PRL 100, 188302 (2008)
PHYSICAL REVIEW LETTERS
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9 MAY 2008
Analytic Expressions for the Statistics of the Primitive-Path Length in Entangled Polymers
Renat N. Khaliullin and Jay D. Schieber
Department of Chemical and Biological Engineering, and Center for Molecular Study of Condensed Soft Matter,
Illinois Institute of Technology, 10 W. 33rd Street, Chicago, Illinois, USA
(Received 22 October 2007; published 5 May 2008)
An analytic expression is proposed for the primitive-path length of entangled polymer chains. The
expression is derived from statistical mechanics of a chain that is a random walk with randomly scattered
entanglements. The only parameters are the number of Kuhn steps in the chain and a dimensionless
parameter that contains information about the entanglement density and Kuhn step size. The expression
is found to compare very favorably with numerical results recently found from examining topological
constraints in microscopic simulations. The comparison also predicts well the plateau modulus of
polyethylene, suggesting that the slip-link model is a viable intermediate in the search for true
ab initio rheology predictions. Since the expression is analytic, it can be used to make predictions where
the simulations cannot reach, and hence is applicable for coarse graining.
DOI: 10.1103/PhysRevLett.100.188302
PACS numbers: 83.80.Sg, 81.05.Lg, 83.10.Kn
Because the primitive path (PP) is a promising intermediate in the multiscale modeling of both linear and
branched polymer chains in the entangled state, determination of its proper statistics is an outstanding problem in
polymer melt rheology [1]. The long relaxation times
exhibited by entangled polymers, particularly branched
ones, make atomistic-level simulations of relaxation dynamics practically impossible. Therefore, current molecular models typically exist on a coarse-grained level, such as
the sum of the distances between successive entanglements
on the chain—the primitive-path length [2 –6].
Hence the recent interest in finding the PP from atomistic simulations [7,8], or other dense-liquid simulations
[9,10]. These simulations are expensive to run, and only
limited information can be obtained about the statistics of
the PP distribution.
In modeling the relaxation of the arms of entangled starbranched polymers, Doi and Kuzuu [11] were likely the
first to estimate the primitive-path potential. This estimate
was accomplished by considering the one-dimensional
problem of a Gaussian chain pulled at its ends by
Maxwell demons. The strength of the demons is determined by the average length of the primitive path, which
is itself fixed by the entanglement density. The Gaussian
chain has a force that increases linearly with path length,
and the demons have a constant force. Hence, the resulting
potential is quadratic in PP length
F
NK
L 2
;
(1)
1
kB T
Ne
Lmp
where NK is the number of Kuhn steps in the arm; Ne is the
average number of Kuhn steps in an entangled strand,
which is determined by the entanglement density; F is
the free energy of the primitive path of length L and
temperature T; Boltzmann’s constant is given by kB , and
Lmp is the most-probable primitive-path length. The parameter was treated as an adjustable constant by Doi and
0031-9007=08=100(18)=188302(4)
Kuzuu and they found that 2=3 fit relaxation data for
stars. Previously Doi and Edwards [12] pointed out that
equilibrium random walk statistics for the chain require
that 3=2. Results are extremely sensitive to the exact
value of , since it appears as an exponent in the probability density for L.
Subsequent attempts to estimate statistics for the primitive path have been from simulations. McLeish ([13],
x 4.2.1) reviews lattice methods used to estimate the potential of the primitive path until 2002. In these on-lattice
simulations, the entanglements were not found from the
conformations of the chains themselves, but rather were
mimicked by a superlattice whose size determines the
entanglement spacing. However, in 2004, based on an
idea of Edwards [14,15], Everaers et al. [9] showed that
the entanglement density could be calculated from the
chain conformations, and that the values agreed with the
entanglement density estimated from the height of the
plateau modulus measured from small-amplitude oscillatory shear flow experiments. They used an annealing algorithm that shrunk the chains (which are off lattice) by
lowering the bonding energy while preserving topological
constraints in the melt. Entanglement theories postulate
that the dominant contribution to stress is from the entropy
of the entangled strands, so that the plateau modulus G0N is
proportional to the thermal energy kB T times the entangled
strand density (the slip-link model has a prefactor of unity).
Those simulations were shown to lose about 15% of
their entanglements [16]. Hence, later work refined this
procedure, and examined more-detailed statistics of the
primitive path, instead of only the entanglement density
[7,8,17]. These later works showed good agreement with
analytic statistics derived for a slip-link model [18].
In parallel with this work, Shanbhag and Larson [10,19]
used the bond-fluctuation model (BFM) to estimate PP
statistics. These are lattice simulations, but this time the
chain-chain interactions were again used to find the entanglements, similar to the method introduced by Everaers
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© 2008 The American Physical Society
PRL 100, 188302 (2008)
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PHYSICAL REVIEW LETTERS
et al. [9]. Their results were well approximated by a
quadratic potential for the free energy, with the prefactor
very close to 3=2, as predicted by Doi and Edwards.
While Foteinopoulou, et al. [7] also found PP length
statistics close to quadratic for atomistic simulation, they
found a prefactor that varied with the average number of
entanglements per chain, with values approaching 1 at
large entanglement numbers. It is not clear why these
two similar detailed simulations apparently give such
qualitatively different results.
It is important to note that Ne is a parameter in the tube
theory, Eqn. (1), although both the atomistic simulations
and the slip-link theory show that it is mildly dependent
upon chain length.
While such atomistic or near-atomistic simulations presumably give the most accurate estimates of the primitivepath statistics, they do have additional shortcomings. First,
they are computationally expensive. Second, they provide
good statistics only where lengths are most likely near the
average length hLieq or the most-probable length Lmp . In
branched systems, however, the most important dynamics
occur when L is extremely small, when all entanglements
have been destroyed by deep fluctuations of the branch tip
towards the branch point. These conformations are very
rare, so statistics are difficult to obtain numerically. For
example, if we assume that the potential is quadratic, the
free energy for complete branch tip withdrawal is FL0
kB T NNKe hZieq , where hZieq is the average number of entangled strands per branch arm. Hence, even moderately
entangled arms are extremely unlikely to relax. Therefore,
analytic expressions for the primitive-path length statistics,
or equivalently the free energy, are necessary for further
coarse graining.
In this Letter we show that a thermodynamically consistent slip-link model can be used to find analytic results
for the PP length statistics. The results are consistent with
what is observed for atomistic simulations and BFM simulations. However, a discrepancy remains between bondfluctuation-model simulations [19] and atomistic simulations [7] in predicting the width of the potential well.
Moreover, we show that the free energy is not quadratic
in the important region of complete branch tip withdrawal,
having important implications for tube models.
For the system of interest in this Letter (amorphous
chains at equilibrium in an isotropic, homogeneous environment), we may assume that the chain’s random walk
statistics are well approximated by a Gaussian expression
for the free energies of individual strands. Hence, the
conformation for a whole chain is given by the modified
Maxwell-Boltzmann relation [18]
P
3=2
Z NK Zi1 Ni Y
3
peq fNi g; fQi g; Z 2
JZ1
i1 2Ni aK
3Q2i
exp ;
(2)
2Ni a2K
where : expent =kB T is a parameter related to the
entanglement density and is approximately equal to the
average number of Kuhn steps per entanglement strand,
Ne , for well-entangled systems. The chemical potential
ent conjugate to the number of entanglements Z 1,
acts as a bath for entanglements on the mean-field chain.
Consequently, depends upon chemical composition,
temperature, and pressure, but, importantly, not on chain
length. The atomistic simulations of [8] suggest that entanglements might show some repulsive force, but this is
not included here.
The parameter J is a normalization constant (its exact
expression depends on whether or not we assume that the
Kuhn step numbers can be treated continuously). The
fluctuating variables are the number of strands Z, the
number of Kuhn steps Ni in strand i, and the conformation
Qi of strand i.
P
The PP length is then L Zi1 jQi j. Using this definition, one can calculate the distribution of PP lengths by
using the expression [20]
NK Z
Z
Z
X
X
peq L L
Qi
Z1
fNi g
fQi g
i1
peq fQi g; fNi g; ZdQZ dN Z ;
(3)
where integrals over dQZ and dN Z represent Z threedimensional and Z one-dimensional integrals, respectively.
If we assume that the number of Kuhn steps in a strand is
discrete, then the integral over the fNi g should be replaced
by summations. Unfortunately, if we insert Eq. (2) into
Eq. (3), the integrations cannot be performed analytically
to give a compact expression for the probability density
peq L. However, moments of L can be found analytically
for continuous Ni . If we multiply Eq. (3) by Ln (where n 1; 2; . . . ), and integrate over all possible conformations
using Eq. (2), we can obtain
ss
s s
hLieq
2N
N
1
NK
K
K
q erf
2
2
N
3
aK
K
s
2NK
N
exp K ;
(4)
3
hL2 ieq
NK 2 NK 2
:
3 a2K
(5)
Higher moments can also be found. We will use these
expressions in the following to compare with the atomistic
simulation results. If we are assuming that the free energy
is quadratic, it is determined uniquely by the first and
second moments, and the parameter can be estimated
from our moment expressions above.
Because the average number of Kuhn steps in an entangled strand Ne hNi ieq depends on chain length, and
because the number of entanglements is not accurately
known in the atomistic simulations, it is better to write
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PHYSICAL REVIEW LETTERS
PRL 100, 188302 (2008)
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1
0.8
0.6
0.004
0.4
0
0.2
-0.003
20
0
20
25
30
L /a K
30
35
40
40
FIG. 1 (color online). The cumulative probability P for the
primitive-path length L found from an ensemble of 106 slip-link
chains, with NK 82, 4:36 (dots). Also shown is the
cumulative probability for the quadratic free energy distribution
Pquad found from the first and second moments (solid line). Inset
are the residuals for the quadratic estimate.
Eqn. (1) as
FL FLmp N L Lmp 2
K
:
L2mp
kB T
(6)
We can then estimate within the quadratic approximation as
L2mp
;
2NK hL2 ieq hLieq Lmp quadratic free energy:
(7)
We checked the quadratic assumption for the free energy in
the neighborhood around hLieq by generating a large ensemble of slip-link chains with conformational statistics
determined by Eq. (2) using an efficient algorithm described elsewhere [21]. In Fig. 1 we plot this Rnumerically
estimated cumulative probability PL : L0 pL0 dL0
and that from assuming a quadratic free energy, Eqs. (6)
and (7),
p
p
LLmp NK =
erf
erf NK =
Lmp
P quad L : (8)
p
1 erf NK =
Inset are the residuals. The cumulative distribution indeed
looks very similar to the approximation. It appears that we
can use the moments to estimate the complete distribution
and therefore to estimate the parameter . Fitting the
cumulative distribution is preferable to fitting the free
energy (as was done in [10]), since the uncertainty is
much smaller and binning is unnecessary.
We now use the slip-link model to analyze the simulations. In particular, we first wish to see if the atomistic
simulations can predict the plateau modulus, whose dependence on the primitive-path length requires a model. We
examine the first moment, hLieq predictions (which we find
FIG. 2 (color online). Renormalized first moment of the PP
length hLieq vs chain size for the theory and the simulations. The
dotted line is the theoretical prediction when the Kuhn step
numbers are taken as continuous and the curve is independent
of , squares (䊐) are for Kuhn step numbers taken as discrete
quantities. The updated atomistic simulation results of
Foteinopoulou et al. are filled circles (䊉). The filled squares
are the bond-fluctuation-model results of [10].
to be nearly identical to Lmp ) with the simulation results. In
Fig. 2 we see very good agreement among the bond-fluctuation-model simulations of Shanbhag and Larson, the
atomistic simulations, and theoretical predictions, by adjusting (independent of molecular weight).
Using the slip-link model to analyze the atomistic simulations, we can then predict that the plateau modulus
of high-molecular-weight polyethylene at T 450 K
should be G0N RT=Me 2RTcos2 =2=
1C1 Mm 3:0 MPa, where Mm is the molecular weight
of a monomer, and is the chain bending angle. This
agrees favorably with the experimentally determined value
of 2.6 MPa at 413 K.
We now turn to the width of the potential. Foteinopoulou
et al. [7] report estimates for in their Fig. 13 based on
atomistic simulations. However, estimation of requires
detailed knowledge of the number of entanglements, which
cannot be determined unambiguously from the simulations. We therefore suggest using plots of the sort in
Fig. 3 to compare results. We use the same values for chosen to fit the first-moment results. The atomistic
simulations predict much higher values of , corresponding to narrower potentials. The published results of
Foteinopoulou, et al. [7] were much closer to the theory
than the results used here; causes for the discrepancy are
unclear. On the other hand, the theoretical calculations
above assumed that the number of Kuhn steps in a strand
can be treated as a continuous variable; such an assumption
is unnecessary, however, and its removal results in the
empty squares for 4:36. We see nearly universal
results independent of chain chemistry, which could otherwise affect our estimate for .
Since the BFM simulations of Shanbhag and Larson are
not atomistic, their results cannot be compared with experimental data. However, they can be checked for agree-
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PRL 100, 188302 (2008)
PHYSICAL REVIEW LETTERS
FIG. 3 (color online). Modified parameter vs the renormalized number of Kuhn steps in the chain, NK =. Symbols are the
same as in the previous figure.
ment with the theory. These BFM results are put on the
same two plots, shown by the filled squares. Previously,
they had reported agreement with the Doi-Kuzuu result,
3=2. The difference here is that we do not treat Ne hNieq as a constant independent of chain size NK . Excellent
agreement is seen between the theory and the moredetailed simulation results for both moments of the primitive path, with an estimate for 22.
An essential difference between the slip-link theory and
traditional tube models involves fluctuations. The slip-link
model treats the atomistic chain as a random walk on all
length scales, and then scatters the entanglements randomly along the chain. This procedure leads to a distribution of entanglement number and entanglement spacing for
monodisperse systems. The resulting distribution of Kuhn
steps trapped between entanglements also leads to an
average that depends on chain length. Previous analysis
of atomistic simulations took only some of these fluctuations into account. Fortunately, the slip-link formalism
allows a more consistent analysis.
Since the theory yields many analytic results, it can be
used to make predictions where the simulations are not
well suited. In particular, it would be practically impossible
to examine statistics for a very short primitive-path length
of even moderately long chains using atomistic models.
Whereas typical analysis using a quadratic assumption
suggests a finite probability of zero path length, the sliplink theory predicts a zero probability: peq L 0 0.
This discrepancy is particularly problematic for tube theories of branched polymers. In tube theories, relaxation is
complete only when the branch tip retracts completely
back to the branch point, resulting in zero primitive-path
length for that branch. One could argue that the tube model
breaks down at such short lengths, and other physics
should come into play. However, since these smalllength-scale dynamics are not accounted for in tube theory,
extrapolation to zero primitive-path length is akin to hoping for a cancellation of two errors.
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No such hope is required for the slip-link theory for two
reasons. First, complete relaxation of a branch is possible
for finite primitive-path lengths; relaxation occurs whenever the final entanglement is destroyed near the branch
point. Second, straightforward statistical mechanical calculations show no significant entropic penalty for a branch
point to slide through an entanglement. Hence, in the sliplink theory, the branch point could retract significantly into
the interior of the chain, leading to a lowered entropic
penalty for total branch relaxation. These ideas will be
examined quantitatively in future work utilizing the sliplink theory.
Finally, the analytic expressions found here for the moments of the primitive path might be useful ingredients to a
further coarse-grained model in branched systems.
Support of this work by the National Science Foundation
under Grant No. NSF-SCI 05063059 is gratefully acknowledged, as are useful discussions with Professors Martin
Kröger and Vlasis Mavrantzas, who graciously provided
their updated simulation results.
[1] R. G. Larson, Q. Zhou, S. Shanbhag, and S. J. Park, AIChE
J. 53, 542 (2007).
[2] F. Greco, Phys. Rev. Lett. 88, 108301 (2002).
[3] H. C. Öttinger, J. Rheol. (N.Y.) 43, 1461 (1999).
[4] J. Fang, M. Kröger, and H. C. Öttinger, J. Rheol. (N.Y.) 44,
1293 (2000).
[5] M. Doi and S. F. Edwards, J. Chem. Soc., Faraday Trans. 2
74, 1789 (1978).
[6] G. Marrucci and G. Ianniruberto, J. Non-Newtonian Fluid
Mech. 128, 42 (2005).
[7] K. Foteinopoulou, N. Karayiannis, V. Mavrantzas, and
M. Kröger, Macromolecules 39, 4207 (2006).
[8] C. Tzoumanekas and D. N. Theodorou, Macromolecules
39, 4592 (2006).
[9] R. Everaers et al., Science 303, 823 (2004).
[10] S. Shanbhag and R. G. Larson, Phys. Rev. Lett. 94, 076001
(2005).
[11] M. Doi and N. Y. Kuzuu, J. Polym. Sci., Polym. Lett. Ed.
18, 775 (1980).
[12] M. Doi and S. F. Edwards, The Theory of Polymer
Dynamics (Clarendon Press, Oxford, 1986).
[13] T. C. B. McLeish, Adv. Phys. 51, 1379 (2002).
[14] S. F. Edwards, Br. Polym. J. 9, 140 (1977).
[15] M. Rubinstein and E. Helfand, J. Chem. Phys. 82, 2477
(1985).
[16] S. Shanbhag and M. Kröger, Macromolecules 40, 2897
(2007).
[17] M. Kröger, Comput. Phys. Commun. 168, 209 (2005).
[18] J. D. Schieber, J. Chem. Phys. 118, 5162 (2003).
[19] S. Shanbhag and R. Larson, Macromolecules 39, 2413
(2006).
[20] N. G. van Kampen, Stochastic Processes in Physics and
Chemistry (North-Holland, Amsterdam, 1992).
[21] J. D. Schieber, J. Neergaard, and S. Gupta, J. Rheol. (N.Y.)
47, 213 (2003).
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