We analyze dynamical heterogeneities in a simulated “bead-spring” model of a nonentangled, supercooled polymer melt. We explore the importance of chain connectivity on the spatially heterogeneous motion of the monomers. We find that when monomers move, they tend to follow each other in one-dimensional paths, forming strings as previously reported in atomic liquids and colloidal suspensions. The mean string length is largest at a time close to the peak time of the mean cluster size of mobile monomers. This maximum string length increases, roughly in an exponential fashion, on cooling toward the critical temperature TMCT of the mode-coupling theory, but generally remains small, although large strings involving ten or more monomers are observed. An important contribution to this replacement comes from directly bonded neighbors in the chain. However, mobility is not concentrated along the backbone of the chains. Thus, a relaxation mechanism in which neighboring mobile monomers along the chain move predominantly along the backbone of the chains, seems unlikely for the system studied.

1.
J.
Jäckle
,
Rep. Prog. Phys.
49
,
171
(
1986
).
2.
W. Götze, in Proceedings of the Les Houches Summer School of Theoretical Physics, Les Houches 1989, Session LI, edited by J. P. Hansen, D. Levesque, and J. Zinn-Justin (North-Holland, Amsterdam, 1991), pp. 287–503.
3.
P.
Lunkenheimer
,
U.
Schneider
,
R.
Brand
, and
A.
Loidl
,
Contemp. Phys.
41
,
15
(
2000
).
4.
H.
Sillescu
,
J. Non-Cryst. Solids
243
,
81
(
1999
).
5.
M. D.
Ediger
,
Annu. Rev. Phys. Chem.
51
,
99
(
2000
).
6.
R.
Richert
,
J. Phys.: Condens. Matter
14
,
R703
(
2002
).
7.
S. C.
Glotzer
,
J. Non-Cryst. Solids
274
,
342
(
2000
).
8.
K.
Schmidt-Rohr
and
H. W.
Spiess
,
Phys. Rev. Lett.
66
,
1991
(
1991
).
9.
U.
Tracht
,
M.
Wilhelm
,
A.
Heuer
,
H.
Feng
,
K.
Schmidt-Rohr
, and
H. W.
Spiess
,
Phys. Rev. Lett.
81
,
2727
(
1998
).
10.
M. T.
Cicerone
and
M.
Ediger
,
J. Chem. Phys.
103
,
5684
(
1995
).
11.
B.
Schiener
,
R.
Böhmer
,
A.
Loidl
, and
R. V.
Chamberlin
,
Science
274
,
752
(
1996
).
12.
M.
Yang
and
R.
Richert
,
J. Chem. Phys.
115
,
2676
(
2001
).
13.
Information about the length scale of this correlation can be directly measured via multidimensional NMR experiments. Typically, it is of the order of 3 nm close to Tg (Ref. 9).
14.
E.
Weeks
,
J. C.
Crocker
,
A. C.
Levitt
,
A.
Schofield
, and
D. A.
Weitz
,
Science
287
,
627
(
2000
).
15.
W. K.
Kegel
and
A.
van Blaaderen
,
Science
287
,
290
(
2000
).
16.
E.
Bartsch
,
Curr. Opin. Colloid Interface Sci.
3
,
577
(
1998
).
17.
W.
van Megen
,
Transp. Theory Stat. Phys.
24
,
1017
(
1995
).
18.
W.
van Megen
and
S. M.
Underwood
,
Phys. Rev. E
49
,
4206
(
1994
).
19.
W.
Götze
and
L.
Sjögren
,
Phys. Rev. A
43
,
5442
(
1991
).
20.
W.
van Megen
,
T. C.
Mortensen
,
J.
Müller
, and
S. R.
Williams
,
Phys. Rev. E
58
,
6073
(
1998
).
21.
R.
Yamamoto
and
A.
Onuki
,
Phys. Rev. Lett.
81
,
4915
(
1998
).
22.
W.
Kob
,
C.
Donati
,
S. J.
Plimpton
,
P. H.
Poole
, and
S. C.
Glotzer
,
Phys. Rev. Lett.
79
,
2827
(
1997
).
23.
C.
Donati
,
J. F.
Douglas
,
W.
Kob
,
S. J.
Plimpton
,
P. H.
Poole
, and
S. C.
Glotzer
,
Phys. Rev. Lett.
80
,
2338
(
1998
).
24.
C.
Donati
,
S. C.
Glotzer
,
P. H.
Poole
,
W.
Kob
, and
S. J.
Plimpton
,
Phys. Rev. E
60
,
3107
(
1999
).
25.
Y.
Gebremichael
,
T. B.
Schrøder
,
F. W.
Starr
, and
S. C.
Glotzer
,
Phys. Rev. E
64
,
051503
(
2001
).
26.
N.
Giovambattista
,
S. V.
Buldyrev
,
F. W.
Starr
, and
H. E.
Stanley
,
Phys. Rev. Lett.
90
,
085506
(
2003
).
27.
B.
Doliwa
and
A.
Heuer
,
Phys. Rev. Lett.
80
,
4915
(
1998
).
28.
B.
Doliwa
and
A.
Heuer
,
Phys. Rev. E
61
,
6898
(
2000
).
29.
B.
Doliwa
and
A.
Heuer
,
J. Non-Cryst. Solids
307–310
,
32
(
2002
).
30.
K.
Vollmayr-Lee
,
W.
Kob
,
K.
Binder
, and
A.
Zippelius
,
J. Chem. Phys.
116
,
5158
(
2002
).
31.
C.
Bennemann
,
C.
Donati
,
J.
Baschnagel
, and
S. C.
Glotzer
,
Nature (London)
399
,
246
(
1999
).
32.
C.
Bennemann
,
W.
Paul
,
K.
Binder
, and
B.
Dünweg
,
Phys. Rev. E
57
,
843
(
1998
).
33.
K.
Binder
,
J.
Baschnagel
, and
W.
Paul
,
Prog. Polym. Sci.
28
,
115
(
2003
).
34.
C.
Bennemann
,
J.
Baschnagel
, and
W.
Paul
,
Eur. Phys. J. B
10
,
323
(
1999
).
35.
M.
Aichele
and
J.
Baschnagel
,
Eur. Phys. J. E
5
,
229
(
2001
).
36.
M.
Aichele
and
J.
Baschnagel
,
Eur. Phys. J. E
5
,
245
(
2001
).
37.
C.
Bennemann
,
W.
Paul
,
J.
Baschnagel
, and
K.
Binder
,
J. Phys.: Condens. Matter
11
,
2179
(
1999
).
38.
C.
Bennemann
,
J.
Baschnagel
,
W.
Paul
, and
K.
Binder
,
Comput. Theor. Polym. Sci.
9
,
217
(
1999
).
39.
S.-H.
Chong
and
M.
Fuchs
,
Phys. Rev. Lett.
88
,
185702
(
2002
).
40.
J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, London, 1986).
41.
S.
Kämmerer
,
W.
Kob
, and
R.
Schilling
,
Phys. Rev. E
58
,
2131
(
1998
).
42.
W.
Kob
and
H. C.
Andersen
,
Phys. Rev. E
51
,
4626
(
1995
).
43.
F.
Sciortino
,
P.
Gallo
,
P.
Tartaglia
, and
S.-H.
Chen
,
Phys. Rev. E
54
,
6331
(
1996
).
44.
S.
Mossa
,
R.
Di Leonardo
,
G.
Ruocco
, and
M.
Sampoli
,
Phys. Rev. E
62
,
612
(
2000
).
45.
J.
Horbach
,
W.
Kob
, and
K.
Binder
,
Philos. Mag. B
77
,
297
(
1998
).
46.
J.
Colmenero
,
F.
Alvarez
, and
A.
Arbe
,
Phys. Rev. E
65
,
041804
(
2002
).
47.
R.
Zorn
,
Phys. Rev. B
55
,
6249
(
1997
).
48.
M. Aichele et al. (unpublished).
49.
V.
Krakoviack
,
J. P.
Hansen
, and
A. A.
Louis
,
Europhys. Lett.
58
,
53
(
2002
).
50.
M.
Guenza
,
Phys. Rev. Lett.
88
,
025901
(
2002
).
51.
M.
Fuchs
,
W.
Götze
, and
M. R.
Mayr
,
Phys. Rev. E
58
,
3384
(
1998
).
52.
M. Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, Oxford, 1986).
53.
K. Kremer and G. S. Grest, in Monte Carlo and Molecular Dynamics Simulations in Polymer Science, edited by K. Binder (Oxford University Press, New York, 1995), pp. 194–271.
54.
Suppose that Gs(r,t)=[3/2πg0(t)]3/2exp[−3r2/2g0(t)]. The fraction of the 6.5% of the most mobile monomers can then be expressed as: 0.065=4π∫r*drr2Gs(r,t)=(2/π)Γ(3/2,u*), where u=3r2/2g0 and Γ(α,x) is the incomplete gamma function. This condition yields u*≃3.614. Similarly, we can write for the MSD of the mobile monomers g0,m(t)=(4π/0.065)∫r*drr4Gs(r,t)=(4/0.065)g0(t)Γ(5/2,u*). Using the identity Γ(α+1,x)=αΓ(α,x)+xαe−x, we obtain Eq. (6).
55.
This finding is not unreasonable. The mean-quartic displacement is more sensitive to large displacements than g0. Thus, it particularly samples the large-r wing of Gs(r,t) similar to g0,m.
56.
W.
Götze
,
J. Phys.: Condens. Matter
11
,
A1
(
1999
).
57.
We have not considered the possibility of loops which would occur if a set of particles replaced each other in a circular fashion. In our analysis loops occurred very seldomly. Mostly, they were of length two (or four) and consisted of contiguous chain segments of length two. This means that bonded, neighboring monomers switch places. Because the loops were so rare, we decided not to analyze the results for open and closed strings separately. We note that in the LJ system studied in Ref. 23, longer loops were found.
58.
G. Strobl, The Physics of Polymers: Concepts for Understanding their Structures and Behavior (Springer, Berlin–Heidelberg, 1997).
59.
Note that we include “strings” of size one in analogy with percolation theory, where clusters of size one are customarily included in the evaluation of mean cluster size, and with equilibrium polymerization, where polymers of size one are customarily included in the calculation of mean polymer size.
60.
Y. Gebremichael, M. Vogel, and S. C. Glotzer (unpublished).
61.
R.
Bellissent
,
L.
Descotes
, and
P.
Pfeuty
,
J. Phys.: Condens. Matter
6
,
A211
(
1994
).
62.
Y.
Rouault
and
A.
Milchev
,
Phys. Rev. E
51
,
5905
(
1995
).
63.
S. C.
Greer
,
Adv. Chem. Phys.
94
,
261
(
1996
).
64.
J. P.
Wittmer
,
A.
Milchev
, and
M. E.
Cates
,
J. Chem. Phys.
109
,
834
(
1998
).
65.
Suppose we have a vector d on the unit sphere S2 connecting the origin to a point on the surface of S2 in direction (θ, φ). θ∈[0,π[ is the latitude and φ∈[0,2π[ the longitude. Furthermore, let be the vector from the origin to the north pole (θ=0,φ=0). Then, the angles defined in Eqs. (16) and (17) correspond to a measurement of the latitude θ, the angle between d and , after integration over φ. Because the sector of the unit sphere for fixed θ is small if θ is close to the poles, but large if it is close to the equator, the φ-integrated probability of an isotropic distribution of vectors on S2,Piso(θ), is maximum at θ=90°.
66.
The symmetry in Fig. 13 arises because we calculate Pd,r(θ[di(t),rij]) by averaging over the displacements di(t) and dj(t) for the same rij. Thus, the rise of Pd,r(θ)/Piso(θ) close to θ=0° results from motions of particle i in direction of rij, whereas that close to θ=180° comes from displacements of particle j against the orientation of rij. Note that the analysis of Ref. 23 only consider the former case. Therefore, the probability distribution is asymmetric (see Fig. 3 of Ref. 23).
67.
An analysis of Pd,d(θ[di(t),dj(t)])/Piso(θ[di(t),dj(t)]) in the time interval 1≲t⩽104 shows that Pd,d(θ[di(t),dj(t)])>Piso(θ[di(t),dj(t)]) for θ≲60°, whereas Pd,d(θ[di(t),dj(t)])→0, as θ→180°.
68.
T.
Franosch
,
M.
Fuchs
,
W.
Götze
,
M. R.
Mayr
, and
A. P.
Singh
,
Phys. Rev. E
55
,
7153
(
1997
).
69.
W.
Kob
and
H. C.
Andersen
,
Phys. Rev. E
52
,
4134
(
1995
).
70.
M.
Fuchs
,
W.
Götze
,
S.
Hildebrand
, and
A.
Latz
,
J. Phys.: Condens. Matter
4
,
7709
(
1992
).
71.
W.
Götze
and
L.
Sjögren
,
Z. Phys. B: Condens. Matter
65
,
415
(
1987
).
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