Why polymer chains in a melt are not random walks

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by  J. P. Wittmer

epl draft
Why polymer chains in a melt are not random walks
J. P. Wittmer1 ∗, P. Beckrich1, A. Johner1, A. N. Semenov1, S. P. Obukhov1,2, H. Meyer1 and
7J. Baschnagel1
0
01 Institut Charles Sadron, 6 Rue Boussingault, 67083 Strasbourg Cedex, France
2
2 Department of Physics, University of Florida, Gainesville FL 32611, USA

n
a
J PACS 61.25.Hq–Macromolecular and polymer solutions; polymer melts; swelling

1 PACS 64.60.Ak–Renormalization-group, fractal
1 PACS 05.10.Ln–Monte Carlo methods

] Abstract. - A cornerstone of modern polymer physics is the ‘Flory ideality hypothesis’ which
t
f states that a chain in a polymer melt adopts ‘ideal’ random-walk-like conformations. Here we
o revisit theoretically and numerically this pivotal assumption and demonstrate that there are no-
s
ticeabledeviationsfromideality. Thedeviationscomefromtheinterplayofchainconnectivityand
.
t the incompressibility of the melt, leading to an effective repulsion between chain segments of all
a
m sizess. Theamplitudeofthisrepulsionincreaseswithdecreasingswherechainsegmentsbecome
more and more swollen. We illustrate this swelling by an analysis of the form factor F(q), i.e.
-
d the scattered intensity at wavevector q resulting from intramolecular interferences of a chain. A
n ‘Kratkyplot’ofq2F(q)vs. qdoesnotexhibittheplateauforintermediatewavevectorscharacteris-
o ticofidealchains. Oneratherfindsaconspicuousdepressionoftheplateau,δ(F−1(q))=|q|3/32ρ,
c which increases with q and only dependson the monomer density ρ.
[

2
v
2 Polymer melts are dense disordered systems consisting a density ρ= const b
n
2of macromolecular chains. Theories that predict proper- c* (s) hai
13tailelys osftacrhtafirnosminthaem‘Felltoroyr icdoenacleitnytrhaytepdotshoelusitsi’on[1s].geTnehre- Segmental c(r,s) total c
1 correlation hole r
hypothesis states that polymer conformations correspond R(s)
6
0to those of ‘ideal’ random walks on length scales much
/larger than the monomer diameter [1–3]. The commonly
t
aaccepted justification is that intrachain and interchain R(s) Repulsion
m
excluded volume forces compensate each other in dense
-systems [2,3]. This compensation has several important
d Fig. 1: Role of incompressibility and chain connectivity in
consequences. For instance, the radius of gyration R(s)
n dense polymer solutions. (a) Sketch of the segmental corre-
of chain segments of curvilinear length s N scales as
o lation hole of a marked chain segment of curvilinear length s.
≤
cR2(s) = a2s, where N denotes the number of monomers Densityfluctuationsofchainsegmentsarecorrelated,sincethe
v:perchainandathestatisticalsegmentlengthofthechain. total density (dashed line) cannot fluctuate in dense polymer
iThis result holds provided s is sufficiently large for all solutions. (b)Self-similarpatternofnestedsegmentalcorrela-
X
local correlations to be neglected. For s = N, it im- tionholesalignedalongthebackboneofareferencechain. The
rplies that the radius of gyration of the total chain obeys large dashed circle represents the classical correlation hole of
a
R = R(s = N) = a√N [3]. A further consequence of thetotalchain(s=N)[2]. Herewearguethatincompressibil-
g
chain ideality is that the intrachain scattering function, ityandchainconnectivitylead toarepulsion ofthesegmental
correlation holes, which increases with decreasing s. The en-
the ‘form factor’ F(q), is given by the Debye formula,
suing swelling of chain segments manifests itself by deviations
F(0)(q) = 2N exp( (qR )2) 1+(qR )2 /(qR )4 [3].
− g − g g of theform factor from Debye’sformula.
For intermediat(cid:0)e wavevectors, 1/R q (cid:1)1/a, F(0)(q)
g
≪ ≪
reduces to the power law F(0)(q) 2/(qa)2. Due to
≈
this power-law behavior, we refer to the latter q-regime
as ‘scale-free regime’ in the following. usually plots q2F(q) versusq (‘Kratkyplot’) to revealthe
Neutron scattering experiments have been extensively existence of a ‘Kratky plateau’ in the scale-free regime.
used to test Flory’s hypothesis [4]. This technique allows In applications, however, the Kratky plateau appears to
one to extractF(q) from the totalscatteredintensity of a be elusive [4]. Possible causes for deviations are effects of
mixture of deuterated and hydrogenated polymers. One chain stiffness and finite chain thickness. In special cases,
the scattering signals from chain stiffness and thickness
∗E-mail: [email protected] may compensate one another, leading fortuitously to an
p-1
J.P. Wittmer et al.
extended Kratkyplateau [5,6]. Apparently, ‘Kratky plots 0.5 u(r,s) = -ln( p(r,s)/p(∞,s) )
have to be interpreted with care’ [4].
Itisgenerallybelievedthatifallpossibleobscuringfac- 0.4 3
BFM c(r,s) R(s) /s
tors are avoided and ‘thin’ flexible polymers are exam-
) BFM s=64
s
ined, no deviations from the Kratky plateau should oc- ( 0.3 BFM s=256
cur. Hereweshowthatthisisnottruebecauselong-range *U BFM s=1024
cpourlrseivlaetiionntesraalcotniogntshoefcchhaaininbsaecgkmboenntesairne dinednusecepdoblyymreer- s) / 0.2 Rep BBFSMM ss==N12=84,0N9=61024
u
, l
systems[7,8]. This effectisrelatedtothe well-knowncor- (r 0.1 si
relationhole[2]. Forchainsegmentsoflength1 s<N, u o
≪ n
the correlation hole leads to non-Gaussian deviations of
0.0
the form factor. Our key claim is that for asymptotically
Weak attraction
longchains(N )thedeviationsaregiveninthescale-
→∞ -0.1
free regime by 0 1 2 3
r/R(s)
1 1 1 1 q 3
δ = | | (1)
(cid:18)F(q)(cid:19)≡ F(q) − F(0)(q) 32 ρ
Fig.2: Pair-distribution function p(r,s) ofthecentersofmass
with ρ being the monomer number density. In the fol- ofchainsegmentsoflengths. Datafordifferentchainandseg-
ment lengths are successfully scaled by plotting u(r,s)/U∗(s)
lowing,we first outline the theory (sketched in fig. 1) and
as a function of r/R(s) with U∗(s) ≡ s/ρR(s)3. The figure
provide a scaling argument for eq. (1). Computer simula-
compares different s for BFM chains of length N = 4096 and
tions of two well-studied coarse-grained polymer models,
one BSM data set for N = 1024 (stars). The function is pos-
both described in detail in Ref. [9], are then utilized to
itive for small r/R(s) where segments repel each other. It is
carry out a critical test of our predictions (figs. 2, 3 and
weakly attractive for larger distances. For comparison we also
4). indicate the self-density c(r,s)R(s)3/s for one BFM example
The general physical idea of our theory and the long- with s=256 (dashed line) which is similar — albeit not iden-
range intrachain correlations it predicts is related to the tical — tou(r,s)/U∗(s).
‘segmentalcorrelationhole’ofachainsegmentoflengths
in d dimensions (fig. 1). Polymer melts are essentially
incompressible (on length scales large compared to the acterizing infinite chains in the melt. In reciprocal space,
monomer diameter), and the density ρ of all monomers this swelling appears in the form factor. As the segment
does not fluctuate. On the other hand, composition fluc- length is related to the wavevector q and the ideal form
tuations of marked chains or subchains may occur, sub- factor F(0)(q) by s(q) 1/q 2 F(0), we find for the
∼ | | ∼
ject to the constraint of constant density. Figure 1 shows deviations of the measured and ideal form factors
a marked chain segment of curvilinear length s within
1 1 F(0) 1 qd
a much larger chain of length N s. The monomer ∗
δ = 1 U (s) (3)
density distribution c(r,s) of the s≫egment (dash-dotted (cid:18)F(q)(cid:19) F(0) ×(cid:18) F − (cid:19)≈ s × ≈ ρ
line) becomes c∗ = c(r 0,s) s/R(s)d close to the
center of mass (r = 0) a≈nd deca≈ys rapidly at distances which agrees with eq. (1). The important feature of this
of order R(s). Since composition fluctuations are cou- q 3-correction is that it depends neither on the strength
| |
pled by the density constraint, a second chain segment of the excluded volume interaction nor on the statistical
(of the same or another chain) feels an entropic penalty segment length. Hence, it must be generally valid, even
U∗(s) c∗(s)/ρ when both correlation holes approach forsemidilutesolutions. Wecheckedexplicitlythatthere-
each ot≈her [7]. (The scaling of U∗(s) is demonstrated be- sult from the renormalizationgroup theory for semidilute
low, fig. 2.) This repulsion when considered between two solutions [11,12] takes the same form as eq. (1) with the
half-segments swells the segment. In d = 3, the effect is amplitude0.03124...thatiswithin0.03%ofour1/32[10].
weak, U∗(s) s/ρR(s)3 1/√s, and a standard per- The scaling of the effective interaction between
≡ ∼
turbation calculation can be performed [3,10]. This cal- two chain segments can be tested numerically by
culation considers quantities which are defined such that computing the correlation hole function u(r,s)
they vanish if the perturbation potential U∗ is switched ln(p(r,s)/p( ,s)) where p(r,s) denotes the pai≡r-
− ∞
off, and are then shown to depend, in leading order, lin- distributionfunctionofthecentersofmassofallsegments
earlyon U∗ (which may be directly checkednumerically). oflengths. Sincethecorrelationholeisshallowforlarges,
For instance, one finds that [8] expansionleads to u(r,s) 1 p(r,s)/p( ,s) c(r,s)/ρ
≈ − ∞ ≈
[7]. Hence the interaction strength at r/R(s) 1 should
1 R2(s) U∗(s) (2) be given by U∗(s) u(0,s) c∗(s)/ρ. Figu≪re 2 shows
− a2s ≈ thata scalingin ter∼ms of the m∼easuredradiusof gyration
∗
describes the swelling of chain segments in real space. R(s) and U (s) is indeed possible. This finding confirms
Note that a is the renormalized statistical segment char- that segments repel each other for small r/R(s) with an
p-2
Why polymer chains in a melt are not random walks
1.0 105 BFM N=512 mit
Kratky plateau ? N) 104 BFM N=1024 e e li
2a)/200..68 Hump B r a g g *q)) / U( 110023 BBBB(QFFFS/MMMMQ NNNN)4====/324812001049928624 s c ale fr Bragg
(q BBFFMM NN==48019962 N/F( 101 |Q|3/302
q) 0.4 BDSebMy eN =fo1r0 N24=8192,R =120 δ( 100
F(0.2 Guinier 1/(1+BBqaSF/MM(1::6 ρρρ==a003.).58)/48,, aa=g=01..5342 m(Q) = 1100--21 G uinier ≈Q6.30 U*(N)=N/ρRg(N)3
-3
10
0.0 0 1 2
0.0 0.5 1.0 1.5 10 10 10
q a Q = q R (N)
g
Fig. 3: Kratky representation of the form factor F(q)(qa)2/2 Fig.4: Scalingoftheformfactordeviationsδ(N/F(q))/U∗(N)
vs. rescaled wavevector qa. The axes are rescaled by the sta-
plotted versus Q = qR (N) as suggested by eq. (4). Here
g
tistical segment length a for the BFM and BSM (as indicated U∗(N) = N/ρR3(N) and δ(1/F(q)) = 1/F(q)−1/F(0)(q),
inthefigure)tomakebothdatasets readily comparable. The g
where F(q) is the measured form factor and F(0) denotes the
values of a for asymptotically long chains were obtained by a
Debye function. For each data set the scaling of the axes
fit to eq. (2). Note that ρa3 ≈0.13 is similar for both models
and the calculation of the Debye function are done with the
which explains the collapse for small q. (The agreement for
measured radius of gyration R (N). Without free parameter
largeqaisfortuitous.) TheDebyeformula(dashedline),com- g
perfect data collapse is obtained for all systems. Deviations
putedhereforN =8192usingthemeasuredradiusofgyration
from data collapse are expected for large Q due to finite per-
(R ≈120), overestimates thedipof theform factor at qa≈1
g sistence lengthandmicrostructureeffects(‘Bragg’ regime). In
by about 20%. The bold line indicates the deviation from the
the Guinier regime (Q ≪Q ), δ(1/F(q)) increases rapidly as
Kratky plateau, eq. (1), with the choice F(0)(q) (qa)2/2 = 1, 0
Q4 (thinline). Most importantly,the non-analytic |Q|3-power
validforidealchainsinthescale-freeq-regime. Theincreaseof
law slope (bold line) can be seen over an order of magnitude
F(q)forthelargest wavevectorsshownisduetonon-universal
in Q — a striking confirmation of the theory. Obviously, the
correlationsonthemonomerscale. Suchmicrostructureeffects
larger thechains, thebetter the agreement with eq. (1) which
are not captured by thetheory.
is valid for asymptotically long chains only.
entropic penalty U∗(s) 1/√s for large s 10.1 the longest chains currently available for both models. It
∼ ≫
Here, as in the subsequent figures, the body of our revealscleardeviationsfromtheidealformfactor(dashed
data comes from the bond fluctuation model (BFM) — line). Instead of reaching a plateau, the simulated F(q)
a lattice model which we study by Monte Carlo (MC) continuously decreases with increasing q in the scale-free
simulations. Using a mixture of local, slithering snake, regime. This difference is expected from our theory. The
anddouble-bridging MC moves [9] we were able to equili- resultforinfinitelylongchains—eq.(1)with1/F(0)(q)=
brate dense systems (ρ =0.5/8) with chain lengths up to (qa)2/2 (boldline) — enables us to understandthe initial
N =8192containedin periodic simulationboxesoflinear depression below the Kratky plateau. The prediction for
size L = 256. Additionally, molecular dynamics simula- infinite N cannot capture the decrease of the form factor
tionsofabead-springmodel(BSM)[9]wereperformedto for qa 0.1 leading to the Guinier regime [4] where F(q)
≤
dispel concerns that our results are influenced by the un- is determined by the finite size of the simulated chains.
derlyinglatticestructureoftheBFM.Forclarity,weshow A clearer evidence for the theory should thus be obtained
only one BSM data set (N = 1024, ρ 0.84, L 62). by a different comparisonbetween theory and simulation,
≈ ≈
All length scales are given in units of the lattice constant which accounts for finite-N effects.
or the bead diameter for the BFM or BSM, respectively. This is achieved in fig. 4 which focuses on deviations
Compared to standard experimental polymers both mod- δ(1/F(q)) = 1/F 1/F(0) from the Debye formula F(0).
−
els have in common that the chains are very flexible and If the axes are rescaled according to
the units (beads) are point-like. They thus represent an
∗
m(Q) δ(N/F(q))/U (N) (4)
ideal tool for testing the theory. ≡
Figure 3 presents a Kratky plot of F(q) obtained for with Q qR (N), a perfect collapse of the simulation
g
≡
data for all N and both polymer models is obtained by
1Strictlyspeaking,u(r,s)coincideswiththeeffectiveinteraction usingonlyonelengthscale,the(independently)measured
potential U∗(r,s) of two independent segments of length s only if
R (N). Eq. (4) is suggested by the fact that F(q) is pro-
s=N. Moregenerally, u(r,s) reflects alsothe interactions of their g
portional to the number of scatterers, N, and is a func-
neighborsalongthechains. However,thisdoesnotaffectthescaling
onshortdistances, r R(s), whichmattershere. tion of qRg(N) only [3]. The master curve m(Q) thus
≤
p-3
J.P. Wittmer et al.
generalizes eq. (1) to finite N. In the Guinier regime effective repulsive interactions. They could, for instance,
(Q Q 6.3) we find m(Q) Q4, as expected from a be responsible for observeddeviations from Rouse behav-
0
≪ ≈ ∼
standard analytic expansion in Q2. (The first two terms ior [19]. Moreover, for thin polymer films the repulsive
in Q0 and Q2 must vanish by construction.) The scale- interactionsare knownto be strongerthan in the bulk [7]
free regime is observed for Q > Q . Since δ(1/F) N0 providing, hence, a mechanism to explain the systematic
0
∼
in this regime, it follows from eq. (4) that m(Q) Q3, trend towards chain swelling observed experimentally for
∼ | |
whichprovidesanadditionalscalingargumentfor eq.(1). film thicknesses below the bulk radius of gyration [20].
Fig. 4 demonstrates for the largest chains available over
more than an order of magnitude in Q a good quantita-
∗∗∗
tive agreement (bold line). We therefore suggest that an
analysis of experimental data should employ this scaling We thank M. Mu¨ller (Go¨ttingen), M. Rawiso,
plot as a diagnostic for the effects discussed here. H.C. Benoˆıt(Strasbourg), andM.D. Dadmun (Knoxville)
In summary, we have identified a general mechanism for numerous discussions. Computer time by the IDRIS
that gives rise to deviations from ideal chain behavior in (Orsay)isalsogratefullyacknowledged. S.P.O.isindebted
polymermelts,evenifthechainsareveryflexibleandthin. to PetroleumResearchGrant PRF# 43923-AC7 and the
This mechanism rests upon the interplayof chainconnec- LEA“MacromoleculesinFinelyDividedMedia”forfinan-
tivity and the incompressibility of the system, which gen- cial support. J.B. acknowledges financial support by the
erates an effective repulsion between chain segments of IUF and from the European Community’s “Marie-Curie
curvilinear length s (fig. 1). A polymer in dense solutions Actions” under contractMRTN-CT-2004-504052(POLY-
maynotbeviewedasonesoftsphere(orellipsoid)[13,14], FILM).
but as a hierarchy of nested segmental correlation holes
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p-4