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John von Neumann Institute for Computing

Entangled Polymers: From Universal Aspects to Structure Property Relations

Kurt Kremer

published in

Computational Soft Matter: From Synthetic Polymers to Proteins, Lecture Notes, Norbert Attig, Kurt Binder, Helmut Grubmüller, Kurt Kremer (Eds.), John von Neumann Institute for Computing, Jülich, NIC Series, Vol. 23, ISBN 3-00-012641-4, pp. 141-168, 2004.

c© 2004 by John von Neumann Institute for Computing Permission to make digital or hard copies of portions of this work for personal or classroom use is granted provided that the copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise requires prior specific permission by the publisher mentioned above.

http://www.fz-juelich.de/nic-series/volume23

Entangled Polymers: From Universal Aspects to Structure Property Relations

Kurt Kremer

Max-Planck-Institut für Polymerforschung, 55021 Mainz, Germany

E-mail: kremer@mpip-mainz.mpg.de

Simulations are very versatile tools to study the relaxations and dynamics of polymer melts and networks. The fact that polymer chains cannot pass through each other poses special difficulties for analytic theories, while on the other hand many experiments are dominated by this fact. The paper discusses some basic concepts and conditions and ways to study such problems by computer simulations.

1 Introduction

In the previous lectures of this school the basic concepts to describe polymer melts or dense solutions were introduced and discussed (cf. chapters by J. Baschnagel et al, B. Dünweg and W. Paul). Among others random walks, excluded volume screening, the Rouse model as well as the reptation concept have been included. In the present chapter I mainly focus on the consequences of the fact that chains cannot pass through but only along each other, which eventually leads to the reptation or tube model. To discuss this I will shortly review the background coming from the Rouse model and then discuss similarities between polymer melts and networks, as they are observed experimentally and as they can be studied in different details in a simulation. Since most of the information gained by simulation is complementary to typical (scattering) experiments an altogether rather coherent picture has emerged over the many years of research1–3.

Dense polymer systems such as melts, glasses, and crosslinked melts or solutions (net- works such as rubber and gels) are very complex materials. Besides the local chemical in- teractions and density correlations, which are common to all disordered liquids and solids the global chain conformations and the chain connectivity play a decisive role for many physical properties. Local interactions determine the liquid structure on the scale of a few or at most a few nm. This question has been examined in detail by the contribution of W. Paul, where simulations of atomistically detailed melts are discussed. When we look at the dynamics of a polymer chain in such a melt local interactions determine the packing and the bead friction but not the generic properties4. It is the main focus of the present contri- bution to discuss generic aspects common to all polymers and then lateron go back to the question to what extent chemistry specific aspects play a role or make a difference. The consequences of the latter are also termed as structure property relations (SPR) in applied research5, 6.

To stick to simple situations we consider polymer melts or networks where the chains are all identical. They can be characterized by an overall particle density ρ and a number of monomers N per chain. As shown in previous chapters, the overall extension of the chains is well characterized by the properties of random walks7–9. With ` being the average bond length we then have (for N >> 1) for the mean square end to end distance

141

〈R2(N)〉 = `K`(N − I) ≈ `K`N (1) and 〈R2G(N)〉 = 16 〈R2(N)〉 for the radius of gyration respectively. `K is the Kuhn

length and a measure for the stiffness of the chain. This gives an average volume per chain of

V ∝ 〈R2(N)〉3/2 ∼ N3/2 (2) leading to a vanishing self density of the chains in a melt. In order to pack beads

to the monomer density ρ, 0(N 1/2) other chains share the volume of the very same chain. These other chains effectively screen the long range excluded volume interaction, since the individual monomer cannot distinguish, whether a non-bonded neighbor monomer belongs to the same chain or not. This general property is firmly established by experiment and many simulations10.

On very large scales polymers diffuse as a whole and the motion is well described by standard diffusion. However over distances up to the order of the chain size, the motion of a polymer chain is more complex, even though hydrodynamic interactions are screened and do not play a role. A detailed discussion of hydrodynamic effects is given by B. Dünweg in this school. The random motion of a monomer is constrained by the chain connectivity and the interaction with other monomers. To a very good first approximation, the other chains can be viewed as providing a viscous background and a heat bath. This certainly is a drastic oversimplification, which ignores all correlations due to the structure of the surrounding. The advantage of this simplification is that the Langevin dynamics of a single chain of point masses connected by harmonic springs can be solved exactly1. This was first done in a seminal paper by Rouse11 and about the same time in a similar fashion by Bueche12. In this model, which is commonly referred to as the Rouse model, the diffusion constant of the chain D ∼ N−1, the longest relaxation time τd ∼ N2 and the viscosity η ∼ N . This describes the dynamics of a melt of relatively short chains, meaning e.g. M ≤ 20 000 for polystyrene [PS] or M ≤ 2000 for polyethylene [PE], both qualitatively and quantitatively almost perfectly, though the reason is not well understood. Only recently some deviations have been observed13. The effects are rather subtle and would require a detailed discussion beyond the scope of this lecture. For longer chains, the motion of the chains are observed to be significantly slower. Experiments show a dramatic decrease in the diffusion constant, D ∼ N−2.414, and an increase in the viscosity towards η ∼ N3.41. The time-dependent modulus G(t) exhibits a solid or rubber-like plateau at intermediate times before decaying completely. Since the properties for all systems start to change in the same way at a chemistry- and temperature-dependent chain length Ne or molecular weight Me, one is led to the idea that this can only originate from properties common to all chains, namely the chain connectivity and the fact that the chains cannot pass through each other. This is what I am going to discuss in the subsequent chapters.

2 Polymer Dynamics and Network Elasticity

The plateau modulus G(t) can be derived from the restoring force of a polymer melt or network after a step strain. Experimentally usually an oscillatory shear is applied. What one finds then is illustrated in Fig. (1).

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Figure 1. Cartoon of the characteristic structure and response of a polymer melt (top) and a polymer network (bottom) after a step strain. For short chains (length N1) the restoring force decays to zero very fast, while for the longer ones with increasing length N, as indicated, a plateau in the time dependent modulus occurs, which is independent N

After a drastic fast initial decay, if the chains are long enough, the modulus G(t) stays almost constant at a value GoN for a long time until G(t) eventually decays to zero. Many related experimental findings can be found in Ferry’s book from 198015. Fig. (1) illustrates the similarities between cross-linked melts (rubber) and non-cross-linked melts. In both cases the value of the plateau modulus eventually becomes independent of N , which is either the chain length of the melt or the average strand length between two cross-links, if only N is large enough. These similarities lead to the famous reptation or tube concept by Edwards16 and deGennes17.

Edwards in his work on cross-linked networks introduced the concept of obstacles created by the other chains, resulting in a “tube” in which the monomers move. Fig. (2) shows a “historical” sketch of the development of this concept. First consider a network. The figure shows one strand of the network in the center marked by a thick line and a rather crude sketch of the surrounding. Edwards discussed how the black center chain could move around subject to obstacles created by all the other chains which in this case are part of the network. He noted that due to the topological constraints the chain is much more localized than expected just by the fact that the two ends are connected to a cross-link. All loops and their links in the system are conserved; they cause the strand to be essentially confined to a tube-like region (Fig. (2), middle part). This hypothetical tube, built by all the other chains, follows the coarse-grained conformation of the chain. The length scale of this coarse graining is called the entanglement length Ne and a sphere of the diameter dT of the tube typically contains d

1/ν T = Ne monomers, where ν = 1/2 is the random

walk exponent. Within this picture the strand can perform a quasi one-dimensional Rouse relaxation along that tube. Later, deGennes realized that the motion and spatial fluctuations of long chains in melts should be governed by the same mechanism (Fig. (2), lower part). When the chains are very long, most of the monomers are far from the chain end. Then, on intermediate time scales, these monomers do not realize that the ends are free. Since the density of chain ends is very small, O(ρ/N), the topology of the surrounding does not change significantly on these intermediate time scales and a chain can only diffuse by reptating out of its original tube. This gives D ∼ N−2, τd ∼ N3 as well as a plateau modulus at intermediate time scales. Considering the simplicity of the concept, the model describes many experimental findings remarkably well. However, in spite of its successes,

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several open questions remain, including how to formulate the reptation concept on a more fundamental basis.

Figure 2. Sketch of the historical development of the tube constraint and reptation concept. Starting from a network Edwards in 1967 defined the confinement to the tube, while deGennes in 1971 realized that for long chains the ends only play a small role for intermediate times.

A quantitative structure based model or theory of what an entanglement really is, re- mained largely unsettled until some very recent progress18–20. Also the discrepancy be- tween the observed viscosity of η ∝ N 3.4 and the predicted power law η ∝ N 3 by now is safely attributed to a very slow crossover towards the asymptotic regime2. This is a little different for the diffusion of constantD, where the most complete and careful collection of data finds D ∝ N−2.4 instead of N−2. Here it is not yet settled whether this is the “same” crossover effect21 or whether this is mostly due to the so called correlation holea effect. A detailed discussion however is beyond the scope of the present contribution.

There have also been a variety of nonreptation/tube phenomenological approaches, which only treat the interactions between the chains either in an averaged mean field ap- proximation or develop a memory function formalism. Though they can reproduce exper- imental data to some degree, they fail when it comes to the microscopic motions of the polymers and the confinement due to the surrounding strands. Here I do not discuss these approaches any further. Rather comprehensive reviews are given by McLeish2 and on the simulation aspects by Kremer and Grest3(and references therein). In a similar way our understanding of networks has improved over the last years. It has been known for a long time, especially also due to simulations, that the noncrossability of the chains plays an im- portant role for the elasticity and related phenomena. Experiments on networks with the

aThe correlation hole is defined by the self density of the chain in the melt. The density ρ = ρ(r)other + ρself (r) with ρself ' N/R3 ∝ N−1/2. Viewing the chains as very soft spheres these spheres like in a liquid of ordinary spheres have to leave their cage when diffusing around. This leads to a back jump correlation and slows down the diffusion beyond the influence of the microscopic bead friction. Note that the center of gravity of a chain not necessarily has to move in order to relax the overall chain conformation.

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same average strand length but cross-linked at different initial concentrations directly prove this point. However, when it comes to the question of identifying the different contribu- tions (cross-links and entanglements) still conceptual problems exist. Also, the comparison of scattering experiments to theory/simulation only very recently made some significant progress. There the relaxation phenomena are only partially the same as for melts, since the cross-links at the chain ends introduce boundary conditions for the tube contour which do not exist for uncross-linked melts22, 23.

From the above it is clear that computer simulations can be a very versatile tool to investigate such problems, since they offer the unique opportunity to have full control over the chain conformations while simultaneously typical experimental observables can be “measured”. Though CPU time intensive, simulations have played an important role over the years and will continue to do so.

Experimental quantities such as the viscosity, diffusion constant and modulus do not directly probe the microscopic motion of monomers on the chain. In contrast neutron spin-echo scattering covers the appropriate length scales, but the time range is rather lim- ited. Pulsed large field gradient spin-echo NMR is able to address the appropriate time and distance scales as well. However, an experiment typically probes one aspect only. Also samples are never really ideal and one must e.g. deal with polydispersity effects. Simula- tions do not suffer from such problems and can now be performed on melts of chains of 15 - 20 Ne, answering a number of unsolved questions.

3 Theoretical Concepts

The Rouse and reptation models are also shortly discussed in the contributions by W. Paul, J. Baschnagel, and B. Dünweg. I now first review some more of the background, restricting myself mostly to quantities which can be investigated directly by simulation. In a melt of homopolymers, the excluded volume interaction is effectively screened. There is no tendency for a chain to swell beyond the ideal random-walk dimension. Only the prefactor, or more precisely the Kuhn length lK (`K = `c∞ in Flory’s terminology24), is governed by the local monomer-monomer interactions.

3.1 Unentangled Chains - Rouse Regime

In the Rouse model, all the complicated interactions are absorbed into a monomeric friction and a coupling to a heat bath. It was originally proposed to model an isolated chain in solution, though it actually works very well for short chains in a melt. For chains in solution we refer to the chapter by Dünweg in this volume. Here I follow essentially the book of Doi and Edwards1 and a recent review by McLeish2. The polymer is modelled as a freely jointed chain of N beads connected by N - 1 springs, immersed in a Newtonian continuum. Hydrodynamic interactions are neglected. Each bead experiences a friction, with friction coefficient ζ. The beads are connected by a Hookean spring with a force constant k = 3kBT/b2, where b2 = ``K . Each bead-spring unit is intended to model a subchain of the real molecule, not a monomer. The equation of motion of the beads is given by a Langevin equation. For monomer i(i 6= 1, N) it reads,

ζ ṙi = k(2ri − ri−1 − ri+1) + fi (3)

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Usually the model is solved for a ring with no free ends. If the chain ends are free, as for all linear chains, the first and the last monomer have to be treated differently. For i = 1, the first term on the right hand side is −k(r1 − r2) and similarly for i = N . The distribution of random forces fi is Gaussian with zero mean and the second moment:

〈fi(t) · fj(t′)〉 = 6ζkBTδijδ(t− t′). (4) Note that this model does not contain any specific interactions between monomers

except those due to the chain connectivity. Since in a melt, the long-range hydrodynamic interactions are screened, it was suggested that this model could describe the motion of those chains, except that ζ arises from other chains rather than the solvent.

The Rouse model can be solved by transforming to normal coordinates Xp(t) of the chain. For a discrete monomer chain these are given by5

Xp(t) = 1

N − 1 N−1∑

i=1

ri(t)cos pπ(i+ 1/2)

N − 1 (5)

and p = 0, 1, 2, . . ., N - 1. Equation (3) can be rewritten as

ζpẊp = kpXp + fp (6)

where ζ0 = Nζ and ζp = 2Nζ for p ≥ 1 and

kp = 8Nksin 2 pπ

2(N − 1) · (7)

For small p/N , one recovers the usual result

kp = 2π 2k2p/N =

6π2kBT

N``K p2 (8)

with Nb2 = 〈R2〉. Since the random forces fp are not correlated, the Xp decouple and the motion of the polymer can be decomposed into independent modes.

For chains in a melt, the Rouse modes are eigenmodes of the chains. This has been verified by MD for melts of short chains. The time correlation functions of the normal modes, p ≥ 1, are

Rp(t) = < Xp(t) ·Xp(0) >

< X2p(0) > = exp(−t/τp), τp =

ζ(``K) 2NN

3π2kBTp2 , (9)

where we have used the small p/N approximation for kp. The longest relaxation time is τR = τ1 ∼ N2. For long chains in a melt, this equation is expected to only describe the relaxation of high p modes with N/p ≤ Ne. The relaxation modulus of the melt is given by

G(t) = ρkbT

N

p

Rp(2t). (10)

146

where ρ is the monomer density. This assumes that the single chain Rouse modes can be taken as eigenmodes of the whole melt. This certainly is an assumption, as briefly discussed in the introduction. For the Rouse model this gives

G(t) = pKBT

N

P

exp(−2tp2/τR) (11)

and the viscosity η reads

η =

∫ ∞

0

G(t)dt = ρkBT

2N

p

τp = π2ρkBT

12N τR =

ρζ

36 < R2 >∼ Nζ (12)

The self-diffusion constant D can be determined from the mean-square displacement of X0 = rcm, the center-of mass of the chain,

g3(t) =< (rcm(t)− rcm(0))2 > (13) Within the Rouse model g3(t) ∼ t for all times and the diffusion constant D(N) =

limt→∞g3(t)/6t is expected to reach the asymptotic value

D = kBT

Nζ (14)

for relatively short times. In simulations often the mean-square displacement of a monomer g1(t) as a function of time t

g1(t) = 1

N

N∑

i

= 1〈[ri(t)− ri(0)]2〉. (15)

is studied. Using the fact that the chain structure is that of a random walk, it is easy to show that

g1(t) ∼

   t1, t < τ0, g1(t) < ``K t1/2, τ0 < t < τR, g1(t) . 〈R2〉 t1, t > τR, g1(t) & 〈R2〉

(16)

For very short times, when a monomer has moved less than its own diameter, it is affected little by its neighbors along the chain. This short time regime, t < τ0 is governed by the local chemical/model properties of the chains. Within the worm-like chain model in which the chain is a continuous flexible path, τ0 is zero. For intermediate times, the motion of a monomer is slowed down because it is connected to other monomers. This can be viewed as the diffusion of a particle with increasing distance dependent mass. The actual mass at time t is just the number of monomers within a sphere of diameter

√ g1(t). This

continues until the chain has moved a distance comparable to its size 〈R2〉1/2. After that is observed free diffusion with a diffusion coefficient D ∼ N−1. It turns out experimentally that this extremely simple model provides an excellent description of polymer dynamics, provided that the chains are short enough. Measurements of η15 as well as NMR25, 26and

147

neutron spin-echo scattering experiments27 which probe the motion of the monomers agree to that. Results for molecular dynamics simulations on short chains also agree surprisingly well. For short chains, it turns out that the noncrossability of the chains, as well as the chain nature and chemical structure of the surrounding of each monomer mostly affects the prefactors in the diffusion coefficient through the monomeric friction coefficient ζ. Why these effects average to such a simple contribution still is not understood.

3.2 Entangled Chains

For chains which significantly exceed the lengthNe, the motion is slowed down drastically. Clearest evidence for this slowing down comes from the diffusion constant D14, 28, 29. For N ≥ Ne,

D ∼ N−2 . . . N−2.4 (17) Several forms for the prefactor of D have been discussed in the literature. Similarly

the viscosity η increases so that

η ∼ N3.4 (18) compared to N for short chains. In the reptation theory the motion of the chains is

viewed as Rouse motion of chains in a tube of diameter dT , which follows the coarse grained back bone of the chain. Since the chain is modelled as a random walk, forces at the ends have to keep a tube contour length LT ∼ N . In the original concept the tube is fixed and the chain had to completely move out of the tube to relax its conformation and any stress linked to the conformation. All other means of relaxation, such as constraint release due to chain ends or fluctuation effects like contour length fluctuations of the tube modify this scheme only somewhat quantitatively, but do not alter the qualitative picture. I thus will here discuss the simplest case only1.

For short time scales the motion of the monomers cannot be distinguished from that of the Rouse model, the motion of the monomer is isotropic and g1(t) ∼ t1/2. Only after the motion reaches a distance of the O(d2T ≡ 〈R2(Ne)〉) the constraints from the tube are showing up. The corresponding time is the Rouse time of a subchain of Ne beads, namely τe ∼ N2e . After this time the monomers can diffuse along the tube only. By this forward and backward motion, the chain explores new space and slowly destroys the original tube. The contour length of the tube LT can be estimated to LT ≈ dTN/Ne ∼ N/N1/2e . For t > τe, the chain performs essentially a one dimensional Rouse motion along the random walk like tube turning the t1/2 power law for g1(t) into a t1/4 power law. However, after the Rouse relaxation time τR, the monomers have only moved a distance of order L

1/2 T  LT for N  Ne. Following this regime the overall diffusion along the tube gives

a second t1/2 regime for the motion in space. There the elemental step is the displacement of a polymer chain along its tube. The longest relaxation time is the mean lifetime of the tube. The initial tube will be destroyed when one of the segments has visited O(N) different contiguous sites. This requires a time τd ∼ N3/Ne. In this time the chain has moved a distance comparable to its own size, therefore the diffusion constant D is expected to scale as N−2. The theory predicts the following general power-law sequence

148

for the mean-square displacement in space, g1(t):

g1(t) ∼

  

t1, t < τ0; t1/2, τ0 < t < τe ∼ N2e ; t1/4, τe < t < τR ∼ N2; t1/2, τR < t < τd ∼ N3/Ne; t1, t > τd

(19)

which is shown schematically in Fig. (3). For the motion of the center-of-mass g3(t) one expects

g3(t) ∼

   t1, t < τe ∼ N2e ; t1/2, τe < t < τR ∼ N2; t1, τR < t

(20)

These power laws are schematically indicated in Fig. 3 Direct experimental evidence for these intermediate time regimes has been seen NMR and diffusion of polymers at an interface, however simulations were the first to observe the crossover into the t1/4 regime3. Indirectly the tube confinement is also seen by scattering experiments.

Figure 3. Schematic plot of the mean-square displacement for a monomer in the Rouse and the reptation model.

The reduced mobility also affects the relaxation of the long-wavelength modes Xp. The relaxation time of mode p, with N/p > Ne is enlarged by a factor of N/Ne, giving

τp,Rep = N〈R2〉 p2

ζ

π2kBT

N

Ne = τR ·

N3

3p2 ∼ N

3

p2 (21)

The plateau modulus G0N which was at the beginning when the similarities between melts and networks have been discussed, is determined from rheological experiments. Within the reptation model, it is related to the tube diameter dT = 〈R2(Ne)〉1/2 and thus Ne via G0N =

4 5 ρkBT Ne

. The prefactor 4/5 arises from tube length fluctuations, which allow for a somewhat more efficient relaxation under deformation, which is not the case for networks. The typical asymptotic properties are again summarized in the table:

149

Rouse Regime Reptation Regime N < Ne, M < Me N  Ne, M Me

Diffusion D ∝ N−1 D ∝ N−2 Relaxation Time τR ∝ N2 τd ∝ N3 Viscosity η ∝ N η ∝ N3 Modulus GoN G

o N = 0 G

o N ∝ kTNe

(t τd) It is one of the central current research topics on entangled polymer systems, how to

determine Ne uniquely and how this actually quantitatively determines GoN . Before we go into that we discuss shortly the connection to classical network elasticity1, 15, 30. For this we go back to the random walk statistics of the chains and the idea of the Rouse model that an individual chain experiences all its surrounding only as a viscous background not affecting the conformations at all. For this we start with the probability distribution of chain ends:

P (R) =

( 3

2σL`K

)3/2 exp

( −3R2/2L`K

) (22)

Since no specific inter-chain interactions are considered, P (R) is the probability of a chain in a melt or a strand between two crosslinks in a network to be separated by a vector R. The total number of possible conformations with a separation of R is the fraction Ω(R) = ΩtotP (R) of all random walk conformations. This defines the entropy for a given R as

S = kB`KΩ(R) = const− 3kBR2/2L`K (23) with the energy U = 0 (chain is viewed as a free random walk)

F = U − TS = 3kBR 2

2L`K (24)

for the conformation dependent contribution to the free energy of a chain with fixed end positions. This leads to an entropic force

f = −∇F (R) = −3kBT L`K

R (25)

the force of a linear elastic spring. The whole chain acts like a bond in the Rouse model. A random walk at a temperature T is a Hookean spring with spring constant ∼ T/N . This almost trivial fact is the basis of all theories of network elasticity as well as a basis for the view of the Rouse chain to be a string of beads with linear elastic springs in between. No let us assume a perfect network with the crosslinks fixed in space. Under deformation λ they move affinely with the macroscopic deformation of the box. Then the force contributed from each network strand is

f = −3kBT L`K

( λ− 1

) R (26)

Each chain contributes with a force constant 3kBTL`K to the restoring modulus of the system. Since there are ρ/N chains per unit volume present, we can use this to estimate

150

the stress in a given volume element of volume V = B3. The probability that one chain cuts a plane with index j(j = 1, 2, 3) is just Rj/B. The force component transmitted through this plane contributes from this subchain to 3kBTRiRj/L`KB3. Summing over all strands passing through our test volume and replacing RiRj by the ensemble average 〈RiRj〉 the microscopic stress tensor σij reads

σij = 3kBTρ

L`KN 〈RiRj〉 (27)

In analytic theory now usually a continuous path is taken for the chain and then taking the crossover to an infinitely small volume element, using L = ` · N and for small N : Ri/N ≈ ∂Ri/∂N one gets for the stress tensor

σij = 3kBTρ

``K

〈 ∂Ri ∂n

∂Rj ∂n

〉 (28)

Transforming from the bead index to the contour index s and taking the statistical segment length (``K) into account, this rewrites to

σij = 3kBT ρ

N 〈∂ri ∂s

∂rj ∂s 〉 (29)

for the microscopic formulation of the stress tensor in a polymer melt or network. In the affine picture this yields

G = kBTρ/N (30)

In networks a variety of modifications are introduced. The above equation assumes an affine deformation of the whole system. This certainly is not the case! In an alternative approach the positions of the network beads at the box surface are fixed and the whole rest is only subject to connectivity constraints. In this phantom network model the modulus is somewhat reduced. In general, disregarding any entanglement contribution one can write

G = ρkT

N [ν − hµ] (31)

where ν is the number of active strands in a network (no closed loops ending in the same point, no free ends ...) and µ is the number of crosslinks. The quantity 0 ≤ h ≤ 1 interpolates between the two models and is often taken as an adjustable parameter.

In the case of very long chains, N  Ne so that the simple Rouse model assumption that the chains or strands act as non-interaction random walks a hierarchy of models trying to identify constraints has been introduced. They range from the constraint junction fluctu- ation approach of Flory all the way to the tube model. Ideas along these lines are employed to understand the spatial fluctuations in randomly crosslinked systems under strain. Taking all this one can view the strands in the Edwards tube as a continuation of entropic springs of length Ne, which leads to15

GoN = ρkBT

Ne (32)

151

for networks and taking additional relaxation mechanisms into account

GoN = 4

5

ρkBT

Ne (33)

for melts. This is the formula commonly used to determine Ne from measurements of the plateau modulus. A typical example in given in Fig. (4).

Figure 4. Experimental determination of the frequency dependent modulusG′(ω) of polystyrene melts for many different chain lengths, from1

Considering the previous discussion simulations nowadays do not focus that much any more on the question whether entanglements exist and whether they are relevant but more on their consequences and on the problem of how to quantify them properly. Typical topics are:

• Crossover to asymptotic power laws for η and D, what additional relaxation mecha- nisms exist and what kind of additional constraints, mechanisms might slow down the diffusion?

• How and why do different measurements ofNe give rather different results? Whatever are the links between different experimental observations?

• What are the effects of dilution and/or solvent quality in semidilute systems? • What does the tube “look like”? • Tube deformation and tube relaxation, quantitative and qualitative differences be-

tween networks and melts

• Swelling behavior with and without the influence of charges (polyelectrolyte gels). • Polydispersity effects, mixtures as well as the influence of branched additives • Structure property relations, can we predict Ne from the chemical structure of the

polymer or any static measurement?

This list is not comprehensive, but it shows that the consensus that entanglements dom- inate many crucial properties is not really the end of a long development but rather defines many new research opportunities. To follow this by simulation we first have to describe how to prepare “good” initial states.

152

3.3 Simulation Models, Equilibrated Melts

Most of the following will be discussed for a freely jointed bead spring chain. The ex- tension to other models is straight forward10, 31, 5. To study entanglement effects one can construct special model situations and study those. However more information applicable to experiment can be gained by studying “well equilibrated samples”. A variety of meth- ods to generate those has been discussed in other chapters. The most direct way would be to set up a system in an arbitrary state and equilibrate by running it for a few relaxation times. This however would need CPU time which by itself is of at least of the order of the running time of the whole project. Employing standard MD one typically can reach around (2 − 4) × 105 particle steps per second on a single processor. A typical (large) time step is about 10−2 τ for a dense Lennard Jones system (see below). For that system with τe ≈ 2000τ (estimated from the mean square displacement of inner beads, taking the stress relaxation after a step strain τe is estimated close to 10000 τ ), one needs about (2− 5)× 104 steps or at least one second of CPU time per particle to reach τe32. Thinking of about M = 100 chains of N ≈ 800, about 10Ne, a system just big enough to com- pete with typical modern scattering experiments, for the fully flexible bead spring system defined below, we get τD ≥ τe · (N/Ne)3 = 103τe, equivalent to 103 sec per particle. Thus the above system on a single processor would require at least M ·N · 103sec, more than 20000 hours. Modern parallel systems reduce this time significantly, however, more complex models, such as semiflexible chains or models which more closely resemble the atomistic structure easily increase the time by several orders of magnitude time. For typ- ical all atom models one can estimate the increase in CPU time roughly by the following. Taking about the same number of monomers for one entanglement molecular weight Ne (which is typically not enough) one can multiply the above time by the number of atoms per monomer (e. g. 4 for PE and 16 for PS ...) and the average ratio of the integration time steps, which is between 103 and 104. To reach the same relaxation, about 104 - 105

times more CPU time is needed, which is beyond reach. This estimate did not take into account the typically higher complexity of the force calculations! These numbers illustrate that the computers of the nearer future also will not solve the problem. Also, the tendency to reach high computing power by increasing the number of processors is not helpful since our systems are moderate in size but need long time runs. To study in very detail the final dependency of the elastic response of a system, one would need roughly 20 Ne and run that system for a time τ  τe and eventually close to τd. Fortunately, for equilibration there are ways out. However, this requires a solid knowledge of the average equilibrium conformation33. In a melt or dense solutions the chains assume random walk statistics for all contour lengths L  lk. Once a “sample” which reaches the asymptotic regime is available, one can use this as a reference state even for much longer chains. Values for internal distances, 〈∆r2ij〉, have to fall on a general master curve. There are two typical options, internal distances or the chain form factor. For the internal distances 〈∆r2ij〉 one finds

〈∆r2i−j〉 ∝ { | i− j |2 | i− j | ` < `k | i− j | | i− j | ` `k (34)

For distances r  l beyond the local sphere or bead packing the crossover between the two regimes is approximated by the worm like chain model:

153

〈∆r2|i−j|=n〉 = nl2 (

1 + 〈cos θ〉 1− 〈cos θ〉 −

1

n

2〈cos θ〉 ( 1− 〈cos θ〉n

) ( 1− 〈cos θ〉

)2

) (35)

where the asymptotic prefactor also is referred to as

c∞ = 1 + 〈cos θ〉 1− 〈cos θ〉 . (36)

with 〈cos θ〉 the average bond angle “measured” between subsequent bonds. This func- tion has to be determined in a complete simulation of shorter chains. Thus in a melt a system specific master plot gives 〈∆r2ij〉/l | i− j | vs l | i− j | the corresponding contour length. A characteristic example is given in Fig (5). This shows characteristic deviations from the theoretically expected curve for small distances, where the bead packing is domi- nant. For larger distances, the more flexible chains still show significant deviations. There the approximate formula of Eq. (5) is only useful as a general guide.

1

1.5

2

2.5

3

3.5

1 10 100 1000

<R 2 (

n) >/

n

n

Figure 5. Normalized internal distances for different chain lengths for a standard bead spring LJ polymer, for different `K ranging from about `K = 1.7σto3.3τ . From33

In a similar way one can apply the analysis of the chain form factor S(k)

S(k) = 1

N

〈 | N∑

j=1

eikrj |2 〉

|k|

(37)

where the index | k | denotes a spherical average. In a melt one finds

S(k) ∝

  

N(1− 13k2〈R2G〉) 2πk  〈R2G〉 k−2 2π`k 

2π k  〈R2G〉

k−1 2π` < 2π k <

2π `k

O(1) 2πk > 2π `

(38)

Besides the initial decay the form factor describes one characteristic, chain length in- dependent curve. Fig. (6) shows a typical example.

Often the so-called Kratky plot, k2S(k) vs k, is shown. Deviations from slope zero in the k−2 regime are easily to detect, even by eye. This helps for a quick check. Master plots of S(k) or 〈∆r2ij〉 can be used to control the equilibration of melts.

154

Figure 6. Chain form factor S(q) vs q for fully flexible LJ chains in a melt. Chain lengths range from N = 25 to N = 350. For fits the Debye function was used. The straight line indicates a slope of q−2, from34

3.4 Preparing an Equilibrated Melt or Network, Specific Systems

A rather straight forward approach is to run a system by a reptation or slithering snake algorithm. This beats the slow realistic dynamics by a bit more thanO(N), however, is not really applicable for dense continuum systems. For semidilute solutions or lattice polymers at moderate densities and for moderate chain lengths this however is very appropriate. For dense continuum systems as well as systems with realistic chemical details such an approach fails. The strategy we meanwhile follow is as follows33:

• Simulate a melt of many short, but long enough chains (N`  `K ) into equilibrium by a conventional method

• Use this melt to construct the master curve or target function for the melts of longer chains (cf. Fig. (6))

• Create non reversal random walks of the correct bond length, which match the tar- get function closely, especially at the longer distances. Introduce, if needed beyond the intrinsic stiffness of the bonds, stiffness via a suitable second neighbor excluded volume potential along the chain. (This might be a bit larger than the one of the full melt!)

• Place these chains as rigid objects in the system and move them around by a Monte Carlo procedure (shifting, rotating, inverting..., but not manipulating the conforma- tion itself) to minimize density fluctuations

• Use this state as starting state for melt simulations • Introduce slowly but not too slowly the excluded volume potential by keeping the

short range interaction, taking care that in the beginning the chains can easily cross each other (see below)

• Run until the excluded volume potential is completely introduced. Control internal distances permanently to check for possible overshoots, deviations.

• Eventually support long range relaxation by so called end bridging35, 36 or double pivot moves

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Independent on the details of the procedures, it is important to continuously compare the actual structure to the master or target curves. Ref.33 also demonstrates some typical deviations from the master curve as they can occur during the set up.

As an example I now discuss the above steps along a recent publication33 on simple bead spring chains37, 38. An extension to other chains is straight forward. We consider bead with a unit mass. All beads interact via a purely repulsive LJ potential, to model the excluded volume interaction.

UrLJ =

{ 4 {

(σ/r)12 − (σ/r)6 + 14 } r ≤ rc

0 r ≥ rc (39)

with a cutoff rc = 21/6σ. The beads are connected by a finite extensible non-linear elastic potential (FENE)

U (r) FENE =

{ −0.5R2ok`n(1− (r/Ro)2) r ≤ Ro ∞ r > Ro (40)

in addition to the Lennard Jones Potential. The parameters are usually taken as k = 30/σ2, Ro = 1.5σ in melt simulations. The temperature T = /kB and the ba- sic unit of time is τ = σ(m/)1/2. Volume and temperature are kept constat and the bead density is ρ = 0.85σ−3. The temperature was kept constant by coupling the motion to a Langevin thermostat. An alternative, which does not screen hydrodynamics is the DPD thermostat (cf. contribution by B. Dünweg). The average bond length with these parame- ters is 〈`2〉1/2 = 0.97σ. Chains of N beads each are considered.b

For this model in a melt of density ρ = 0.85σ−3 the bond length is 〈l2〉1/2 = 0.97σ with c∞l2 = llk = 1.7σ2. Since, in our case, there are no torsional barriers, the monomer packing relaxes quickly. It locally depends very sensitively on the ratio of bond length to effective excluded volume of the beads. Both do not allow any conclusion for the overall relaxation of the melt. The local packing only characterizes an equilibration on the smallest length scale considered and very small systematic deviations on that scale can lead to significant deviations from equilibrium at the large length scales on the order of the chain extension. For this system chains of N = 350 beads (which was not really needed in that case) were equilibrated by brute force simulations running in the background. Fig (5) also includes some data for stiffer chains. From these data we get the target function:

〈R2(| i− j |, N)〉/ | i− j |∝ ``k, | i− j | ` `k (41)

Having this reference state, we can proceed as mentioned before. The crucial steps are the last two. To introduce the exclude volume we proceed as follows:

1. Use a force-capped-Lennard-Jones-potential

2. increase the push-off time to make the procedure “quasi static”.

bIn addition a bending stiffness can be introduced via an effective three body interaction. With cosΘi = (r̂i,i−1 · r̂i,i+1), with r̂i,i−1 = (ri − ri−1)/ | ri − ri−1 | and accordingly r̂i,i+1 the bending potential reads Ubend(Θ) = kΘ(1− cosΘ) By variation of kΘ the model can be tuned from very flexible to very stiff.

156

The force capped potential we use reads :

UFCLJ(r) =

{ (r − rfc) ∗ULJ′(rfc) + ULJ(rfc) r < rfc

ULJ(r) r ≥ rfc (42)

In the present case, rfc is gradually reduced from the Lennard-Jones cut-off radius rc = 21/6σ to 0.8σ which is significantly smaller than the relevant interparticle distances, as illustrated in Fig. (7).

Figure 7. Illustration of the “force capped” Lennard Jones interaction used to introduce step by step the full excluded volume of the chains.

Force-capping has the advantage that this form of the soft potential systematically ap- proaches the true potential. The typical time we used to introduce the full potential was about 50000τ , which is of the order of twice the Rouse time of a chain of N=100 beads. Such a procedure relatively safely equilibrates the internal conformations of the chains as Fig. (8) shows. There the results of a too fast and a proper push off procedure are shown for illustration. The too fast introduction shows the characteristic overshoot of the internal distances for small n. It is however also often rather important that the average end to end

1

1.2

1.4

1.6

1.8

2

1 10 100 1000 10000

<R 2 (

n) >/

n

n

a)

1

1.2

1.4

1.6

1.8

2

1 10 100 1000 10000

<R 2 (

n) >/

n

n

b)

Figure 8. Mean square internal distances for chains of length N = 25 (+), 50 (x), 350 (*) and 7000 () for a too fast pushoff procedure (a) and the slower version as described in the text (b), from33.

distance is close to the desired value, even though the overall number of chains might be relatively small. Then methods, which break and recombine chains, called the double-pivot or bridging algorithms are used. Note that they very effectively relax and manipulate large

157

distances, however affect small scales only extremely slowly! Having prepared melts by the above procedure we now can either perform standard simulations to study the melt or crosslink them in order to create networks by different methods.

The preparation of networks deserves another short comment.In the literature a number of different crosslinked systems has been studied extensively34, 39–43. A typical range of systems in illustrated in Fig. (9)

Figure 9. Typical crosslinked systems ranging from randomly crosslinked melts via various versions of endlinked polymer networks to the idealized lattice structure systems and the theorist’s dream of an “olympic gel”

The randomly crosslinked system certainly is closer to experiments however encoun- ters many difficulties39. Especially the dangling chain ends cause the relaxation to slow down dramatically (The time grows exponentially!), because it is linked to a retraction of the arms. Such effects are also known from other situations where branched polymers play an important role2. In addition the disorder or fluctuations are quenched. Thus simply running a system longer does not improve the data in comparison to experiment. For this one would need “many” medium sized systems or single extremely large ones. Thus we mostly stick to melts in the following.

3.5 Entanglement Analysis: Melts and Networks

In the following I will give a few examples of numerical experiments for melts and in some cases networks as well. The first almost trivial question is, whether the tube exists, the chain motion is confined and whether the noncrossability clearly makes a difference. Experimentally the determination of confinement is somewhat indirect and does not easily reveal the shape of the confining volume. Here simulations offer a unique way of visual inspection of the chain displacements. To do so one simply can plot the initial conformation of the chain and on top of that subsequent conformations. This is shown in Fig. (10) for network chains.

From other investigations we know that 35 < Ne and 100 > Ne, which is nicely confirmed by the pictures. Depending on the view onto the initial conformation (the typical conformation of a random walk is a flattened thick cigar with a ratio of principal moments of inertia of R211 : R

2 22 : R

2 33 : ∼= 11.8 : 2.5 : 1) one clearly can identify the confinement.

The diameter of the confining tube actually nicely fits results from earlier simulations on the mean square displacements. In a similar way one can also visualize the motion in polymer melts. There, however, due to a continuous release in the number of constraints, this for shorter chains, is more difficult to visualize. Thus one uses a small trick. What we now plot is not the bare conformation, but the back bone of the statically averaged chain contour, namely

158

Figure 10. Time evaluation of the conformation of chains in an endlinked network of functionality f = 4,N = 100, where all network strands have the same length. The chains are, for clarity shown without the network. The indicated tube diameter was determined in an independent melt simulation, from40.

Ri = 1

n+ 1

i+n/2∑

j=1−n/2 ri (43)

This smoothens the contour and by a proper choice of n should be close to the back bone of the tube. Here n = 35 was used. These contours can then be plotted vs time and compared to a different system, where the chains can easily cross each other. For the full LJ interaction the barrier height is around 70kT . One can introduce a potential, where the chains can cut through each other, but the pressure, 〈R2〉 and the bead friction ζ remain unchanged. These two systems are compared in Fig. (11) for chains of N = 350. While the confinement is not as easily visible as for the networks, the difference between the two

Figure 11. Time evolution of the coarse grained back bone of the chains for a melt of N = 350 bead chains for times up to the Rouse time τR. The left panel shows chains with the full excluded volume, while for the right panel the crossing barrier is only a few kT, from34.

cases is striking, and nicely supports the tube idea.

159

Another direct test which shows the relevance of the noncrossability of the chains/network strands is shown in Fig. (12) For these networks with diamond lattice

Figure 12. Elongated diamond lattice networks, where the only source of disorder are random links between net- work loops. The strands are colored due to their stretching (similar to the stress they carry) from small stretching (blue) to strong stretching (red). From3

connectivity were prepared. In order to fill space with the condition that the strands obey random walk statistics several interpenetrating networks are needed. This is the source of random linking of the different subnets. By this, “short topological paths” through the net are created, which under strong elongation carry most of the stress. Note that in this special example all “short chemical paths” have the same length.

After these intriguing pictures we can move to the data analysis which demonstrates the confinement not that directly, however is more closely related to possible experiments. The first to look at is the mean square displacement of the middle beads and the chains. Many studies on this have been performed38, 44–47, for an overview see3 and computer simulations actually were the first to observe the slowing down of the motion clearly38, 48. Fig. (13) shows results from a system of fully flexible polymers. More complicated models have

0.1

1

10

100

102 103 104 105 106

g 2 (t)

, g 3(

t) [σ

2 ]

t [τ]

slope 1/2

slope 0.26

slope 1/2

slope 1

Figure 13. Mean square displacements g2(t) (open symbols) and g3(t) (closed symbols) for chain length N = 350 (), 700 (◦) and 10000 (4). The straight lines show power laws as guide to the eye. The local reptation power laws g2(t) ∝ t1/4 and g3(t) ∝ t1/2 are verified with remarkable clarity32.

160

been studied by Faller and Müller-Plathe49, 50 and45 for instance. Fig. (13) shows that the expected power laws are reproduced with a remarkable accuracy. From this a tube diameter of 7σ ≤ dT ≤ 8τ can be deduced, corresponding to an Ne of about Ne ≈ 30, which is very small. To check the overall consistency the onset of the t1/4 regime, the g1, g3 values at that time etc. are analyzed. In all cases the results roughly agree! Experiments also by now are able to analyse the chain motion directly and display the expected power laws. They however typically only cover a too small time window. To overcome this problem rather different temperatures are employed. It should however be noted that the shift in temperature not only affects the ratio kBT/ζ but via conformational changes also Ne! Here simulations, despite of their many shortcomings are still superior.

Another important experimental technique is the neutron spin echo (NSE) method27. In NSE experiments the motion of the segments can be obtained by measuring the expectation value of the time-dependent single-chain structure function

S (k, t) = 1

N

i,j

〈exp(ik · (ri(t)− rj(0)))〉 . (44)

For reptating chains deGennes calculated an expression for S(k, t) neglecting the initial Rouse-like motion51. In recent simulations a slightly modified version is applied52, 53 which is identical to de Gennes result when a Gaussian model of the tube is explicitly assumed. We used this formula together with a correction for very long times (however, for the experimental NSE results or our present set of data this modification may be neglected):

S(k, t)

S(k, 0) = {[

1− exp ( −(kd/6)2

)] · f ( k2b2

√ 12Wt/π

) (45)

+ exp ( −(kd/6)2

)} × 8 π2

∞∑

p=1,odd

exp ( −tp2/τd

)

p2 ,

where f(u) = exp(u2/36)erfc(u/6). Note, that the derivation of deGennes does not take into account chain end effects and thus the above formula may only be applied in a narrow range of k-space 2πRG . k .

2π dT

. Also since the initial Rouse-motion is not explicitly described, this formula only applies for t > τe (The short time motion only enters through the inverse friction coefficient W = kT /ζ``K ).

What one finds there is a rather strong dependency of chain length N or the value of dT deduced from the plot. Even for N = 10000 one finds Ne significantly larger than the result from the mean square displacements. Though these are, to my knowledge, the best data available, there might be some problems with the statistics (N = 10000). Note that for these data the conformations of the longest chains were not prepared by the procedure discussed before, while all the shorter chains simply ran long enough!

The standard experimental method to determine Ne,p (for clarity we index Ne deter- mined from the plateau-modules with an additional index p) is by measuring the plateau modulus under oscillatory shear15, 1. Alternatively, it is also possible to measure the normal stress decay σN (t) in a step strain elongation. Since the latter is much simpler to perform in a simulation, volume conserving step strain runs were performed for four different am- plitudes λ = 1.25, 1.5, 1.75 and 2.0. The normal stress was σN was determined by the

161

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 10000 20000 30000 40000 50000 60000 70000

S (k

,t) /

S (k

,0 )

t [τ]

q=0.2σ-1

q=0.4σ-1

q=0.6σ-1

Figure 14. S(k, t) for different chain lengths: N = 10000(M), N = 2000(), and N = 700() and the centered subchain of length 550 of the same N = 700 chains (). Continuous curves are simultaneous fits to the N = 2000 data corresponding to dT = 9.6σ. The dotted curve is a simultaneous fit to the N = 700, 550 monomer subchain with a tube diameter of dt = 12.9σ.32, 54

10

20

30

40

50

60

70

80

1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0

N e

lk

g1(t)/D S(q,t)

G0

Figure 15. Ne as a function of the Kuhn length of the chains `k determined from the mean square displacements, the scattering function and the modulus.

microscopic virial-tensor. The plateau-values of the stress were fitted to the stress-strain formulas for classical rubber elasticity (CRE)55 σN = G

( λ2 − 1λ

) and to the Mooney-

Rivlin (MR) formula56 σN = 2G1 ( λ2 − 1λ

) + 2G2

( λ− 1λ2

) to determine G0N . MR gives

G0N = 0.0105kBTσ −3 while CRE gives G0N = 0.008kBTσ

−3. It is known experimen- tally that MR slightly overestimates the modulus while CRE always underestimates it. The standard formula1 to calculate Ne,p,

G0N = 4

5

ρkBT

Ne,p , (46)

gives Ne,p ≈ 65 for the MR fit and Ne,p ≈ 80 for the classical formula. Both values

162

0.01

0.1

1

0.1 1 10 100

D (N

)/D R

(N )

N/Ne,p

slope -1

Figure 16. Scaled diffusion constant D(N)/DR(N) vs. scaled chain length N/Ne,p for polystyrene (•) (Me,p = 14600, T = 485K), polyethylene () (Me,p = 870, T = 448K), PEB2 (N) (Me,p = 990, T = 448K), the present bead spring model (4) (Ne,p = 72), the bond-fluctuation model for Φ = 0.5 () (Ne = 30) and tangent hard spheres at Φ = 0.45 (◦) (Ne = 29). All data are scaled withNe,p from the plateau modulus or with 2.25Ne from g1(t). From32

are much higher than our previous estimate, Ne = 32. From a theoretical point of view this discrepancy might not be relevant, since the prefactors in the reptation model are not rigorously determined. For practical issues, however, like comparing results from various experiments and simulations the difference is relevant.

At this point we can conclude that we are able to clearly observe and demonstrate en- tanglement effects. However the different values originating from different experiments prohibit to give an overall consistent picture. Fig. (15) shows the outcome of such investi- gations as we plot the entanglement molecular weight as a function of stiffness for different “experimental” tests. This illustrates that one has to decide on one definition ofNe. Taking the original discussion on the chain confinements, the results taken from the modulus be taken as a reference state. If done like this, one can use this to scale other measurements. Fig. (15) shows the scaled diffusion constants for a variety of experiments. The actual dif- fusion constant D(N) is normalized by the hypothetical Rouse regime diffusion constant DR(n) as it can be extrapolated from short chain simulations or experiments. However, ut should be noted that a similar plot employing e. g. S(k, t) will lead to different fits for Ne for different polymers or models systems.

3.6 Structure and Property Relations

Though a general picture emerges, there is still the problem of the structure property rela- tion. Is there a way to predict Ne or at least one value for a specific experiment based on the conformations, meaning the chemistry alone?

We saw that with increasing stiffness, the value of Ne decreased, however, the ratio of the different values obtained from the mean square displacements, scattering and modulus did not remain constant. Also, the modulus always predicted the largest value. This seems to hold for simple bead spring chains, while there are chemical systems where this is not

163

the case. Polycarbonate as an example of a technically very important polymer (e. g. for compact discs) has an Ne ≈ 6 from the modulus, while the mean square displacement suggests via a scaling from Fig. (15) a value of Ne ≈ 1257. Obtaining just one measure- ment obviously is not sufficient! Thus there is a significant need for a predictive theory or ansatz, which determines a value of Ne from the chain conformations, which then can be used to predict the other quantities, especially the modulus. How different Ne, or Me re- spectively is shown in the table. A typical ambient temperatures (note that theMe depends on temperature just as the overall chain extension does) one obtains the following values:

PE PS PDMS BPA PC (Polyethylene) (Polystyrene) (Polydimethyl-Siloxane) (Polycarbonate)

Ne 100 170 135 5 - 7 Me 1400 18000 10000 ≈1700 Even within one class of materials significant differences can be observed, as indicated

in Fig (17). There different modifications of polycarbonate are shown, which within ex- perimental accuracy have about the same Kuhn length `K but rather different entanglement lengths, as indicated58, 59.

Figure 17. Structure of different polycarbonate modifications, the respective entanglement molecular weight are Ne ≈ 6, 10, 15 from top to bottom.

These results demonstrate a key problem for a successful link between the more basic science, generic phenomena, oriented research and modern materials science. Our un- derstanding has to become more quantitative. This holds for many fields in the science of microscopically or nanoscopically designed molecular structures as well as for more conventional bulk systems. Because of the huge relaxation times a factor of 2 in the en- tanglements change the viscosity also by a factor of 21.4 ≈ 2.6 and similar reduces the

164

diffusion constant. Uncertainties of this order in predictions are relatively small, however, are of significant experimental and technological relevance.

Is there a way to understand these deviations as well as the sensitivity to chemical structure and conformation? Over the years several criteria have been developed to identify entanglements. One set of publications deals with an analysis of the topological structure of a network and related approximations for melts60. For this the problem however is that multi chain / multi ring effects can in all cases (Gauss integrals, Alexander polynominals ...) can only be taken into account up to a finite order. How important many chain effects can be is illustrated by Fig. (18) for an example of three chains only42, 61.

Figure 18. Constraints between two chains (red, black) depend on the presence of the 3rd chain.

In principle one would need for long chains an “infinite” hierarchy of such interactions. On the other hand, in order to observe Ne is dependent on N for N  Ne a constant average density of constraints along the back bone of the chain is needed to provide the tube. Iwata and Edwards61, 62 tried to overcome this by looking at the local values of the Gauss integral upon numerical integral. All this suggests that the local chain-chain packing plays a crucial role. Therefor relatively early first attempts were made to relate Ne to the number of different cains within the value d3T

63, 64. All these attempts were not really satisfactory. More recently a collection of many moduli data of different polymers Fetters et al18, 19 suggest that the modulus is related to the so called chain packing length p = N/ρ〈R〉, which is a good measure for the spatial packing, in a way that G ∼ p−3. Many polymers seem to follow that within a ±20% corridor. This however still does not explain how this packing is able to produce the entanglement mesh. Beyond that Everaers et al.20 just recently performed a topological analysis of a number of different polymer melts and dense solutions. They studied simple bead spring chains of different stiffness as well as simplified model for polycarbonate. In all cases the data of G0N derived from the topological determination of the primitive path and the analysis from analyzing normal stress differences after a step strain quantitatively agree. Thus it seems that (1) there is a way to relate purely static quantities to the modulus with reasonable accuracy and (2) the tube model can quantitatively be linked to a topological analysis of the state of the melt. With this we arrive at the question of the very beginning, namely to what extent does the fact that the chains do not pass through each other, influence the relaxational properties of these fascinating ultra soft solids or viscoelastic liquids. Based on recent

165

progress in various fields extensions to more complicated systems such as mixtures of linear and branched polymers or more stiff polymers, as they can be found in biological systems, are underway.

4 Summary

In the first part of this contribution certain aspects concerning the dynamics and stress relaxation in polymer melts or networks, complementary to the other contributions has been discussed. In addition typical “measurements” to study reptating polymer melts have been mentioned. From this a basic ingredient to study the dynamics of polymer melts and networks, namely the equilibration and preparation of suitable starting conformations has been presented in more detail. This is a crucial condition for the next step, namely to study both the dynamic development as well as the conformational properties of a system. Then a few typical examples of numerical tests performed over the last years illustrated, what can be done by simulations and dense model systems employing modern numerical methods and computers.

Acknowledgement

I would like to thank many collaborators over many years of research. The topics described above benefitted especially from longstanding fruitful interaction with G. S. Grest and R. Everaers.

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