Linear Viscoelastic Behavior of Unentangled Polymer

Melts via Non-Equilibrium Molecular Dynamicsa

Jose Gines Hernandez Cifre,*1b Siegfried Hess,1 Martin Kroger2

1 Institut fur Theoretische Physik, Technische Universitat Berlin, D-10623 Berlin, Germany2 Polymer Physics, ETH Zurich, Wolfgang-Pauli-Str. 10, CH-8093 Zurich, Switzerland

Received: March 18, 2004; Revised: August 30, 2004; Accepted: October 14, 2004; DOI: 10.1002/mats.200400021

Keywords: linear viscoelasticity; molecular dynamics; non-equilibrium; polymer melt

Introduction

The molecular dynamics (MD) method applied to meso-

scopic models for macromolecules is known as a powerful

approach enabling the simulation of the dynamics of poly-

meric fluids of arbitrary architecture. Often, results are in

remarkable good agreement with known rheological and

structure-resolving experimental findings. Extensive MD

simulations of polymer melts have been performed ear-

lier[1,2] in order to study the different regimes of diffusive

motion of polymer chains in melts. These studies allowed to

determine a critical chain length (/molecular weight)

characterizing the dynamical crossover from the qualita-

tively different Rouse[3] to reptation[4] regimes. Further, the

flow behavior of polymer melts, subjected to simple

shear,[5,6] uniaxial elongational flow,[7,8] and stress relaxa-

tion[9] has been investigated using non-equilibrium mole-

cular dynamics (NEMD). For reviews on this method we

refer to ref.[5,10] In ref.[6], a rheological crossover from

Rouse to reptation had been obtained via NEMD. Based on

these findings, we know that the polymer melts to be studied

in this article (N< 100 beads) should be in the unentangled

regime. The viscoelasticity of unentangled polystyrene

melts has been investigated in detail in ref.[11,12] Computa-

tional work devoted to the study of the model polymer melts

under oscillatory shear flow is scarce, although the linear

viscoelastic response is of great practical importance.

Frequency-dependent properties of simple fluids have been

studied via NEMD[13–15] (linear viscoelasticity under elon-

gational flow) and extensively discussed in ref.[13,14,16]

Summary: We present and assess the use of non-equilibriummolecular dynamics (NEMD) simulation method for thedirect study of the linear viscoelastic behavior of polymermelts. The polymer melt is modeled by a collection of repul-sive, anharmonic multibead chains subjected to small amp-litude oscillatory shear flow. We present results for chainlengths below the critical entanglement length and obtaingood agreement with theoretical results for the viscoelasticbehavior of melts of low molecular weight. The range ofoscillation frequencies attainable in the simulation is of a fewdecades. Thus we use, as in experiments, a time-temperaturesuperposition rule to extend the frequency domain. As a sideresult, we confirm the so-called Cox-Merz rule.

Snapshot from a non-equilibrium molecular dynamics (3D)simulation of a polymer melt with 100 chains and 40 beads.

Macromol. Theory Simul. 2004, 13, 748–753 DOI: 10.1002/mats.200400021 � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

748 Full Paper

a PACS: 83.10.Mj,61.25.Hq,83.50.Ax.b Present address: Dep. de Quımica Fısica, Universidad de

Murcia, Campus Espinardo, 30100 Murcia, Spain.E-mail: jghc@um.es

(non-linear response under periodic external fields). Experi-

mentally, the linear viscoelastic properties of polymeric

systems are usually obtained by subjecting the sample to a

small strain amplitude oscillatory shear flow characterized

by the dynamic moduli (G0 and G00). This article is devoted

to show the use of NEMD to study polymer melts under

oscillatory shear flow. Relaxation time of melts strongly

increases with chain length and therefore frequencies cor-

responding to the terminal zone of dynamic moduli where

the viscous behavior dominates are such small that direct

simulations targeting this regime are still very time con-

suming. This work is devoted to summarize our findings

using unapproximated MD. The results should serve to test

approximate methods and single segment theories. Alter-

nate methods (variance reduction,[17] transient time corre-

lation function,[18] average stress fluctuations,[19–21]

beyond-equilibrium MD[22]) particularly useful at low

shear rates (and certainly also useful at low frequencies)

have been proposed and should be mentioned, but would

not be employed in this brief article.

Model and Simulation Technique

The dynamical model consists of an ensemble of interacting

linear, multibead polymer (FENE) chains contained in a

cubic box with periodic boundary conditions. Each chain

consists of N beads connected by anharmonic, finitely

extendable non-linear elastic (FENE) springs, UFENEðrÞ ¼�1

2kR2

0lnð1 � r2=R20Þ, between adjacent beads separated by

a distance r. With the choice for the spring coefficient k¼ 30

and the maximum extensibility of the spring R0¼ 1.5, we

follow ref.[1,5,6] The spring coefficient is strong enough to

make bond crossings energetically and virtually impossi-

ble. In order to model excluded volume, all beads interact

with a repulsive Lennard-Jones (LJ) potential, ULJðrÞ ¼4e½ðr=~ssÞ�12 � ðr=~ssÞ�6 þ 1=4�, for r� rc, and ULJ(r)¼ 0

otherwise. Here, r is the distance between pairs of beads, ~ssis the distance where the LJ energy vanishes, e is the

minimum energy value, and rc ¼ 21=6~ss is the cutoff radius.

Along this work, all the quantities are reduced to LJ units:

the energy and length parameters of the LJ potential, e and

~ss, and the particle mass, m. Thus, reference units for time,

density, temperature, and pressure are tref ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffim ~ss2 =e

q,

rref ¼ ~ss�3, Tref¼ e/kB, and pref ¼ e ~ss�3, respectively. To

transform our dimensionless results to dimensional results

suitable for comparison with experimental data, our results

must be just rescaled depending on the physical units of the

quantity of interest and the polymer system at hand. For

a representative set of experimental data for different

polymers along with the corresponding simulation data

(FENE model) in dimensionless form, we refer to Table 1 of

ref.[6] In order to study bulk properties, periodic boundary

conditions and the nearest image convention together

with Lees-Edwards boundary conditions are employed.

Newton’s equations of motion for the system are integrated

by a velocity-Verlet algorithm and thermostatted using

velocity rescaling as in ref.[5] Thus, the macroscopic velo-

city of the beads due to oscillatory shear is superimposed to

their peculiar (thermal) velocities. While we employed a

simple velocity rescaling method, we should mention

alternate methods, which for the simulation at hand, lead to

practically the same results. There is the ‘‘van Gunsteren-

Berendsen’’ thermostat,[23,24] which uses a factor at each

time step that depends on the deviation of the instantaneous

kinetic energy from its average value, corresponding to the

desired temperature. Although this method does not repro-

duce the canonical ensemble it is widely used, and usually

gives the same results as the rigorous method discussed

below, although caution must be taken using it. It reprodu-

ces the correct average energy but the distribution is wrong.

Therefore, averages are usually correct but fluctuations are

not. The ‘‘Hoover thermostat’’[25] reproduces the canonical

ensemble, even though it also involves velocity rescaling.

The idea is to introduce an additional degree of freedom

describing the external bath and its corresponding velocity.

Additional kinetic and potential energy terms, coupled to

the particles momenta, are added to the Hamiltonian. The

whole system is conservative and obeys Liouville’s equa-

tion. An alternate, similar method called ‘‘Nose thermo-

stat’’ also uses an additional variable for the momenta

rescaling. It was introduced earlier[26] and works in the scal-

ed coordinates, and reproduces canonical distribution for

positions and scaled momenta. The spring coefficient k

chosen for the present simulations is small enough to use a

reasonable integration time step of typically Dt¼ 0.001–

0.005 (LJ units) while integrating Newton’s equation of

motion. In practice, we adjust the time step to ensure that the

maximum interparticle force stays below an upper limit,

and sample data for each separate frequency over an interval

of 100 periods.

Linear Viscoelasticity andTime-Temperature Superposition

Among the several small strain experiments used in rheo-

metry to study linear viscoelasticity phenomena, sinusoidal

(or oscillatory) rheometry is often employed to characterize

the frequency dependence of polymer solutions and

melts.[12] In this controlled-strain test, a sample is deformed

in a oscillatory shearing flow up to reach a maximum strain

g0 (amplitude of strain), which must be small enough to be

in the linear regime, i.e., dynamic moduli are independent

of g0. Then, the time-dependent shear stress s(t) that arises

because of the deformation is monitored. This controlled

strain experiment is usually represented as follows: g¼ g0

sin ot and s¼ s0 sin(otþ d) where o is the frequency of

the oscillation and s0 the stress amplitude. For viscoelastic

fluids, the resultant stress is delayed in time with the strain

Linear Viscoelastic Behavior of Unentangled Polymer Melts via Non-Equilibrium Molecular Dynamics 749

Macromol. Theory Simul. 2004, 13, 748–753 www.mts-journal.de � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

by a phase angle d. The stress wave can be deconvoluted

into two waves of frequency, o, and amplitudes, s00 and s000,

the former in-phase and the latter by an angle p/2 out-of-

phase with the strain. In the simulation, the instantaneous

shear stress, s, is computed by adding kinetic and potential

contributions (virial formula),[5] due to the peculiar velo-

cities and positions of all the beads. From the stress-strain

relationships, two dynamic moduli are defined: (i) the

elastic, storage, or in-phase modulus related with the elastic

energy stored by the sample, therefore being a measure of

the solid-like behavior of the sample, G0 ¼ s00=g0, and (ii)

the viscous, loss, or out-of-phase modulus, related with the

energy dissipated by the viscous flow, therefore being a

measure of liquid-like behavior of the sample, G00 ¼ s000=g0.

Usually, at low frequencies one reveals the liquid-like

behavior and the material has sufficient time to respond to

the perturbation. In practice, the steady-state values of the

dynamic moduli are computed as: G00 ¼ (s0/g0)cos d and

G00 ¼G0tan d, wherein s20 ¼ 2 sh i=p, the brackets h i

standing for time average and the angle d is obtained by a

least squares fit of the stress data. The complex or dynamic

viscosity is then defined as Z*:�iG*/o, where the

complex modulus is defined as G*¼G0 þ iG00.A practical semi-empirical rule is the so-called time-

temperature superposition rule,[11] based on the fact that the

relaxation time of polymeric system is a decreasing func-

tion of temperature. The validity and failure of the time-

temperature superposition rule for polymers has been

extensively discussed, and related to the molecular para-

meter, cf. ref.[12,27] In the Rouse model, thermorheological

simplicity holds because all relaxation processes, no matter

how slow, involve relaxations of submolecules each of

which is controlled by the same drag coefficient. The slower

Rouse relaxation processes require the coordinated move-

ment of more submolecules than do faster processes, but

since the degree of coordination does not depend on

temperature, the rates of all modes change proportionately

when the temperature is raised or lowered. For real poly-

mers, time-temperature shifting works for relaxation modes

that involve motions of portions of the chain large enough to

average out small-scale heterogeneities in the chain or in the

viscous environment through which the chain moves.[27]

Thus, increasing the temperature produces identical system

response than decreasing the oscillation frequency and vice

versa. Accordingly, pairs (T, o) exist, that give rise to the

same stress. This allows one to choose the best work

conditions and, after the suitable data manipulation, results

merge into a single curve. More precisely, values of G0 and

G00 measured in a range of frequencies at different temper-

ature are converted into the corresponding values at a given

reference temperature, T0, by making suitable shifts on both

coordinate axis. The vertical shift affecting the complex

modulus reads G*(T0)¼G*(T)T0V0/(TV), where V is the

volume of the system (simulation box) at the working

temperature T and V0 the analogous volume at T0. Volumes

are slightly different in making experiments on a given

sample at different temperature and same pressure. On the

other hand, the horizontal shift along the frequency

scale reads o(T0)¼ aTo(T). Thus, to make the horizontal

shift we multiply frequencies by the so-called horizontal

shift factor, aT, which depends on the temperature and is

also determined empirically. Following the above proce-

dure, a single master curve corresponding to T0, and com-

prising a large frequency range can be obtained. Thus, the

time-temperature superposition rule in principle allows to

appreciate all kinds of behavior of polymeric material

(glass, rubber, liquid) from a limited frequency interval.

Results

We studied the linear viscoelastic behavior of FENE poly-

mer melts, characterized by bead number density n¼ 0.84

and (reference) temperature T¼ 1 (LJ units). Concerning

initial conditions, a total number of Nt¼ 3 500 beads

allocated to FENE chains of length N are placed into the

periodic simulation box along the lines indicated in ref.[28]

As in experimental works, a suitable working strain amp-

litude, g0, must be chosen; low enough in order to be in the

linear regime but not so low that values of the properties

are largely affected by noise. A good value turned out to be

g0¼ 0.1 which is also a common value used in experiments.

The inset of Figure 1 shows the amplitude dependence of

the loss modulus, G00, for a melt of linear FENE chains of

N¼ 20 using a value of oscillatory frequency o¼ 0.1. As

observed, for g0¼ 0.1, we are fully in the linear region but

far from the noisy region appearing at very low g0. Thus, all

plots presented in this work correspond to that selected g0.

Figure 2 shows the frequency dependence of the storage

modulus, G0 (black symbols) and G00 (empty symbols) for

different melts of linear FENE chains with N¼ 5, 20, 70.

Figure 1. Loss modulus, G00, versus strain amplitude, g0, for amelt of linear multibead chains with N¼ 20 beads each, at theparticular frequency o¼ 0.1. Search of suitable work strainamplitude.

750 J. G. H. Cifre, S. Hess, M. Kroger

Macromol. Theory Simul. 2004, 13, 748–753 www.mts-journal.de � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

This kind of representation helps to appreciate the fre-

quency evolution of the solid-like and liquid-like behaviors

of the material under study as well as their relative impor-

tance. The chain lengths simulated in this work are below

the critical entanglement length Nc� 100.[6] For chain

lengths below the rheological crossover length, the G0 and

G00 in the low frequency zone are expected to conform well

to the modified Rouse theory for undiluted polymer solu-

tions as long as the temperature is well above the glass

transition temperature, cf. ref.[11] In addition, when en-

tanglements are not formed, there is a transition from the

terminal zone (smallest o values in our dynamic moduli

curves) directly to the glassy zone, without an intermediate

rubbery range.[11] Let us first compare the behavior of G0

(i.e., black symbols only) by varying the chain length N.

Figure 2 shows that at the low frequency range swept, the

shorter the chain, the smaller is the value ofG0, while at high

frequencies all G0 curves tend to converge. Differences in

the G0 curves start to occur in reaching the terminal zone

(predominant liquid-like behavior). In the transition zone to

the glassy consistency, viscoelastic properties are domi-

nated by configurational rearrangements of chain segments

which are not very extended (chains remain near the coil

equilibrium conformation because of the small g0 applied).

There possible crosslinks play a minor role. As a result, the

behavior is not much affected by differences in chain

length. For N¼ 20 and N¼ 70, because of their relatively

large relaxation times,[1] the simulated frequencies are

located ‘‘above’’ the terminal zone and therefore their G0

curves in general, superimpose. For N¼ 5, however, the

work frequencies are close to the terminal liquid-like region

and at the lowest frequencies simulated itsG0 values diverge

from those of the larger chains. In contrast, the quantity G00

(empty symbols only) is much less sensitive to the transition

from the glassy to the terminal zone since the characteristic

slopes of each zone in the log-log plot are closer than in the

case ofG0. Therefore, in the whole frequency region, values

for different chain lengths practically superimpose.

Let us compare the relative behavior of G0 and G00 for a

given chain length (i.e., black and empty symbols of a given

type). As appreciated G00 >G0 for the whole frequency

range and all chain lengths. It means that the liquid-like

(viscous) behavior predominates over the elastic behavior

for the chain model simulated and, as expected for polymers

of low molecular weight, there is no intermediate rubbery

region and therefore no crossover between the G0 and G00

curves. However, in increasing the chain length, the behav-

ior seems to tend progressively to that of entangled poly-

mer melts. Thus, it can be appreciated that for the chains

with N¼ 70 (triangles), curves of G0 and G00 tend to

converge at the small frequency corresponding to the

crossover between a hypothetical rubbery region and the

glassy one. It also indicates that, at the frequencies em-

ployed, these melts are fully in the glassy zone. On the other

hand, for the shortest chains, N¼ 5 (circles), the work

frequency region is in transition between the terminal zone

and the glassy zone. In this case, the expected Rouse

behavior in the terminal zone, i.e., G0 /o2 and G00 /otends to hold, as is better visible in Figure 3. It is interesting

to observe how the dynamic moduli in the frequency range

0.01<o< 10 (cf. plots showing results for N¼ 5) are

similar to those reported for dilute polymer solutions.[11,29]

Thus, not only in the terminal zone (obtained for N¼ 5)

slopes of G00 and G0 tend to be 1 and 2, respectively, but in

the transition to the glassy region their curves tend to reach

slopes about �2/3 (lower frequencies) and �1/2 (higher

frequencies) similar to the theoretical values predicted for

dilute polymer solutions.[11] This finding is established

through the G00 curve (N¼ 20), which is fully in the tran-

sition to the glassy region. Accordingly, we confirm[12] that

the Rouse behavior of dilute solutions is applicable, with

slight modifications, to melts of low molecular weight. This

is not surprising since for the amplitude and frequencies

studied, chains are merely non-stretched and a Hookean

Figure 2. Effect of chain length, N (/ molecular weight) onfrequency dependence of the dynamic moduli,G0 andG00 for meltsof linear finitely extendable non-linear elastic (FENE) chains.

Figure 3. Application of the time-temperature superpositionrule for a melt of linear FENE chains with N¼ 5.

Linear Viscoelastic Behavior of Unentangled Polymer Melts via Non-Equilibrium Molecular Dynamics 751

Macromol. Theory Simul. 2004, 13, 748–753 www.mts-journal.de � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

bond potential rather than a FENE potential should yield

similar results. Because hydrodynamic interaction is not

relevant for polymer melts, Rouse rather than Zimm model

is expected to be more suitable in describing the investi-

gated system.

An informative picture is obtained by plotting the dyn-

amic viscosity versus the oscillatory frequency in a log-log

plot, cf. Figure 4. Only for the melt with N¼ 5, the New-

tonian plateau is fully reached and the zero-shear viscosity,

Z0, can be extracted. The onset of shear thinning occurs at a

frequency (inverse of the ‘‘small’’ relaxation time) inside

the range explored. Viscosity values of melts formed by the

longer polymer chains (N> 20) fall inside the shear thin-

ning region. In Figure 4, we observe the validity of the

empirical Cox-Merz rule jZ*(o)j ¼ Z(o). Notice how the

dashed line in Figure 4 (corresponding to values of the shear

viscosity obtained via NEMD with N¼ 20 subjected to a

simple shear flow) superimposes with the empty circles

(oscillatory flow). As appreciated in the steady simple shear

experiment, we can get values in the Newtonian plateau

because these simulations are less time consuming. All

curves superimpose in the shear thinning region with a slope

of ��0.55, in overall agreement with experimental find-

ings.[11] By subjecting samples with N¼ 2, 5, 10, and 20 to

simple shear flow, we obtained the dependence of Z0 on

the chain length (i.e., molecular weight). As observed in

Figure 5, the slope in the log-log plot of Z0 versusN, i.e., the

exponent of the power law Z0/Na, is a¼ 0.91� 0.02

close to the value 1 expected for Rouse-like behavior of the

polymer chain. That scaling relationship and the Z0 values

obtained agree well with previous results for short chains

using the same methodology.[6] If chains were longer, repta-

tion behavior sets in and the power law exponent increases

to about three (experimentally the value 3.4 is found).[1]

The representation in the complex plane of Z00 versus Z0

(Cole-Cole plot) gives a circular arc for a simple Maxwell

fluid. From this plot, Z0 is extracted as the intercept of the

circular arc with the Z0 axis. Figure 6 presents such a repres-

entation for melts with chains of N¼ 5 and N¼ 20. For

N¼ 5, for which some values in the Newtonian regime are

available, the interception is observed, giving Z0� 4.5.

However, for N¼ 20, we are still in the uprise of the curve

(non-Newtonian part in the flow curve, Figure 4). Another

parameter that can be extracted from the Cole-Cole plot is a

characteristic relaxation time of the melt, t0, corresponding

to omax, where the function Z00 approaches its maximum:

t0¼ 1/omax. From Figure 5 we can get that relaxation time

for the melt of chains with N¼ 5 and t0ffi 7. This value

compares well with the one obtained for the Rouse relaxa-

tion time of a FENE melt in ref.[1] It is worth mentioning

that the shape of graphs in the Cole-Cole and related

representations, e.g., so-called ‘‘Van Gurp-Palmen plot,’’

have been used to classify polymeric fluids by their

architecture (linear, branched, star etc.).[30]

Similarly to experiments, the frequency range available

for simulation is limited. This is because simulations at very

small frequencies are prohibitively time consuming and at

Figure 4. Both (absolute) dynamic viscosity jZ*j and steady-shear viscosity, Z, versus frequency, o, and shear rate, _gg,respectively (both in reduced Lennard-Jones (LJ) units). Cox-Merz rule verified.

Figure 5. Effect of chain length on the Newtonian shearviscosity of polymer melts with chain length N¼ 2, 5, 10, and20 in the unentangled regime. Data obtained from steady shearflow NEMD simulations.

Figure 6. Cole-Cole plots of melts of linear FENE chains oflengths N¼ 5 and N¼ 20.

752 J. G. H. Cifre, S. Hess, M. Kroger

Macromol. Theory Simul. 2004, 13, 748–753 www.mts-journal.de � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

high frequencies become unstable, both resulting in large

inaccuracy. Therefore, analogously to experiments, we de-

cided to perform simulations at temperatures lower and

higher than the reference one (T¼ 1) and make use of the

time-temperature superposition rule exposed in the theore-

tical section to extend slightly the range of frequencies

embraced. Figure 3 shows the resulting master plot obtained

for the melt with N¼ 5 at different temperatures after the

shift to the reference temperature T¼ 1. Black and empty

symbols correspond to G0 and G00 values, respectively, and

different symbol types were used to distinguish values of a

given modulus obtained originally at different temper-

atures. Simulation parameters were considered independent

of the temperature provided that the temperature range used

was very small. Care was also taken to keep the system at

approximately the same pressure in varying the temper-

ature. Thus, by shifting the corresponding curves along the

coordinate axis (as outlined in the previous section) we got

the plot in Figure 3. In that plot, the shift factor, aT, was

determined in order to get a ‘‘nice’’ superposition of the

curves obtained at different temperatures. Extension of the

curve at higher frequencies by lowering the work temper-

ature gave better results than extension at lower frequen-

cies. In the terminal zone, Z0 can be obtained from the slope

of the G00 curve since G00 ¼ Z0o. In this case, in agreement

with the result obtain from the Cole-Cole plot (Figure 6) and

from the extrapolation in the Newtonian regime in Figure 4,

Z0� 4.5.

Conclusion

Small strain oscillatory rheometry is widely used to char-

acterize the mechanical behavior of viscoelastic systems.

Nevertheless, computational studies devoted to reproduce

this experimental setup are scarce. We have shown the

adequacy of NEMD of a coarse-grained polymer (FENE)

model to simulate oscillatory flow in spite of its limitations

in sweeping a broad frequency range due to the enormous

CPU time required to get results in the terminal zone. In

order to extend the frequency range, we make use of the

time-temperature superposition rule, widely used in experi-

mental works. In simulations, care must be taken if applying

that empirical rule, because variations of the work temper-

ature entail changes in the temperature-dependent model

parameters (for instance, the spring coefficient) and in the

thermodynamic properties of the system (pressure, volume,

density). We think that as long as temperature keeps closed

to T0, these changes play a negligible role in simulation

results.

Results obtained for chain lengths under the critical

entanglement length give rise to results which, at least

qualitatively, agree with experimental findings for melts of

low molecular weight.[12] Furthermore, we were able to

check the validity of the empirical Cox-Merz rule. Apart

from CPU time requirements, NEMD has, in principle, no

limitations to simulate and thus study and understand the

dynamics and conformational evolution of melts formed by

polymer chains of arbitrary topology, such as branched

polymers.[31] Therefore, beside studying longer chains, the

next step is to investigate melts composed of chains with

different topology as well as polydisperse melts with the

methodology outlined here.

Acknowledgements: J. G. H. C. is the recipient of a postdoctoralfellowship from the Spanish Ministerio de Educacion Cultura yDeportes. This work has been performed under the auspices of theSfb 448 ‘‘Mesoskopisch strukturierte Verbundsysteme’’, DeutscheForschungsgemeinschaft (DFG).

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Macromol. Theory Simul. 2004, 13, 748–753 www.mts-journal.de � 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim