**UFRJ**

# Entangled Polymers

(Parte **1** de 4)

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 Grubm¤uller, Kurt Kremer (Eds.), John von Neumann Institute for Computing, J¤ulich, NIC Series, Vol. 23, ISBN 3-0-012641-4, p. 141-168, 204.

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 pro t or commercial advantage and that copies bear this notice and the full citation on the rst page. To copy otherwise requires prior speci c 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 f¤ur 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 dif culties 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. D¤unweg 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 (networks such as rubber and gels) are very complex materials. Besides the local chemical interactions 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 A 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 contribution to discuss generic aspects common to all polymers and then lateron go back to the question to what extent chemistry speci c 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 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 and hR2G(N)i = 16hR2(N)i 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 leading to a vanishing self density of the chains in a melt. In order to pack beads to the monomer density , 0(N1=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 rmly 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. D¤unweg 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 rst approximation, the other chains can be viewed as providing a viscous background and a heat bath. This certainly is a drastic oversimpli cation, which ignores all correlations due to the structure of the surrounding. The advantage of this simpli cation is that the Langevin dynamics of a single chain of point masses connected by harmonic springs can be solved exactly1. This was rst 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 d N2 and the viscosity N. This describes the dynamics of a melt of relatively short chains, meaning e.g. M 20 0 for polystyrene [PS] or M 20 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 signi cantly slower. Experiments show a dramatic decrease in the diffusion constant, D N 2:414, and an increase in the viscosity towards

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 nds then is illustrated in Fig. (1).

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 ndings 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 gure 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 con ned 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= 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 uctuations 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( =N), the topology of the surrounding does not change signi cantly on these intermediate time scales and a chain can only diffuse by reptating out of its original tube. This gives D ∼ N−2; d N3 as well as a plateau modulus at intermediate time scales. Considering the simplicity of the concept, the model describes many experimental ndings remarkably well. However, in spite of its successes, 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 de ned the con nement 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, remained largely unsettled until some very recent progress18 20. Also the discrepancy between the observed viscosity of η ∝ N3.4 and the predicted power law / N3 by now is safely attributed to a very slow crossover towards the asymptotic regime2. This is a little different for the diffusion of constant D, where the most complete and careful collection of data nds 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 eld approximation or develop a memory function formalism. Though they can reproduce experimental data to some degree, they fail when it comes to the microscopic motions of the polymers and the con nement 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 important role for the elasticity and related phenomena. Experiments on networks with the aThe correlation hole is de ned by the self density of the chain in the melt. The density ρ = (r)other + self(r) with self ' 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 in uence 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.

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 contributions (cross-links and entanglements) still conceptual problems exist. Also, the comparison of scattering experiments to theory/simulation only very recently made some signi cant 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 limited. Pulsed large eld 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. Simulations 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. D¤unweg. I now rst 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 D¤unweg 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 coef cient . 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,

Usually the model is solved for a ring with no free ends. If the chain ends are free, as for all linear chains, the rst and the last monomer have to be treated differently. For i = 1, the rst 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:

Note that this model does not contain any speci c 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 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

For small p=N, one recovers the usual result

with Nb2 = hR2i. 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 veri ed by MD for melts of short chains. The time correlation functions of the normal modes, p 1, are

where we have used the small p=N approximation for kp. The longest relaxation time is R = 1 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

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 brie y discussed in the introduction. For the Rouse model this gives

and the viscosity reads

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

Within the Rouse model g3(t) t for all times and the diffusion constant D(N) = limt!1g3(t)=6t is expected to reach the asymptotic value

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

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

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 < 0 is governed by the local chemical/model properties of the chains. Within the worm-like chain model in which the chain is a continuous exible path, 0 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 p g1(t). This continues until the chain has moved a distance comparable to its size hR2i1=2. After that is observed free diffusion with a diffusion coef cient 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 15 as well as NMR25,26and 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 coef cient through the monomeric friction coef cient ζ. Why these effects average to such a simple contribution still is not understood.

3.2 Entangled Chains

For chains which signi cantly exceed the length Ne, the motion is slowed down drastically. Clearest evidence for this slowing down comes from the diffusion constant D14,28,29. For

Several forms for the prefactor of D have been discussed in the literature. Similarly the viscosity increases so that 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 xed 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 uctuation effects like contour length uctuations 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 hR2(Ne)i) the constraints from the tube are showing up. The corresponding time is the Rouse time of a subchain of Ne beads, namely e 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.

(Parte **1** de 4)