Baixe Polymer Chains in Melts: Rouse and Reptation Models e outras Notas de estudo em PDF para Engenharia de Produção, somente na Docsity!
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 3-00-012641-4, pp. 141-168, 2004.
© c 2004 by John von Neumann Institute for Computing
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〈R^2 (N )〉 = K(N − I) ≈ KN (1)
and 〈R^2 G(N )〉 = 16 〈R^2 (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 ∝ 〈R^2 (N )〉^3 /^2 ∼ N 3 /^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 simulations^10. 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 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 exactly^1. This was first done in a seminal paper by Rouse^11 and about the same time in a similar fashion by Bueche^12. 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 ∼ N 2 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 observed^13. 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 ∼ N 3.^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).
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 1980^15. 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 Edwards^16 and deGennes^17. 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 T/ν = 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 ∼ N 3 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,
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. D¨unweg. 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 terminology^24 ), 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 Edwards^1 and a recent review by McLeish^2. 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 = 3kB T /b^2 , where b^2 = ``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 r˙i = k(2ri − ri− 1 − ri+1) + fi (3)
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(r 1 − r 2 ) and similarly for i = N. The distribution of random forces fi is Gaussian with zero mean and the second moment:
〈fi(t) · fj (t′)〉 = 6ζkB T δij δ(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 by^5
Xp(t) =
N − 1
N∑ − 1
i=
ri(t)cos
pπ(i + 1/2) N − 1
and p = 0, 1, 2,.. ., N - 1. Equation (3) can be rewritten as
ζp X˙p = kpXp + fp (6) where ζ 0 = N ζ and ζp = 2N ζ for p ≥ 1 and
kp = 8N ksin^2
pπ 2(N − 1)
For small p/N , one recovers the usual result
kp = 2π^2 k^2 p/N = 6 π^2 kB T N ``K
p^2 (8)
with N b^2 = 〈R^2 〉. 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) > < X^2 p (0) >
= exp(−t/τp), τp = ζ(``K )^2 N N 3 π^2 kB T p^2
where we have used the small p/N approximation for kp. The longest relaxation time is τR = τ 1 ∼ N 2. 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)
neutron spin-echo scattering experiments^27 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 length Ne, 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 ∼ N 3.^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 only^1. 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 g 1 (t) ∼ t^1 /^2. Only after the motion reaches a distance of the O(d^2 T ≡ 〈R^2 (Ne)〉) the constraints from the tube are showing up. The corresponding time is the Rouse time of a subchain of Ne beads, namely τe ∼ N (^) e^2. 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 ≈ dT N/Ne ∼ N/N (^) e^1 /^2. For t > τe, the chain performs essentially a one dimensional Rouse motion along the random walk like tube turning the t^1 /^2 power law for g 1 (t) into a t^1 /^4 power law. However, after the Rouse relaxation time τR, the monomers have only moved a distance of order
L^1 T/ 2 LT for N Ne. Following this regime the overall diffusion along the tube gives a second t^1 /^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 ∼ N 3 /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
for the mean-square displacement in space, g 1 (t):
g 1 (t) ∼
t^1 , t < τ 0 ; t^1 /^2 , τ 0 < t < τe ∼ N (^) e^2 ; t^1 /^4 , τe < t < τR ∼ N 2 ; t^1 /^2 , τR < t < τd ∼ N 3 /Ne; t^1 , t > τd
which is shown schematically in Fig. (3). For the motion of the center-of-mass g 3 (t) one expects
g 3 (t) ∼
t^1 , t < τe ∼ N (^) e^2 ; t^1 /^2 , τe < t < τR ∼ N 2 ; t^1 , τR < t
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 t^1 /^4 regime^3. 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 X p. The relaxation time of mode p, with N/p > Ne is enlarged by a factor of N/Ne, giving
τp,Rep =
N 〈R^2 〉
p^2
ζ π^2 kB T
N
Ne
= τR ·
N 3
3 p^2
N 3
p^2
The plateau modulus G^0 N 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 = 〈R^2 (Ne)〉^1 /^2 and thus Ne via G^0 N = 45 ρk NBe^ T. 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:
the stress in a given volume element of volume V = B^3. 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 3 kB T RiRj /L`K B^3. Summing over all strands passing through our test volume and replacing RiRj by the ensemble average 〈RiRj 〉 the microscopic stress tensor σij reads
σij =
3 kB T ρ L`K N
〈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 =
3 kB T ρ ``K
∂Ri ∂n
∂Rj ∂n
Transforming from the bead index to the contour index s and taking the statistical segment length (``K ) into account, this rewrites to
σij = 3kB T ρ N
∂ri ∂s
∂rj ∂s
for the microscopic formulation of the stress tensor in a polymer melt or network. In the affine picture this yields
G = kB T ρ/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 to^15
GoN = ρkB T Ne
for networks and taking additional relaxation mechanisms into account
GoN =
ρkB T Ne
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 modulus G′(ω) of polystyrene melts for many different chain lengths, from^1
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 of Ne 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.
〈∆r^2 |i−j|=n〉 = nl^2
1 + 〈cos θ〉 1 − 〈cos θ〉
n
2 〈cos θ〉
1 − 〈cos θ〉n
1 − 〈cos θ〉
where the asymptotic prefactor also is referred to as
c∞ =
1 + 〈cos θ〉 1 − 〈cos θ〉
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 〈∆r^2 ij 〉/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
2
3
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 aboutK = 1. 7 σto 3. 3 τ. From^33
In a similar way one can apply the analysis of the chain form factor S(k)
S(k) =
N
∑^ N
j=
eikrj^ |^2
|k|
where the index | k | denotes a spherical average. In a melt one finds
S(k) ∝
N (1 − 13 k^2 〈R^2 G〉) (^2) kπ 〈R^2 G〉 k−^2 2 πk (^2) kπ 〈R^2 G〉 k−^1 2π < (^2) kπ < (^2) πk O(1) (^2) kπ > (^2)π
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, k^2 S(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 〈∆r ij^2 〉 can be used to control the equilibration of melts.
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 , from^34
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 than O(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 follows^33 :
- 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
The force capped potential we use reads :
UFCLJ(r) =
(r − rf c) ∗ ULJ′(rf c) + ULJ(rf c) r < rf c ULJ(r) r ≥ rf c
In the present case, rf c is gradually reduced from the Lennard-Jones cut-off radius rc = 21 /^6 σ to 0. 8 σ 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 50000 τ , 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
2
1 10 100 1000 10000
<R 2 (n)>/n
n
a)
1
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), from^33.
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
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 difficulties^39. 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 role^2. 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 R^211 : R^222 : R^233 :∼= 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
