other beads of the same chain (or other chains) as well as all images (or parts thereof) of the same chain inside the 26 neighboring cells ("minimum image convention,,30). On the other hand, for computing end-to-end distance, gyration radius and static structure factor of single chainS, we have to use the chain not restricted to cell I (which then was "cut into pieces" due to the periodic boundary condition); in order to avoid ambiguities we use these "images" for which point r/ is inside cell I to compute these quantities, which are defined as usual31 : end-to-end distance (R~),

(R~) = «r1 - rN+1)2) , (9a)

gyration radius (S~) and static structure factor S~~b(q) N N+1

<S~)= (N!1)2& j~1 «r/ _rj)2),

S~~b(q) =\N: 11~ exp[iq' r,l 12).

(9b)

Similarly, we also consider dynamic structure factors S~gb (q, t), Sine (q, t)

N+1 N<1

S~gh(q, t) =\(N: 1) ~ ttexP{iq . [riCO) - rit)]}) , (9c)

N<1

Slnc(q, t) = (N: l)tt exp{iq • [rl (0) - r/ (t)])). (9d)

The angular brackets in Eq. (9) denote both a time average [according to the master equation, Eq. (4)] and an average over the n chains. The single chain (sc) coherent dynamic structure factor is experimentally accessible by inelastic neutron scattering from a system where (ideally) one chain is deuterated while all others are protonated (or vice versa). 26 S~~b(q, 0) '" S~~b(q) is just the standard static structure factor of single chains in the polymer melt. On the other hand, we may also consider the collective coherent scattering from all chains (as it would show up in x-ray or light scattering from the dense polymer system),

ScOb(q, t)

=([n(N\ 1)] ~ ~exP{iq . [r/ (0) - rj(t)]}) - [n(N + 1)] .

(10) Note that in Eq. (10) the sums run over all pOints r/ of all polymers within the basic cell (1), different from Eq. (9), and the angular brackets denote the time average only. Due to the periodicity of the system, Eq. (10) is defined for discrete values of q only {qx, qy, qz=m(21f/L), with m = 0, 1, ... , L -1}, while Eq. (9) is well defined for arbitrary q.

Apart from these structure factors, we are interested in the center of gravity Rec of the chains,

1 N+1

Rec(t) = N + 1 ~ r l (t) , (11)

and their mean-square displacement which yields the diffusion constant DN of the chains,

In Eq. (12) we again perform both a time average and an average over the n chains. In order that Eqs. (9c), (9d), and (12) are meaningful for a finite system with periodic boundary conditions, the positions rl(t) in Eqs. (9c), (9d), and (11) are understood as not being restricted to the basic cell [as in Eq. (10)] or the basic cell and its nearest-neighbor images [as in Eqs. (4)(9b)] but one must let the polymers diffuse into arbitrarily distant images of the basic cell and keep track of the labels of these image cells, when computing Rec(t) from Eq. (11) or for computing the averages in Eqs. (9c) and (9d).

Finally consider two kinds of dynamic segment selfcorrelations

N+1

glnc(t) =(N: 1 &[rl(O) -rl (t)J2) , (13a)

N+1 )

gr(t) =(_l-L {[r/(o) - Rcc(O)] - [rl(t) - Rcc(t)W '(13b) N + 1 1=1




where the r,(t) have the same meaning as in Eq. (11). Equation (l3a) yields the total mean-square displacement of a bead; this quantity is related to Slnc(q, t} via the expansion

(14)

Equation (l3b) measures the mean-square displacement of a bead in a coordinate system in which the center of gravity of the chain is at rest. Looking for the difference between glnc(t} and gr(t) is a convenient tool for estimating the time scale needed that the diffusive motion of the polymer as a whole has a significant effect on Slnc(q, t}. An expansion similar to Eq. (14) also holds for the coherent structure factor,

S~~b(q, t) = N + 1 _ ~q2 N+1 N+1

X(N~1)ttf.t[r,(0}-r/(t}]2)+O(q4}, (15a)

or in normalized form

S~b(q, t}/S~~b(q, O} = 1 - ~q2 gcob(t} + o(l} ,

where N+1 N+1

gcob(t} == (N! 1}2 6= tt g, I(t) ,

g,it} ==([r,(O) -rit)]2 -[r,(O} -riO}]2)

(15b)

(15c)

is the dynamic correlation function related to the gyration radius, Eq. (9b). Equation (15) holds for q2(S~)« 1 only. An expression valid for larger wave vector is obtained if we assume that the variable q[r,(O} -r/(t}] is distributed according to a Gaussian. Then Eq. (9c) be-comes

N+1 N+1

S~~b(q, t} = N ~ 1 L L exp{-~q2([r,(0) -rp)]2)} '=1 1=1

N+1 N+1 (16a)

= N ~ 16= f.t exp{ -iq2([r,(0} - riO} F)} exp [- ~2g, it}]. For such times where S~~b(q, t} is not much smaller than S~gb(q, O} one must have 19'1(t)« 1, and then exp [- ~q2g1 it}] '" 1 - q2g1 I(t}. Hence,

S~gb(q, t} -1 _ q2/6 _1_ SCOb(q, O} - SCOb(q,O} N + 1

N+1 N+1

xL LglJ(t} exp{-!q2([r,(0} _r/(0}]2)} . '=1 1=1

(16b)

This expression will turn out to be useful for the later analysis of numerical data.

III. DYNAMIC STRUCTURE FACTORS IN THE DILUTE LIMIT

In this section we are concerned with the dilute solution case where the box contains a Single polymer chain only. Clear ly, in this limit our model is not at all realistic-it corresponds to the Rouse mode11 since no hydrodynamic effects2 are included; but this limit is a useful reference case and checks our programming procedures: some results on the dynamics of isolated chains (not confined to a box) were obtained earlier23,24; the good agreement which we obtain with the previous re-

suits show that the confining box does not affect the conformations of our chains significantly.

Theoretical predictions for this model are available only in the limit where interactions between the beads are neglected, and hence one can treat the chain as Gaussian. 27 One then finds, assuming a very long chain for which one can apply a continuum limit, 27 N- 00

Slnc(q, t} = exp( - !q2 ginc(t)] , (17a)

f +· ~ gtnc(t}==Z2 2 [1-exp(-2Wp2Itl}]! _. 11

(17b)

where W is a factor setting the time scale and Wit I » 1. Similarly, the coherent structure factor is given by Eq. (16a), where gilt} becomes, for N-oo(s=j - i), 27

gl/(t} ~ 2z2 ~ g [s2 /(4 Wi tl)],

g(u} == f" ~ e-uT2 = e-u - 2 -Iii f" dx e-,,2 , 1 T ./U

(18)

g(u} - const., for t - 00 •

Unfortunately, we are not aware of corresponding results valid for finite N, in which case the crossover from from the bead diffusion, ginc(t} cc lit! [Eq. (17)] to the diffusion of the chain as a whole [glnc(t) ==~DNt] would be exhibited, which is distinctly seen in our numerical results (Figs. 3-5). While in the continuum limit InSlnc(q, t}/q2 t 1/2 is independent of q [Eqs. (14)], Fig. 4(b} shows that for our model this is only correct for qZ$1. Clearly, for large q the fact that our model consists of rigid straight links of length Z joined together is seen in the structure factor. A similar enhancement of Stnc(q, t}/ q2t 1/2 is expected for more realistic models, too, reflecting some degree of rigidity on the microscopic length scale (monomer linear dimension), but the detailed behavior at such large values of q must depend on the chemistry of the polymer; clearly our model is not

10

10-'

• 10-2 • isolated chain

9cc;lt)

10-3

1 10 102 103 10' t

FIG. 3. Mean-square bead displacement gr(t) and meansquare displacement of the center of gravity geG (t) at kBT/E =3andkB T/E=O.4. for an isolated chain. N=16. Timeunit is one attempted rotation per bead.




a reliable description for the structure factor of real polymers at such large values of q. Hence in the following we will concentrate on the behavior for q1'!51. Fitting the parameter W to the proportionality constant of the observed glne(t) ex: v'ItT law, we have evaluated Eqs. (17a) and (17b) numerically. Figure 5 shows that in this manner a nearly quantitative fit of Seoh(q, t) is obtained, for q1< 1 and accessible times [for larger times the diffusion of the chain as a whole, which is not included in Eqs. (16)-(19), becomes important]. Thus it is shown that our model gives a reasonable account of the Rouse model even in the continuum limit, although N is not very large.

The results in Fig. 3 are in qualitative agreement with previous work, 23,24 the only difference being a change in time scale by a factor of about 2 due to the additional restriction, Eq. (8c), in the transition probability in the present study. This fact shows that the entanglements do not restrict much the motions of swollen chains in dilute solutions, as expected. Of course, this conclusion should be altered when we would study temperatures below the collapse transition temperature23

or the glass transition temperature24 of the single chain.

3

kaT =3 0.4 E j9J.

-t 112 .f3 -, 0.2 CT

u ~

(/)

c 10" 0.1 I

~ isolated chain

~ 0.1 -a-u .£

(/)

.5

0.01 0.1 10

ql/V3

FIG. 4. (a) Sine (q, t) of an isolated chain plotted vs time for several values of q and N=16, at kBT/E =3. (b) ZnSlne(q,t)/ q2 fft79ir plotted vs q for times t ~ 20. Full curve represents a (1 + q2/30)-1 law.

=:10 0. CT

.5 N

'CT I

symbol qlIVJ ~ 1

• 0.4 o 0.2

isolated chain

o 0.1 L..-__ --'LL.. ___ --i.".-___ ~----.J

1 10 102 • 103 104

t

FIG. 5. q-2 ln [S .... (q, t)/Scdl (q, 0») of an isolated chain plotted vs time for several values of q and N=16, at kBT/E =3. Full curves represent a numerical solution of Eqs. (18) and (19) with N=16, W=0.25, q2 Z2=3, as well as the asymptotic t1/ 2

behavior.

IV. MEAN-SQUARE DISPLACEMENTS AND STRUCTURE FACTORS FOR DENSE POLYMER SYSTEMS

Simulations have been performed for two temperatures (k BT/€=3 and k BT/€=0.4) and two concentrations (c = 2.5 and c = 10, which corresponds to n = 10 chains in box sizes L = 41 and L = 2.51, respectively; c =Nn/ (L/l)3. Since including the restriction Eq. (8c) slows down the rate of conformation changes by about an order of magnitude, it is clear that the other chains surrounding a chain are quite effective in forming a "tube" restricting chain motions. On the other hand, the slow relaxation required to go to about 105 steps per bead throughout, which needs about 102/h of IBM 370/168 computer time. Therefore it is neither possible to go to larger chain lengths N nor to explore the temperatureconcentration parameter space more systematically.

First we have studied how static properties of the chains change in the more dense system (for this purpose, the restriction Eq. (8c) forbidding link intersection is unnecessary and has been omitted, and by this quicker algorithm we then also found reasonable initial configurations for our dynamic studies). Table I summarizes our results for end-to-end distance and gyration radius.

Note that one expects from theoretical grounds21 that asymptotically both (R~) and (S~) are proportional to N as in the noninteracting case for dense polymer systems. This has also been observed experimentally. 32 However, this fact does not mean that also the prefactors in the laws (R2) ex: N, (S2) ex: N assume their "ideal" (noninteracting) value. We conclude that the prefactors depend on both temperature and concentration.

Let us now turn to dynamic quantities. A log-log plot of segment mean-square displacements and center of gravity displacements (Fig. 6) reveals that the behavior is still qualitatively similar to the Rouse behavior of isolated chains: one still observes laws gr(t) ex: t 112 and gcc(t) ex: t; only the proportionality constants in these laws are now much smaller. Only for high concentra-




TABLE I. Normalized end-to-end distance and gyration radius at various concentrations and temperatures for N=16.

k BT!E=3

c=O c= 2. 5 c=10

~ 1. 05± O. 01 1.12 ± 0.01 O. 92± O. 02 Nl2

(S}.rh !l.. (N+2) 1. 05± 0.01 1.13± O. 01 1. 24± 0.02 6 (N+1)

tions or low temperatures there is some evidence for a different behavior at intermediate times: while the loglog plot of gCG(t) exhibits much curvature, gr(t) ex: t 1/4• This behavior has been found earlier24 for an isolated chain below its glass transition temperature; it has been interpreted24 as an evidence of a "reptation" mechanism. We must note, however, that the absolute magnitude of the displacements in that regime is extremely small, gr(t) "" 1O-2Z2: hence we observe only very small readjustments of the bead positions in the glassy state of the dense polymer system, basically the bead positions are frozen in over these time scales. For larger time scales we again observe Rouse-like behavior gr(t) ex: t 1/2:

there is no evidence of a reptation dynamics where the displacements gr(t) should be of order Z2 in our model. 33

These findings are consistent with the behavior of the structure factors, Figs. 7-11. At not too large densities and not too low temperatures, (Fig. 7) pronounced t 1/2_laws are observed over several decades in time, while at high densities (Fig. 8) or low temperatures (Fig. 9) there is an intermediate regime (of about two decades in time) where one can observe a t 1/4 behavior,

(=2.5

~=04

( =10

~=3 ./ ....... .

.. '

......... ............... ~ ...... .

FIG. 6. Log-log plot of gr(t) andgCG(t) vs t for N=16 and several temperatures and concentrations.

kBT!E = 0.4

c=O c=2.5

O. 04± O. 01 O. 80± 0.02

0.14±0.01 1. 21 ± O. 01

both in the coherent and incoherent structure factor. The initially stronger increase of the coherent structure factor in Fig. 7 is due to corrections to the asymptotic behavior {resulting from the variation of g(u) [Eq. (18)], if one again uses the Rouse model for interpreting the data (cf. Fig. 5)}. The behavior of -In[Scob(q, t)/SCOb(q, 0)] proportional to t (instead of ..ft) at long times is due to the diffusion of the chain as a whole, as seen from a consideration of the structure in the coordinate system where the center of gravity of the chain would be at rest (Fig. 7, open circles).

It is also interesting to analyze the wave-vector dependence of the structure factors (Figs. 10 and 11). It is convenient to define the functions [cf. Eq. (16b)]

'0 !i

.<:; o

Vl_u

-.10"' 0--:c o u

Vl

.!f 10-2

0.7 0.4

FIG. 7. Log-log plot of Slnc(q,t) (upper part) and SS~(q,t)! SS~(q,O) (lower part) vs time for kBT!E=3, c=2.5, and various q. Solid curves represent the Rouse model [Eq. (9)J where the parameter W was adjusted (W lies in the range from 0.02 to 0.03 in our time units). Open circles represent the structure factor in a coordinate system in which the center of gravity of the considered chain would be at rest.




10-' ~ g

C' v1 c:

'10-2

c =10 kBT/e:= 3

2 t ql IfJ

07

2

.... 16 .. '

1.3

FIG. 8. Log-log plot of Sta. (q, t) (upper part) and s-s~ (q, t)/ S:~(q,O) (lower part) vs time for kaT/E =3, c=10, and various q.

rtnc(q, t) =q-2InSiDc (q, t)/[limq-2lnStnc (q, t)). (19b) q-O

From Figs. 10 and 11 it is seen that both r~~b(q, t) and

2

10-1 c::2.5

kaT/E::O.1.

( ::2.5 kaT/E::3

~oh (q.t)

qllV3

eo (=25 00

•• 200 o •• /"

kaTIE:: 3 • ,0'

10

qllV3 10

FIG. 10. Log-log plot of I"';" (q, t), I'f;. (q, t), and S'';'' (q, 0) 0.7 xI"~(q,t)/(N+1) vs q for kaT/E=3, c=2.5, and several

times. The static structure factor S'';'' (q, OJ/IN + 1) (open

g ~ 10-2

If)

E I

10-3

1 q, .:r ~ u

If)

~1O-1 &. 0 U

If)

E I

10-2

102 103

.... . .

"

. • . .. .... ... .

.. .

•• • tI' ."

10' t

1.6

1.3

FIG. 9. Log-log plot of St.e (q, t) (upper part) and S'~ (q, t)/ S'~ (q, 0) (lower part) vs time for kaT/E = O. 4, c = 2. 5, and various q.

circles) is included for comparison.

r Inc(q, t) are very close to unity as long as «q' RN)2) =tq2(R1>::s 1, as expected. For larger q the coherent structure factor is increasing while the incoherent structure factor is monotonically decreasing. The maximum

( =10 r;o, (q.tl ks T IE= 3

01

__ 1_. ,+q2130

O.Ol~--'--L..J-.u..w-,,-~-,---,-LW=

0.1 qtlV3 10

FIG. 11. Log-log plot of I"~(q,t) and ~i.(q,t) vs qforkaT / E

= 3, c = 10, and several times.

J. Chem. Phys., Vol. 75, No.6, 15 September 1981



of r~~b(q, t) is an effect of the normalization of the structure factor S~~b(q, t) by its starting value S~~h(q, 0), as seen from considering the quantity S~~(q, O)r~~b(q, t). If the Gaussian approximation, Eq. (16b), would be invoked, the latter quantity can be recast in the form

S~gb(q, O)r~gh(q, t)

N+1 N+1 IN+1 N+1

=L Lexp{-~q2<[rl(O) -r/O)]2)}glJ(t) L Lgllt) . ;;1 J;1 . 1;1 $;1

(19c) In the very initial stages, the movements of the individual beads are not yet correlated with each other, and hence glit) "" ol/[r;(O) -rp)]2), and hence initially the right-hand side of Eq. (20c) would be q independent. Therefore, Eq. (19c) implies r~gb(q, t) would vary with q just as [S~gb(q, 0) ]-1, and hence the minimum of S~~b(q, 0) at larger q implies a maximum of r~~b(q, t), which is removed by considering the quantity S~~b(q, O)r~gb(q, t). Of course, at large q the Gaussian approximation is not really adequate in our case, it would imply q;c (q, t) '= 1, independent of both q and t, which is not borne out by the data. In fact, the q-dependence of q~c(q, t) and S~~b(q, O)~gb(q, t} are rather Similar at initial times. This similarity indicates that the Gaussian approximation [Eq. (16)] is poor both for the coherent and incoherent structure factor.

It turns out that rlnc(q, t) can be represented in Ornstein-Zernike form,

(20)

where the "effective correlation length" ~(t) depends on time only weakly during the initial stages, and reaches a time independent value at later stages. We find that ~2(oo)"'l2/6 for c=2.5, k BT/(=3 while ~2(oo)"'l2/30 for c = 10, k BT / ( = 3. Thus this length is also of the order of (J (or even smaller), and hence again represents the tight packing of the beads in the dense polymer system.

V. MOTION OF A CHAIN IN A DENSE "NETWORK" OF FROZEN-IN CHAINS: A CASE FOR REPTATION

As mentioned above, our simulations gave clear indication that a dense polymer system by increasing the

101.----.----~----.---~----~--~~--~

network ••.• (=25

kaT _ 3 E -

FIG. 12. Mean-square bead displacementgr(t) and meansquare displacement of the center of gravity of a chain at kBT/( = 3 in an environment of other frozen-in chains at c = 2. 5.

3

1.3 -. 07 E <) c 0.4 U)

c I

-1 10

1/4

ca- - 16 = ~112 -t .2

.g ~13 0. 1 -1 .' • • 1 chain j 0 ~ 07 &~wo,k 'i16'/ 0.'

FIG. 13. Log-log plot of SllIe (q, t) (upper part) and ~fn (q, t)/ ~~(q. 0) (lower part) vs time for a chain in a frozen-in network of other chains at k BT / I: = 3 and c = 2. 5, for several values of q.

density or decreasing the temperature undergoes a transition from a fluid-like phase, to a glass-like phase. In the fluid-like phase, the chains are rather mobile and relax qualitatively similar to free chains in the Rouse model. In the glass-like phase, over many decades of time the displacements of the monomers are much smaller than the interparticle-distances. In neither of these phases we find a clear evidence that "reptation inside a fixed tube,,12-16 is the dominant mechanism by which polymer conformations relax.

At first sight, an obvious explanation for this failure to detect reptation would be that our chains are by far too short, and hence the chains are not enough entangled with each other. In fact, Klein15 has proposed that varying the chain length N at constant concentration in a polymer solution one would have a sharp onset of reptation at a critical value Nc' very much like a second-order phase transition. For N<Nc' the relaxation is predicted to be more or less Rouse-like. 15

In order to check whether with such short chains as available in our simulation one can see reptation at all we went back to the original problem12 of conSidering the movement of a chain in a frozen-in environment. In our Simulation, this is done very easily by using an equilibrium configuration of our system as initial configuration as above, but keeping all chains fixed in their initial pOSitions, apart from one mobile chain. This chain then has to move without intersecting any of the other chains of this frozen-in network. Figure 12 demonstrates that now a law glnc(t) - gr(t) <X t 114 does hold over such a broad range of times, that gr(t) in fact reaches values of order l2. As a result, the interpretation of the relaxation in terms of a reptation process is obvious. The law




+ z

CT .c 8

Vl

chom & network

choln • network

lIN

ql/Y3 10

FIG. 14. Log-log plot of rs~ (q, t), I1:ic (q, t), and Scab (q, 0) r~~b(q,t)/(N+1), vs q for a chain in a frozen-in network of other chains at c=2.5, k B T/€=3, and several times. The static structure factor ~ah (q, O)/(N + 1) (open circles) is included for comparison.

gco(t) 0: t 1l2, which is also quite distinctly observed, is

also consistent with the reptation mechanism. 12 These results are also confirmed by our finding for the structure factors (Fig. 13), which show t 1/4 laws after an initial t 112 law for short times. The behavior is qualitatively similar to the behavior of the dense polymer system with all chains mobile at very high densities (Figs. 6 and 8) or low temperatures (Figs. 6 and 9), with the important distinction that here we observe reptation over large distances, glnc(t) _12

, while previously glnc(t) - 10-212• In fact, for such small displacements of the beads of the other chains the physical situation is clearly equivalent to holding the other chains precisely fixedand hence we must recover a reptation-like relaxation in this initial regime. Thus, our results suggest that one can observe a reptation mechanism for very short chains if either the environment consists of fixed obstacles or the times considered are so short that all dis-

placements in the environment are just very small.

The coherent structure factor shows a behavior 0: t 1/4

only for very large q (which effectively probe nearestneighbor distances or distances less than the "tube" diameter, and hence behave similar to the incoherent structure factor; upper part of Fig. 13); for ql< 1 we find a behavior cc t 1/2, which is in agreement with a very recent prediction of de Gennes. 34

For completeness, Fig. 14 shows also the wavevector dependence of the reduced functions r~~b(q, t) and r Inc (q, t). It is seen that they are hardly different from the case with mobile environment (Fig. 10).

VI. SCALING ANALYSIS OF THE COHERENT STRUCTURE FACTOR OF ONE REPTATING CHAIN

For a better understanding of the results of the previous section, we develop a formal scaling analysiS of the coherent structure factor of one mobile reptating chain in a frozen environment. For the sake of clarity, we start from the Gaussian approximation for the structure factor S~gh(q, t) (which will be generalized subsequently), i. e., Eq. (16a). As a next step, we perform a continuum approximation, N - 00,

N+l N+l

1 1~ -LL'" =2 ds"', s= Ij-il , N + 1 1=1 1=1 0

(21)

which should become accurate in the limit of very long chains. Furthermore we can assume that static mean square distances behave then, for s» 1, according to

(22)

where 10 is a microscopic length of the order of (but not identical to) our link distance 1. From Eqs. (16a), (21), and (22) we obtain an expression, valid for (1 0 -INt1

«q« lot, S~~b(q,t)=2 f~ dsexp[-~(qlo)2slexp[-~q2glj(t)l. (23)

o The crucial assumption of our treatment is now the assumption that all displacements scale with the same characteristic time dependence 0: (wt)1I4, where w is a factor setting the time scale for reptation processes,

(t)-12( t)1I4'f{([rj(0) -r/(O)J2>} gj 1 - 0 w 1~(wt)1I4

(24)

withj(u) is a "scaling function" which can only be determined by a more explicit calculation. The motivation for this assumption, Eq. (24), is the fact that we expect ([rj(t) -rj(O)p>=([r/t) -r/O)p>o: 1~(wt)1!4 (Fig. 12), but at the same time must have «(rj(O) -rl{O)F>=([rj(t) - r it)]2> = l~s independent of time t. This condition would not be fulfilled, if ([ r l (0) - r j(t) F> would gl'OW with time according to a quicker law (such as 0: (wt)1!2) than ([ rj (0) - r l (t)]2> itself.

Inserting Eq. (24) into Eq. (23) we find, changing integration variables as u = s(wtt1

/\

S~~b(q, t) = 2(wt)1I41~ du exp [- ~(qlo)2(wt)1I4ul o

x exp (-! (qlo)2(wt)lIy(u) J • (25)

J. Chem. Phys., Vol. 75, No.6, 15 September 1981



We consider first the case! (q10)2(wt)1I4« 1. Expecting that I'D duj(u):= C < 0() exists, we note j(u) « u for large u, and hence we always may expand the second exponential, exp [-~ (ql 0)2(wt)1/y(u)] ",1 -~ (q10)2(wt)lIy(u), to find

sc ( ) 12 1.( )1/2 ( )2· ScOh q, t = (q 10)2 - 3 wt q10 C , (26a)

S~gh(q, t) = 1 _ .!...-(wt)1I2( 1 )4C sse (q 0) 36 q 0 ,

cob ,

(26b)

i. e., we find a result consistent with the variation a: t 112,

as observed (Fig. 13). In the opposite limit, ~ (q10)2(wt)1I4 »1, it is clear that the main contribution to the integral in Eq. (25) comes from u- [~(qlO)2(wt)1/4]-1_ O. Hence we may approximate the second factor as exp [- ~ (q10)2(wt)1/4 x j(u)] '" exp [- ~ (q10)2(wt)1/y(0)] , and obtain

S~~h(q, t) = (:1~)2 exp [-~(qlO)2(wt)1/y(0)] , (27a)

(27b)

As a result, we can predict that the relaxation rate of the coherent structure factor crosses over from a behavior a: t 112 [for! (q10)2(wt)1I4« 1] to a: t 1/4 [for ~ (q10)2(wt)1I4» 1]. For a description in the crossover region [for ~ (q1o)2(wt)1I4 '" 1] the explicit form of the function j(u) would be needed.

Of course, the use of the Gaussian approximation Eq. (16a) is a serious drawback of this derivation. Thus we generalize this approximation by considering the more general cumulant expansion (of which the Gaussian approximation keeps the first nontrivial term only),

(exp{iq' [ri(O) -rj(t)]})= (exp{A}) .,

= II exp( (A2")c/(2v)!] , (28) v:l

where we have already used the fact that (A) = (A 3) = (A 5) = ... = 0, and hence only even cumulants (A2")c need to be considered [(A2)c=(A2), (A4)c=(A4) -3(A2l, ... ]. Averaging over the angle between q and r/(O) - rj(t) in (A 2")c = ({q. [r/(O) - rj(t) ]2v})c' noting (cos2v

,<) = (2v + 1t1, we obtain a generalized cumulant expansion

(exp{iq' [r/(O) -rj(t)]})

(29)

where

([rl(O) -rit)]2)c' = ([r/(O) -rj(t)]2) ,

([rj(O) -rj(t)]4)c' =([rj(O) -rit)]4) -~([rl(O)-rJ(t)F? ,

(30)

and hence the structure factor becomes

S~~h(q, t)

= 2 ~'" dS:g exp {- (2vQ

: 1)! ([r/(O) - rJ(t)PV)c-}- (31)

This expression can be analyzed by suitably generalizing the scaling assumption Eq. (23) as follows:

([r/ (0) -- r J(t) ]2v)c' = Cv.l (z~st + 1~v(WW/YJS(wttl/4] (32)

[thej(u) in Eq. (23) is now interpreted asjl(u) of this more general formula]. In order to have the standard structure factor in the static limit, we have allowed for a static contribution in Eq. (31) only for the lowest cumulant, v = 1. Thus Eq. (25) is now generalized as

S~~h(q, t) = 2(wt)1I4j" du exp [-! (q1o)2(wt)1I4u] o

II r (q1o) v ( vl4 ( '" 2 ] X v:l exp L- (2v + 1)! wt) 'Iv u) . (33)

While the results for ~ (q1o)2(wt)1I4« 1 are still given by Eq. (26), we now find for ~ (q1o)2(wt)1/4» 1 that

'" 2 sc ( ) 12 II [ (q1o) ( )vI4'f (0)]

Scoh q, t = (q10)2 v=l exp - (2v + 1)! wt v

12 { ~ jv(O) [( )2( )1/4]V} () = (q 10)2 exp - f;t (2v + 1)! q10 wt . 34

In order to interpret this result, it is convenient to define a function F(x) in terms of its derivatives as follows

1 dVF(x) I 1 V!~ x=o= (2v+1)! jv(O), v=1,2, •.. ; F(0):=0.(35)

Then we finally obtain

S~~h(q, t) = (:l~)2 exp { - F[ (q1o)2(wt)1/4]} , (36a)

-In[S~~h(q, t)/S~~h(q, 0)] == F[ (q10)2(Wt)1/4] . (36b)

In the Gaussian approximation the function F(x) is linear, F(x) = [j(0)/6]x [Eq. (27b)]; deviations from linearity (for large x) hence measure the non-Gaussian contribution to the relaxation rate. Unfortunately, we are not able to make more explicit predictions for this function .

Recently de Gennes34 proposed that in the limit (q10)2(wt)1I4» 1 a nonexponential decay of the structure factor, S~~h(q, t) a: 1/[ (q1o)2(wt)1/2]. Such power-law decay would follow from the above analysis if F(x) a: In x2:

in this case we would get a q dependence, S~gh(q, t) a: 1/ (q8t 112), very different from Ref. 34. More work is needed to clarify this discrepancy.

VII. DISCUSSION

In this work, the dyanmics of a system of freely joined polymer chains was studied, segment mean-square displacements, center of gravity displacements, as well as coherent and incoherent structure factors were calculated. While for small wavevectors q the structure factors are qualitatively similar to that of the Rouse model in the continuum limit, the behavior at larger wave vectors exhibits pronounced deviations from the Gaussian approximation implicit in Eq. (16)-(18). We find that both the normalized relaxation rate SCOh(q,O)r~~h(q, t) and r Inc(q, t) are monotonically decreasing with increasing I q I. At larger concentrations and/ or lower temperatures, the relaxation of polymer conformations is slowed down by several orders of magnitude. Since then the segment mean-square displacements are small in comparison to the link length for the




full time of the simulation, we then interpret the response of the dense polymer system as glass-like over these time scales.

When one wishes to compare our findings to experiment, several problems arise: (0 The Monte Carlo time scale is defined only up to a factor which is the characteristic time for a stochastic jump of two neighboring links of our model to a new position (Fig. 1). Since our link 1 effectively represents a group of neighboring monomers, our time unit should correspond to about 10-10 sec, and hence the physical time scales studied corresponds to the regime from 10-10 sec to 10-5

sec. This is a regime which is accessible with recent neutron scattering techniques. 26 However, as our model is of rather qualitative character, a more precise assignment of time scales is hardly poSSible, and a similar ambiguity concerns the precise aSSignment of scales for wave vector, concentration and temperature. (ii) Our model is of strictly stochastic nature, there are no velocities of the links and hence no hydrodynamic interactions in the problem. While hydrodynamic interactions are known to be very important for dilute polymer solutions, their influence for dense polymer systems is thought to be negligible due to the high value of the viscosity of the system. In other words, the hydrodynamic interactions are screened over relatively short distances, and hence should have little influence on the asymptotic properties of very long chains. It is not clear, however, whether the influence of hydrodynamic interactions really is negligible for the properties of not so long chains, particularly when studying the structure factors at wave vectors not much smaller than ir).verse interparticle distances.

While we find a reptation mechanism for a mobile chain inside a network of frozen-in chains of the same length [i. e., ginc(r)a: t 1l4 over time scales large enough to reach displacements larger than a link length), we find reptation-like laws for the dynamics of the glassy state only over such short time scales where all the segment displacements are much smaller than a link length (and hence there is not much phYSical difference between the actual environment of each chain and a frozen-in environment). For longer times crossover to Rouse-like relaxation [i. e., ginc(t) a: t 112 and gcoh 0: t 112, etc.] occurs. Clearly, our chains may be too short and hence the topological constraints of chain entanglement are not really effective. On the other hand, both our observations and recent experiments26 suggest relaxation mechanisms which are not explicitly included in current theories on the dynamics of entangled polymer systems, which emphasize the reptation process. Within the latter picture, these relaxation mechanisms can be incorporated only if one abandons the treatment of the "tube" in which the chain reptates as a basically rigid object, and takes full account of local fluctuations in the shape of the tube ("tube rearrangement"), and the coupling of these fluctuations to the motion of the chain itself. While the necessity of such fully self-consistent treatments certainly has been realized, 11-16 no complete treatment which yields the structure factors considered in this paper is available. A first step would be to treat tubes as elastic rather than rigid objects (Fig. 15).

FIG. 15. Effective diffusion of the excess density of segments along the direct path from A to B.

Consider a thermal fluctuation which induces a slight change of conformation of the chain in region A of the tube. The local density of this chain in the part of the tube belonging to A will then also change slightly. This density change must lead to an elastic distortion of the tube via the microscopic couplings between the segments of the chain in the tube and the segments of other chains forming the tube. As a result, a density fluctuation which consists of the enhancement of the local density in region A and a decrease of density in region B can not only relax by diffusion of the density excess along the (typically rather long) path inside the tube: there is a competing relaxation mechanism in which the tube expands a bit in region A and contracts in region B, via the interaction over the chains in between A and B (Fig. 15). As a result, one would obtain an effective diffusion of the excess density along the direct (and much shorter) path from A to B, which would show up as a relaxation of the type exp(-DNq2t) in S~~b(q, t) for very large times, and as exp( - const q2t 1/2) at shorter times. Since parameters such as effective tube diameter, density (and microscopic nature) of entanglement points along a tube, etc., cannot be easily given quantitatively for any microscopic model such as the one studied in this paper, it is hard to check these ideas quantitatively. We hope that our study will stimulate more experiments to study the dynamics of polymer melts over a wide range of chain lengths, which is not yet possible by Simulations, as well as appropriate analytic theories. As a first step, we have proposed a "scaling" analysis of the coherent structure factor of chains reptating in a frozenin network, which seems to be consistent with our simulations but differs from previous predictions.

ACKNOWLEDGMENTS

We are grateful to D. Richter for stimulating discussions and for informing us about his experimental results prior to publication. We thank P. G. de Gennes for sending stimulating preprints (Refs. 16 and 34).

Ip. E. Rouse, J. Chern. Phys. 21, 1272 (1953). 2B. Zimm, J. Chern. Phys. 24, 269 (1956). 3J. D. Ferry, R. F. Landel, and M. L. Williams, J. Appl.

Phys. 26, 359 (1955). 4F. Bueche, The Physical Properties 0/ Polymers (Intersci

ence, New York, 1962); J. Chern. Phys. 25, 599 (1956). 5W. W. Graessley, J. Chern. Phys. 43, 2696 (1965); 47, 1942

(1967). '6A. Lodge, Trans. Faraday Soc. 52, 120 (1956); Elastic

Liquids (AcademiC, London, 1964).




7M • Green and A. Tobolsky, J. Chern. Phys. 14, 80 (1946). ~. Yamamoto, J. Phys. Soc. Jpn. 11, 413 (1956). 9G. Ronca, Rheol. Acta 15, 149, 156 (1976). lOA. J. Chompff and W. Prins, J. Chern. Phys. 48, 235 (1965);

A. J. Chompff and J. A. Duiser, J. Chern. Phys. 45, 1504 (1966).

11S. F. EdwardsandJ. W. V. Grant, J. Phys. A6, 1169, 1186 (1973).

12p. G. de Gennes, J. Chern. Phys. 55, 572 (1971). 13M. Doi and S. F. Edwards, J. Chern. Soc. Faraday Trans.

2 74, 1789, 1818 (1978). 14p. G. de Gennes, Macromolecules 9, 587, 594 (1976). 15J . Klein, Macromolecules 11, 852 (1978). lG p . G. de Gennes, J. Chern. Phys. 72, 4756 (1980). 17H• Hervet, L. L6ger, and F. Rondelez, Phys. Rev. Lett.

42, 1681 (1979). 18K• Osaki and M. Kurata, Macromolecules 13, 671 (1980). 19J . Klein, Nature 271, 143 (1978). 20W. Graessley, Adv. Polym. Sci. 16, 1 (1974). 21p. G. de Gennes, Scaling Concepts in Polymer Physics

(Cornell University, Ithaca, 1979), p. 219. 22A. Baumgartner and K. Binder, J. Chern. Phys. 71, 2541

(1979). 23A. Baumgartner, J. Chern. Phys. 72, 871 (1980). 24A. Baumgartner, J. Chern. Phys. 73, 2489 (1980). 25K• Kremer, A. Baumgartner, and K. Binder, Z. Phys. 40,

331 (1981). 26 D. Richter and B. Ewen (to be published).

27p. G. de Gennes, PhysiCS 3, 37 (1967). 28An alternative procedure involving a nonlocal dynamiCS is ob

tained by removing a link from one of the ends of the chain and putting it in a randomly chosen direction at the other end of the chain [see Ref. 25 and F. T. Wall and F. Mandel, J. Chern. Phys. 63, 4592 (1975); 1. Webman, J. L. Lebowitz, andM.H.Kalos, J. Phys. (Paris) 41, 579(1980)]. Thismechanism can be viewed as a coherent motion of the whole chain in an infinitesimally narrow tube by one link length per step, and is also denoted as an (artificial) "reptation." Although this mechanism is a valid procedure for calculating static averages, it is not at all an even qualitatively faithful description of the internal dynamics of a chain, and hence must not be confused with the physical reptation dynamics, with which we are concerned in the present paper.

29N. Metropolis, A. N. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chern. Phys. 21, 1087 (1953).

30 Monte Carlo Methods in Statistical Physics, edited by K. Binder (Springer, Berlin, 1979).

31p. J. Flory, Statistical Mechanics of Chain Molecules (lnterSCience, New York, 1969).

32D. G. Ballard, J. Schelten, and G. D. Wignall, Eur. Poym. J. 9, 965 (1973); R. G. Kirste, W. A. Kruse, J. Schelten, Makromol. Chern. 162, 299 (1973).

33Note that gr (t - 00) is of order l2 NX const, with a constant smaller than unity. Therefore for N = 16 g r (t - 00) cannot exceed a few times l2.

34p. G. de Gennes, J. Phys. (Paris) 42, 735 (1981).



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Dynamics of entangled polymer melts: A computer simulationhp0242/P_ABBinder_JCP81.pdf · there is no hope to bridge, in a computer simulation, the many decades of time from the microscopic