Feature Article
NMR Reveals Non-Distributed and Uniform
Character of Network Chain Dynamics
Kay Saalwächter,* Jens-Uwe Sommer
Results of different NMR investigations of elastomers are reviewed with respect to their
significance for statistical models of rubber elasticity. In contrast to earlier work based on
lineshape analysis and relaxometry, results of recent multiple-quantum experiments indicate
that the NMR-detected dynamic chain order parameter, which reflects the conformational
space of individual monomer units at which
the signal is detected locally, is a rather narrowly distributed quantity. Constraints to the
dynamics and the conformations of a network
chain thus act uniformly and appear as a
dynamic average over chains of different
length and with different end-to-end separations. All our findings are in good agreement
with large-scale computer simulations. Anomalies on swelling such as chain desinterspersion
at the early stages and the appearance of heterogeneities, are also discussed.
Introduction
Gaussian statistics is one of the central paradigms in the
theory of dense polymer systems.[1] Specifically, the
assumption that the well-documented (close-to) Gaussian
end-to-end distribution of network chains and sub-chains
is reflected in their conformational space and thus in the
overall entropy, represents a cornerstone assumption
when thermodynamic or elastic properties of networks
K. Saalwächter
Institut für Physik, Martin-Luther-Universität Halle-Wittenberg,
Friedemann-Bach-Platz 6, D-06018 Halle, Germany
E-mail: [email protected];
URL : www.physik.uni-halle.de/nmr
J.-U. Sommer
Leibniz-Institut für Polymerforschung Dresden e. V., Hohe Straße
6, D-01069 Dresden, Germany
E-mail: [email protected]
Macromol. Rapid Commun. 2007, 28, 1455–1465
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far above Tg are to be calculated on the basis of classical
models of rubber elasticity.[2–4] The quantitative interpretation of NMR experiments on elastomers was so far
also mainly based on the same assumption of a (frozen-in)
Gaussian distribution of the end-to-end vectors,[5–12] and
the qualitative agreement between theoretical and
experimental spectra or relaxation curves appeared to
support the traditional picture of rubber elasticity.
In contrast, our recent results based on multiplequantum NMR are much more sensitive to the actual
distribution of chain conformations, and indicate that an
influence of a frozen-in Gaussian distribution on NMR
observables is largely absent. The conformational space
that is probed in time by any given chain segment is found
to be much more uniform than expected theoretically. This
is even true for vulcanized rubbers where, in addition to
the distribution of the end-to-end vectors of the strands, a
high chain-length polydispersity has to be taken into
account.[13] In a recent paper,[14] we have shown that
DOI: 10.1002/marc.200700169
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K. Saalwächter, J.-U. Sommer
Kay Saalwächter studied Chemistry at the Universities of Mainz and Freiburg, Germany, until
1997. After his PhD in Physical Chemistry with H.
W. Spiess at the Max-Planck Institute for Polymer Research (Mainz) in 2000, including postgraduate research with K. Schmidt-Rohr at the
Polymer Science and Engineering Deptartment,
University of Massachusetts (Amherst, USA),
focusing on solid-state NMR methodology, he
joined the group of H. Finkelmann at the Institute of Macromolecular Chemistry in Freiburg.
He obtained his Habilitation in 2004 and was
appointed Full Professor of Experimental Physics
at the Martin-Luther-Universität Halle-Wittenberg in 2005. His research is concerned with
structure and dynamics of mainly polymeric
materials, with current topics such as polymer
crystallization, dynamics and swelling of elastomers, transport in complex polymer systems,
phase behavior of liquid crystals, and chain
dynamics in confined geometry, using NMR
and other spectroscopic techniques.
Jens-Uwe Sommer studied physics in Merseburg
and Jena and obtained his PhD in 1991 working in
the group of G. Helmis on dynamical models of
polymer networks. After post-doctoral research
in Regensburg and Saclay, he joined the group of
A. Blumen in Freiburg were he obtained his
Habilitation in 1998. In 2000, he became a staff
scientist of the CNRS in France, where he worked
at the ‘‘Institut de Chimie des Surfaces et Interfaces’’ in Mulhouse. In 2006 he was appointed
Full Professor for Theory of Polymers at the
Technische Universität Dresden and is since then
heading a research group at the Leibniz-Institute
of Polymer Research in Dresden. His research is
focused on the field of statistical physics of soft
condensed matter using both analytical and
simulation methods. His current research interests include polymers at surfaces and interfaces,
polymer dynamics, networks, and crystallization
and structure formation far from equilibrium.
computer simulations of end-linked polymer networks are
in good agreement with the results of our NMR experiments. Deviations from Gaussian end-to-end vector
statistics in the range of short separations can be explained
by ‘‘selective’’ entanglement effects. We have argued that
in almost closed loops of chains that would explore a
particularly large conformational space, entanglements
have a greater impact on the number of conformations
swept out by the segments as compared to chains which
are rather stretched.
The NMR properties of elastomers are governed by the
fact that the fast fluctuations of network chains between
cross-links and other topological constraints are not
completely isotropic.[15] The remaining degree of local
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segmental order is for instance also reflected in classical
strain birefringence experiments, which make use of the
break of isotropic symmetry of the end-to-end vector
orientation distribution by application of uniaxial deformation.[16] NMR offers the advantage that the local chain
order parameter can be quantified in the undeformed
state; the corresponding (molecular and localized!) observable is well known to be directly proportional to
macroscopic measures of the cross-link density such as
the elastic modulus or the equilibrium degree of swelling.
Traditionally, NMR methods based on transverse
relaxation of proton or deuterium nuclei are the established approaches to assess the local chain order,[9,15,17–27]
and, as mentioned, theoretical analyses of such data were
based on the assumption of a broad distribution of order
parameters, related to the Gaussian distribution of chain
end-to-end separations.[5–12] Applications of multiplequantum (MQ) NMR[28] recently gained momentum for
the investigation of polymer chain dynamics and
order,[29–32] and we found that it offers multiple advantages.[33–36] First, the quasi-static (temperature-independent)
chain order phenomenon and the timescale of chain
motions can be investigated reliably and separately.
Second, the distribution function of local chain order is
accessible directly, which offers the possibility to assess
the conformational statistics. Third, the technique is
applicable on inexpensive low-field instrumentation and
is therefore amenable to high-throughput screening
applications.[34]
In this contribution, we review the implications of our
previous investigations of different elastomer systems,
comprising end-linked PDMS ‘‘model’’ networks and
vulcanized rubbers, with respect to their bearing on
rubber elasticity concepts, without detailed recourse to the
technical NMR aspects. In addition, as yet unpublished
natural-abundance deuteron NMR data and an interpretation in terms of MQ NMR results obtained for the same
network are presented in order to support our critique of
the numerous previous, yet incorrect, interpretations of
proton and deuteron NMR data of rubbers. Support to
our interpretation of the NMR observable in terms of
conformational entropy is also provided by our computer
simulations and experiments on swollen elastomers,
where unexpected trends are observed in agreement with
the well-known anomalies of the osmotic modulus.[37–39]
Conceptual Basics and Experiments
Network Chain Order
The theory outlined in this section has been presented
before in the many cited publications concerned with the
NMR response of elastomers. However, most of these
DOI: 10.1002/marc.200700169
NMR Reveals Non-Distributed and Uniform Character . . .
works ultimately focus on the calculation of the NMR
signal, and for the non-NMR specialist it is usually difficult
to appreciate the link to the physics of polymers and
understand the models used. Therefore, the following
detailed account is meant to fill this gap by omitting most
technical details.
The principle underlying the NMR observation of
polymer dynamics is sketched in Figure 1(a). Typically,
the spectroscopic fine structure is dominated by a dipolar
or quadrupolar interaction, whose angular dependence
follows the second Legendre polynomial, P2(cos u). The
NMR signal is detected at the locus of a specific nucleus
in the monomer unit, and the orientation of the NMR
interaction tensor is specified by the chemical structure. In
order to drop any molecular detail, the concept of the Kuhn
segment can be invoked and a number of monomers
can be combined to form what is typically referred to as
‘‘NMR submolecule’’.[5,32,40–42] The intra-submolecule
dynamics is assumed to be too fast to exert any measurable effect, but leads to a pre-averaging of the localized
coupling. In effect, the (dipolar) coupling constant is
reduced to a model-dependent effective value Deff, and
the angle u then points along the segmental axis, i.e., the
polymer backbone.
The NMR response is ultimately given by the timeintegral over the interaction, which fluctuates randomly
on account of the network chain dynamics,
fNMR Deff
Z
t
P2 ½cosuðtÞdt;
and has the dimension of a phase angle (since Deff is given
in rads1). When the chain dynamics is fast on the
timescale defined by D1
eff ð 20 msÞ, then the interaction is
‘‘quasi-static’’, i.e., the spectral response resembles that of a
system without motion, but with an averaged, residual
coupling constant
Dres ¼ Sb Deff :
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(2)
Sb ¼ Dres =Deff is the local (dynamic) segmental order
parameter of the polymer backbone, and reflects the
conformational space that is set by the presence of
cross-links or topological constraints, and it is the central
observable that links NMR and the polymer physics of
elastomers.
In reality, however, the segmental dynamics extends
into the NMR timescale, introducing relaxation effects
(signal decay, line broadening). These are commonly
treated in terms of the time autocorrelation function of P2,
Cðjta tb jÞ ¼ hP2 ðcosuta ÞP2 ðcosutb Þi;
Figure 1. (a) Observables of network chain dynamics. Orientation
fluctuations of the segment vector b(t), accessible by NMR as
angular fluctuations u(t) relative to the magnetic field B0, are
locally anisotropic. (b) They can be described by an orientation
autocorrelation function C(t), whose plateau value is related to a
local backbone order parameter Sb. Formally, Sb corresponds to a
$
local nematic director n, which, however, assumes a different
orientation for each subchain.
(1)
0
(3)
which is depicted in Figure 1(b). In simple words, C(t)
describes the probability to find a certain segment in one
and the same orientation again after a time t has passed,
i.e., the loss of orientational memory. Within this scenario,
D1
eff does not usually correspond to a time well in the
plateau region, where CðtÞ S2b . In order to analyze actual
data such as T2 relaxation curves or quadrupolar spectra
(where Deff is of course associated with quadrupolar
coupling), the fitting function must embody a model for
C(t). This can be subject to ambiguities (often related to
multiple fitting parameters), which calls for advanced
approaches to extract the plateau value of Sb. MQ NMR has
proven as a unique approach in this regard, as it provides
an experimental signal function, the so-called normalized
double-quantum (DQ) build-up curve, that depends only
on Dres but not on the segmental dynamics.[32,33,36] From
such curves, even a potential distribution of Sb can be
extracted, and the proof that Sb really represents the
long-time average over the full conformational space is
easily evidenced by its temperature independence over
tens of Kelvin.[33] Another signal function, the MQ sum
decay curve, solely reflects the dynamic processes, and
careful analysis showed that the relaxation processes are
in fact dominated by the segmental dynamics.[36] Slow,
cooperative processes on the milliseconds to seconds
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K. Saalwächter, J.-U. Sommer
timescale, that could lead to a decay of the plateau of C(t) at
longer times, and that were earlier held responsible for the
major relaxation effect, were shown to be unimportant.[36,43]
Sb, which is in NMR obtained as a time integral
R t0
Sb ¼
0
P2 ½cosuðtÞdt
t0 P2 ½coshui
(4)
extending to a time t0 well in the plateau region of C(t),
provides the link to polymer physics. hui is the time$
averaged orientation of the director n connecting the
constraint points, and the actual NMR response (see below)
is obtained as a powder average h. . .ihui that takes into
account the isotropic distribution of hui in an unstretched
sample. Sb was first calculated by Kuhn and Grün in the
context of a theory for strain birefringence:[16]
Sb ¼
3 r2
5N
(5)
where r is the ratio of the end-to-end vector of the chain
segment to its average, unperturbed melt state (r2 ¼ r2 =r02 ),
and N is the number of statistical (Kuhn) segments
between cross-links or constraints (e.g., entanglements).
The latter establishes the connection to entanglement
theories or theories of rubber elasticity and swelling.[2] The
1/N relationship and thus the direct proportionality of
Sb Dres to the equilibrium degree of swelling or to the
elasticity modulus is very well supported by a variety of
NMR experiments, as reviewed in ref.[32]
Phenomenologically, Sb is thus directly related to
conformational entropy, and the changes of its mean
value and its distribution upon deformation and swelling
of different kinds of rubbers are highly valuable sources of
data that can be used to test specific models of rubber
elasticity. It should however be noted that a direct
calculation of the entropy from a given Sb is a non-trivial
and model-dependent undertaking, that has only been
attempted for some cases of localized dynamics in
proteins.[44,45]
To date, the models used to analyze NMR data are based
on single chains and assume either (somewhat unphysically) no distribution of the order parameter or the residual
coupling[20–22,24,25] or use a static average over a Gaussian
end-to-end distribution,[5–12,25]
4 3=2 2
PðrÞ ¼ pffiffiffi 32
r exp 32r2
p
(6)
Gaussian statistics is a cornerstone assumption in most
theories of polymer dynamics, and its potential effect on
the measured data is easily evaluated by combination
with Equation (5), from which the distribution of Sb is
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easily obtained as
rffiffiffiffiffiffisffiffiffiffiffiffiffiffiffiffiffi
27 jSb j 3jSb j=ð2hSb iÞ
PðjSb jÞ ¼
e
2p hSb i3
(7)
The explicit form of this gamma distribution was first
published in ref.[33] Its positive average hSb i is directly
related to its standard deviation, s 2g ¼ 23hSb i2 . In earlier
publications, analytical results were only given for the
final time-domain NMR signal or the associated spectral
line shape.[5–12] The latter is given by FTfhcos32fNMR iSb ;hui g,
and may be referred to as ‘‘super-Lorentzian’’; its observation in deuterium spectra (or corresponding transverse
relaxation data) of elastomers was interpreted as a
confirmation of the relevance of Gaussian statistics. Below,
new evidence will be presented that suggests that this
lineshape is not generally observed, and that this
conclusion is not correct.
It should be kept in mind that the distribution given by
Equation (7) neglects any influence of a potentially serious
network chain polydispersity, which we therefore address
here for the first time. For a randomly cross-linked rubber,
the network chain length distribution is exponential.[13]
For large average hNi, the corresponding N-weighted
distribution (necessary since NMR detects each monomer
unit) approximates to
PðNÞ ¼
N
hNi2
eN=hNi
(8)
Substituting hSb i ¼ 3=ð5NÞ into Equation (7) and integrating over N gives
PðjSb j; hNiÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7=2
75 5
5
3
:
¼
jSb jhNi 1 þ jSb jhNi
8 2
2
(9)
Note finally that the traditional models of rubber
elasticity, such as the affine network model and the
phantom model, as well as versions with statistically
distributed but fixed constraints such as the constrainedjunction model or the diffused-constraint model (see ref.[3]
for a review), when taken literally, would all lead to
observations of a significant Sb distribution, reflecting
chains with different but fixed lengths and end-to-end
separations.
Let us point out some interesting features of the order
parameter distribution in Equation (9). First, the expectation value of the order parameter is given by
hSb i ¼ 3=ð5hNiÞ. However, the second moment and all
higher moments do not exist (i.e., they diverge). Thus, the
distribution Equation (9) has a formal singular character.
DOI: 10.1002/marc.200700169
NMR Reveals Non-Distributed and Uniform Character . . .
The fluctuations of the order parameter can become
arbitrarily large. This originates from the assumption of a
continuous strand length parameter N which is formally
extended to zero. In reality the distribution is limited by a
smallest strand length which regularizes the distribution.
On the other hand this means that fluctuations of the order
parameter are determined by the smallest strand lengths.
NMR Spectroscopy
MQ NMR experiments were performed folllowing previously published procedures.[32,33] The corresponding
data and results discussed in this paper were published
before, and the reader is referred to the respective
literature for details.[33,35,46] In short, we used a dedicated
MQ pulse sequence of Baum and Pines, which is applied to
initial z magnetization twice for equal times tDQ, first for
the excitation of mainly DQ coherences, and then their
reconversion into final z magnetization that is detected
after a final 90 8 pulse. Phase cycling of either the excitation
or the reconversion block is used to filter out either the DQ
coherences or a reference signal that comprises all signals
that do not contribute to DQ coherences. The former gives a
tDQ-dependent build-up function that depends on the
phase factor given by Equation (1) as IDQ hsin2 fNMR i,
while the reference signal follows Iref hcos2 fNMR i (in
contrast, transverse relaxometry just gives a decay
function cos32fNMR , where h. . .i denotes a time and
ensemble (powder) average. Incoherent relaxation effects
due to fast and intermediate molecular motions are
normalized away by calculating the normalized DQ
build-up signal InDQ ¼ IDQ =ðIDQ þ Iref Þ, which can be
analyzed in terms of residual dipolar coupling distributions only. On the other hand, the decay of the MQ sum
signal IDQ þ Iref reflects only the relaxation effects.
A natural-abundance deuteron NMR spectrum of
an end-linked long-chain PDMS network sample
(Mc ¼ 47 kgmol1) previously investigated by MQ
NMR[33] was measured on a Bruker Avance 500 solid-state
NMR spectrometer operating at a magnetic field strength
of 11.7 T and a corresponding deuterium resonance
frequency of 76.773 MHz using a single-pulse experiment.
The X-channel of a commercial Bruker static doubleresonance probe was used with a 908 pulse of 4.5 ms,
acquiring 262.144 transients with a 1 s recycle delay over
the course of 3 d. The sample amount was about 50 mg
centered in a 5 mm RF coil. Proton decoupling was not
applied, as it is not necessary due to the weak relative
strength of the 2H–1H dipolar coupling as compared to the
2
H quadrupolar coupling. As the signal is rather narrow it
was not necessary to use the customary solid echo before
detection (i.e., the dead time problem is of very minor
importance).
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Results and Discussion
Distributions in Bimodal Model Networks
Bimodal end-linked networks composed of cocross-linked
long and very short telechelic chains are known to be
phase-separated systems composed of nanometer-sized
clusters of short chains interconnected by long chains, as a
result of mere statistics, since the short chains always
contribute the major fraction of cross-linkable ends.[33,47]
They represent an ideal test case for the ability of the MQ
technique to detect a heterogeneous distribution of
cross-link density in an elastomer.
Results from ref.[33] are summarized in Figure 2. The
bicomponent character of the build-up curves in (a) is
clearly evidenced by the fact that the curves for the
bimodal networks can be modeled by mere superposition
of the experimental curves for the pure-component
networks, weighted by their respective fractions. Fitting
Figure 2. (a) Normalized DQ build-up curves of bimodal endlinked PDMS networks. Long and short chains have Mn ¼ 47
and 0.8 kgmol1, respectively. The solid lines are weighted
superpositions of the experimental pure-component curves,
and the dashed and dash-dotted lines are fits using different
distribution models. (b) Order parameter distributions for the
same networks, obtained by numerical regularization analysis of
the build-up curves in (a). Data from ref.[33]
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K. Saalwächter, J.-U. Sommer
of these curves can be performed by using semi-analytical
build-up functions that are reviewed in ref.[32,35] One can
either use build-up functions derived on the basis of
specific distribution models such as a Gaussian distribution, or a gamma distribution according to Equation (7). A
third option is to use a numerical Tikhonov regularization
procedure to estimate the distribution without assumptions on the shape. The corresponding results in Figure 2(b)
nicely demonstrate the two-component nature of these
distributions for bimodal networks. Notably, the maximum of the more weakly ordered (¼cross-linked) longchain component does not change appreciably by addition
of short chains, indicating that these chains are not
hindered by the presence of the short ones.
Order Parameter Distributions and Comparison
with Theory
The distributions determined for the monomodal longchain network from Figure 2 are compared in Figure 3. The
distributions obtained by imposing a Gaussian shape and
the regularization result agree rather well. The latter only
captures an additional small fraction of more highly
cross-linked regions that is associated with some heterogeneities. Now, the observation of central importance is
that these distributions are rather narrow, and that the
gamma distribution that would be expected on the basis of
single-chain models does not fit the data well at all. Note
that the gamma distribution does not yet consider the
polydispersity of the precursor (PD 1.86), which should
lead to an even broader distribution.
In order to support this reasoning, Figure 4(a) presents a
natural-abundance 2H spectrum of the same network. In
Figure 4. (a) Natural-abundance deuteron NMR spectrum of the
long-chain end-linked PDMS sample (see Figure 2) compared with
simulated spectra based on different order parameter distributions (see Figure 3): (b) regularization result, (c) Gaussian
distribution, and (d) gamma distribution (super-Lorentzian). In
(a), the scaled and broadened regularization result is shown as
dotted line.
the literature, super-Lorentzian line shapes (or corresponding transverse relaxation functions) have frequently
been discussed with respect to the relevance of Gaussian
end-to-end distributions.[5–12] Obviously, the long-chain
PDMS network does not exhibit such a line shape. The
spectra in Figure 4(b)–(d) were calculated on the basis of
the distributions given in Figure 3. The conversion from
the Sb derived from 1H dipolar data to 2H residual
quadrupolar frequency,
vQ;res =2p ¼ 125 kHz
P2 ðcos109:5 ÞP2 ðcos90 ÞSb ;
Figure 3. Distribution functions for the order parameter of a
long-chain end-linked PDMS network (see Figure 2) obtained
by the different fitting strategies.
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(10)
is based on the well-known rigid-limit quadrupolar
splitting of 125 kHz for an aliphatic CD bond, scaled by
fast methyl rotation, P2 ðcos109:5 Þ, and by another factor
of P2 ðcos90 Þ ¼ 12, which represents the model assumption of a 908 orientation of the CD3 group with respect to
the backbone direction.[33]
Under these assumptions, the theoretical spectra are too
broad, but scaling by 50% yields perfect agreement of the
theoretical spectrum based on the regularized distribution
with the experiment. The slight splitting that is exhibited
in the undamped spectrum (Figure 4(b) is not observable
due to an expected line broadening. Note that the MQ
results are free of relaxation effects, while in 2H spectra,
some degree of exponential line broadening due to
segmental dynamics must be considered.
DOI: 10.1002/marc.200700169
NMR Reveals Non-Distributed and Uniform Character . . .
The deviation in the width is straightforwardly
explained by the simplistic model assumption of the fixed
908 orientation of the CD3 in the Kuhn segment.
Intra-segmental dynamics can easily reduce the expected
coupling further—the necessary scaling could for instance
be modeled by an average 668 instead of a 908 orientation.
Note that the proton dipolar reference value needed to
convert Dress into Sb suffers less from such changes in the
model, due to a balance of multiple intra- and inter-group
couplings along the backbone.[35] In any way, our
qualitative argument is not affected, as the line shape
in Figure 4(b) is the only one that does give a good match
with the experiment after scaling and broadening. The
super-Lorentzian line in Figure 4(d) can by no means
describe the experimental data.
A possible explanation why previous data did exhibit
features hinting at the significance of the gamma
distribution is explained as follows. Most randomly and
end-linked networks exhibit a considerable (10%) fraction of sol, dangling chains, and/or loops, which are
isotropically mobile and always contribute a sharp
spectral center that dominates the line shape. Our
end-linked networks have only about 1% or less of such
defects,[33] and the 2H spectrum is thus more flat in the
center. Such a defect fraction can either be extracted from
the MQ data, or by performing a (less reliable) decomposition of relaxation curves. This was not done in the
papers mentioned above, nor is any information given on
the fraction of mobile chains, and we thus conclude that
this contribution is to be made responsible for the
erroneous interpretations.
We now address the potential significance of network
chain length polydispersity for the example of randomly
cross-linked (vulcanized) natural rubber, for which the
strand length distribution is well-known to be exponential, see Conceptual Basics and Experiments, in particular
Equation (8) and (9). Figure 5(a) displays the experimental
distributions for rubbers that span an Mc range from about
20 down to about 5 kgmol1 (based on swelling
experiments), thus covering networks with strands that
are longer as well as shorter than the rheological
entanglement value of about 6 kgmol1.[35] All distributions are rather narrow, with standard deviations that
never exceed 25% of the average couplings. It should be
mentioned that the apparent Mc that can be obtained from
the NMR data are significantly lower than those from
swelling as a result of the packing/entanglement contributions, yet nc 1=Mc 1=N determined in both ways
are always perfectly proportional.[35]
Neither the gamma distribution result shown in
Figure 5(b), nor the even broader distribution that
considers the additional polydispersity [Equation (9)],
are anywhere near the experimental observations. The
very narrow distribution of the order parameter is thus in
Macromol. Rapid Commun. 2007, 28, 1455–1465
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Figure 5. (a) Order parameter distributions for natural rubber
vulcanized with different amounts of sulfur. (b) Gamma distribution (Equation (7)) for the same average Sb as measured for the
10 phr sample, as compared with the distribution according to
Equation (9) taking into account the additional chain length
polydispersity. The latter was simply scaled to the same maximum position. Data in (a) from ref.[35]
contrast to the picture of an affine network model where
the junction points are frozen-in (pinned to the elastic
matrix) and the strand segments are freely fluctuating
under the given constraints.
We should make a final comment with respect to the
fact that Equation (9) does not have well defined moments.
This is due to a definition problem of the lower end of the N
range, which corresponds to an ill-defined upper cut-off for
the range of possible Sb. Implications of problems associated with the pecularities of exponential distributions of
N will be dealt with in an upcoming publication.
Single-chain Treatment versus Segmental Averaging
in Simulations
In order to stress the advantage of NMR as a segmentbased technique, this section highlights the significant
differences that are observed when simulation data, obtained by large-scale bond-fluctuation Monte-Carlo techniques,[14] are simply analyzed on the basis of end-to-end
(cross-link) separation distributions, and by performing a
proper segment-based time average.
First, consider only the consequences of time averaging
on the cross-link separations. Figure 6(a) displays simulated end-to-end distributions, comparing a quasi-static
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K. Saalwächter, J.-U. Sommer
the true conformational entropy, and thus the elasticity, of
the network. A distribution based on a running time
average according to Equation (4) is shown as the dashed
line. In accordance with the experimental results shown
above, it is remarkably narrower. Reasons for this
deviation as well as implications are discussed further
below.
Order Parameter Distributions on Swelling
Additional support to our finding of largely uniform chain
statistics is provided by our experiments and simulations
for swollen elastomers. In increasingly swollen networks,
one can directly observe the trend for the conformational
space available to the individual segments. The data in
Figure 7 clearly show that the experimental trend does not
follow what would be expected for the (affine) Flory–
Rehner model, that would predict a continuous increase in
the average chain order upon isotropic dilation. In contrast,
a shift of the maximum the was observed, corresponding
Figure 6. (a) Simulated distributions of cross-link separations r in
a dry end-linked network with N ¼ 24, comparing the direct,
instantaneous distribution with a vectorial long-time average.
(b) Corresponding order parameter distributions, obtained using
the ‘‘naı̈ve’’ aproach represented by Equation (5), compared with
the distribution obtained after a proper segment-based time
average (dashed line), scaled to the same maximum position.
See ref.[14] for details on the simulations.
‘‘snapshot’’ with a (vectorial) orientation time average on a
timescale that would be relevant for an NMR experiment. Cross-link orientation fluctuations are seen to lead
to some narrowing, but both distributions are still close to
Gaussian in shape. Only a small exception is manifest in
the sharp central peak of the time-averaged distribution,
which arises from cross-links that are very close in space,
and whose connecting vector undergoes an almost
isotropic motion.
Using Equation (5), both r distributions consequently
convert to gamma-type order parameter distributions, as
described by Equation (7) and shown in Figure 6(b) (again,
the small contribution from isotropically reorienting
end-to-end vectors leads to a hardly visible sharp peak
at the origin for the time-averaged data). Such an approach
is the basis of the criticized traditional interpretation of
NMR data, yet it is neither sufficient to properly describe
the outcome of an NMR experiment, nor does the
so-obtained distribution reflect the true conformational
space available to each subchain. The latter can only be
probed by following the orientation excursions of the
individual segments, and only such a procedure can reveal
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ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Figure 7. (a) Change of the order parameter distribution of an
end-linked PDMS network with Mn 5 kgmol1 upon differential
swelling with octane (Q ¼ V/V0 is the degree of volume swelling).
(b) Sb distributions from computer simulations of a dry and an
equilibrium-swollen end-linked network with N ¼ 24. Data from
ref.[46]
DOI: 10.1002/marc.200700169
NMR Reveals Non-Distributed and Uniform Character . . .
to the most probable chain order, to lower values together
with a broadening of the distribution towards higher order
parameters. In fact, distributions of swollen networks are
close to the expected theoretical gamma distribution
(except for the very low values of the order parameters
that are highly affected by trapped entanglements[14]).
The observed effects suggest that the entropy loss
associated with the most highly ordered fraction that
appears on swelling is thus responsible for attaining the
swelling equilibrium. We have interpreted the appearance
of the broad distribution in terms of swelling heterogeneities, that are by now well accepted and supported by
numerous scattering studies. See ref.[46] for a detailed
discussion. In how far the reappearance of single-chain
connectivity or topology effects (towards a gamma distribution) also plays a role in this regard is a matter of
ongoing work. Note also that the phenomenology
observed here for the evolution of the conformational
space available to the chains upon swelling might explain
the trends for corresponding anomalies in the reduced
osmotic modulus determined by differential swelling
experiments.[37–39]
All these observations are again in nice agreement with
large-scale Monte-Carlo simulations of networks with
comparable topology [Figure 7(b)], proving the universality of the observations made by NMR.[14] It is a straightforward exercise to calculate the order parameter of each
simulated segment according to Equation (4) and thus to
determine its distribution for the whole simulated network structure. In this regard, it is interesting to consider
the ensemble-averaged variation of Sb along the chains
that is plotted in Figure 8. While the data, not unexpectedly, show that the order parameter is on average somewhat lower in the middle of the chains, the overall variation is in fact rather minor in the dry network, and still not
very remarkable in the swollen system. Most importantly,
the averaged variation along the chain is always much
Figure 8. Simulated average variation of the segmental order
parameter along the chains in a dry (a) and an equilibrium
swollen (b) end-linked network with N ¼ 24. Data from ref.[14]
Macromol. Rapid Commun. 2007, 28, 1455–1465
ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
smaller than the width of the overall distributions shown
in Figure 7(b), which proves that the latter are dominated
by larger-scale heterogeneity.
It can be expected that computer simulations of
networks with longer chains would exhibit even flatter
profiles, yet it is an open question whether (rheological)
entanglements or perhaps packing effects on shorter
length scale are responsible for the surprising homogenization of chain dynamics, that is most drastically observed
in the vulcanized systems (Figure 5). We hypothesize that a
chain-length effect might account for the relatively broad
order parameter distribution of the short-chain end-linked
network shown in Figure 2(b) (note that this distribution is
still significantly narrower than a gamma distribution
with the same average).
Discussion and Conclusions
In summary, the application of MQ spectroscopy to
elastomers, and in particular its strength in providing
access to the chain order parameter distribution in a
model-free way, made it possible to directly validate
assumptions concerning the chain statistics for the first
time. As a bottom line, we found that the local order in dry
polymer networks imposed by cross-links, and topological
constraints such as (trapped) entanglements, is rather
uniform, even in rubbers with a high fraction of network
chains below Me. Distribution effects of end-to-end
separations are apparently masked by the cooperative
nature of the chain dynamics, and not even the
polydispersity of chain lengths (e.g., the exponential
distribution in vulcanized elastomers) exerts an appreciable influence on the NMR observable. This suggests a
significant spatial averaging of chain dynamics.
Thus, the simplified ‘‘single chain’’ description according
to the affine network model fails. As mentioned in the
Introduction, deviations from Gaussian end-to-end vector
statistics in the range of short end separations of longer
chains can straightforwardly be explained by the fact that
in almost closed loops, for which simple theory assumes a
particularly large conformational space, topological links
(entanglements) have a great impact on the number of
possible conformations. In addition, fluctuation effects of
cross-links and even larger substructures further serve to
homogenize the segmental order parameter distribution.
Cross-link fluctuations have a particularly strong averaging effect on the orientation of short chains, for which
simple theory predicts rather large order parameters.
The homogenization of the segmental order parameter
may also be interpreted in terms of lateral chain ‘‘packing,’’
or a locally anisotropic nematic-like potential. It appears
that the actual segmental dynamics partially adopts a
mean-field character that is the essential ingredient of
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1463
K. Saalwächter, J.-U. Sommer
modern models of rubber elasticity.[3,48,49] At this point it
should be noted, however, that the remaining distribution
of the segmental order is difficult to explain in a true
mean-field model. Moreover, computer simulations display large distributions of other observables such as the
mean-square displacement of monomers.
Our observations are particularly significant in that the
local order parameter is related to the number of accessible
conformational microstates and thus to the entropy of the
network, for which we conclude that it cannot straightforwardly be described in terms of single-chain concepts
embodying fixed random chemical or topological constraints. Notable distributions appear only on swelling,
where packing constraints are relieved (‘‘desinterspersion’’) and the complex topological structure unfolds.[50]
Indeed, single-chain behavior as well as swelling heterogeneities then start to dominate the complex entropic
state of the network.
Again, the non-observation of any significant distribution effect in the dry state means that the chain dynamics
and conformational statistics are dominated by cooperativity, and that the local chain order results from an
average over a region in space spanning several network
chain dimensions. In this regard, it is interesting to
consider the result of a recent transverse relaxation study
of Cohen-Addad,[51] in which the influence of chain ends
on orientation correlations in entangled melts was studied
by comparing fully hydrogenous chains with selectively
end-deuterated ones. The central result was that the long
tail of the relaxation curves, which is commonly attributed
to the isotropically mobile unentangled end parts, is still
observed when extended endsections are deuterated. This
was interpreted in terms of mobilization of the inner parts
of chains that are in proximity to the ends of other chains,
confirming a picture where the detected chain order is
averaged over a certain region in space. This interpretation
now provides a rationale for the non-observation of
substantial distribution effects on residual couplings
between protons measured by MQ spectroscopy, and such
observations may provide a crucial test criterion for
refined theories of chain dynamics and rubber elasticity.
An earlier model of rubber elasticity that might in fact
conform with our findings is that of Jarry and Monnerie,[52]
who assumed that the chain entropy is influenced by
nematic-like inter-chain interactions. In their model, only
chain orientation (as observed in NMR, or by strain
birefringence) is affected, but not the stress–strain
behavior. Deloche and Samulski[53] have later proposed
a variant in which the nematic-like mean field does
depend on strain, whereby not only the mechanical
behavior changes, but also dilution effects on the chain
statistics appear upon swelling. Thereby, NMR observations as well as the anomalous maximum in the reduced
dilation modulus could be qualitatively explained.
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ß 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
In conclusion, NMR experiments and computer simulations reveal distributions of local segmental order which
are in marked contrast to the predictions of single-chain
affine network models. The rather homogenous distribution of local segmental order may be considered to be a
consequence of topological (entanglement) effects and the
large fluctuations of junction points and clusters (topologically connected groups) of network strands. These results
are also a challenge for modern network theories such as
replica mean field models[48,49] or double-tube models[49]
that embody homogeneous confining potentials for all
monomers. Other chain-based candidates are slip-tube[3]
or sliplink models,[54] in which entanglement or packing
effects are assumed to be dynamic, whereby a homogeneous (NMR) response could arise as a result of slip
motions that are at least as fast as the conformational
fluctuations of whole subchains.
Future work will have to be dedicated to critically
evaluate the model assumptions central to the abovementioned modern theories in the light of the NMR and
simulation data discussed in this work. Specifically,
predictions for the conformational space available to the
individual chain segments and the associated time scales
have to be derived, and be compared to the corresponding
order parameter distribution and the relaxation behavior
that is accessible by NMR. To date, theoretical predictions
of scattering functions are rather customary, while NMR
observables are hardly treated. This is particularly
regrettable in view of the fact that the NMR experiments
discussed herein can be performed with only little effort
even on low-field instrumentation, with minimal requirements for sample purity and preparation.
Acknowledgements: Funding for this work was provided by the
Deutsche Forschungsgemeinschaft (SFB 418) and the Fonds der
Chemischen Industrie. Boštjan Zalar is thanked for the idea to
perform 2H NMR in natural abundance.
Received: March 2, 2007; Revised: May 10, 2007; Accepted: May
11, 2007; DOI: 10.1002/marc.200700169
Keywords: chain order parameter; elastomers; natural rubber;
networks; polysiloxanes; residual dipolar couplings
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