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Mott-1996-Elasticity-RubberNetworks-Macromolecule+
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Macromolecules 1996, 29, 6941-6945
6941
Elasticity of Natural Rubber Networks
P. H. Mott and C. M. Roland*
Chemistry Division, Code 6120, Naval Research Laboratory, Washington, D.C. 20375-5342
Received February 6, 1996; Revised Manuscript Received June 17, 1996X
ABSTRACT: Stress, strain, and optical birefringence were measured for cross-linked natural rubber in
compression and tension over a range of strains. The experimental data were compared to the constrainedjunction theory (Flory, P. J.; Erman, B. Macromolecules 1982, 15, 800) and to the constrained-chain theory
(Kloczkowski, A.; Mark, J. E.; Erman, B. Macromolecules 1995, 28, 5089). These elasticity theories were
found to fit both the stress and birefringence data quite well in tension. However, discrepancies arise
when both compression and tension data are analyzed together. The differences between the models
were minor, compared to their deviation from experiment.
Introduction
The study of cross-linked rubber offers unique insights into polymer physics because equilibrium data
can be obtained. Much effort has been devoted to the
development of theories able to predict the elastic
behavior of networks. A proper assessment of elasticity
theories requires measurements over a range of experimental variables, such as network structure, temperature, concentration, and deformation type, and ideally
includes data beyond simply the strain dependence of
the stress.
Particularly informative are stress measurements
made in the transition region around λ| ) 1, where the
stretch ratio, λ|, is the ratio of the deformed and
undeformed sample lengths parallel to the applied force.
Such experiments were carried out by Rivlin and
Saunders1 on natural rubber networks; however, two
different techniques were employedsinflation for λ| <
1 and uniaxial extension for λ| > 1sintroducing artifacts
into the data near λ| ) 1. Pak and Flory2 used a similar
method for poly(dimethylsiloxane) (PDMS), obtaining
data in both tension and compression, although the
measurements exhibited substantial hysteresis, implying nonequilibrium. By using glued cylinders, Wolf 3
obtained data on cross-linked natural rubber in both
tension and compression; however, the latter were
limited to λ| g 0.88. A problem therein was the
inhomogeneous strains arising in uniaxially deformed
cylinders with bonded ends. Erman and Flory4 also
used a cylindrical geometry, obtaining data on PDMS
for strains of about 10% in compression and tension. In
both these studies,3,4 the data were not corrected for the
inhomogeneous strain.
While the strain dependence of the stress is the most
common quantity used to assess elasticity theories,
measurements of the optical birefringence are equally
informative. The stress optical coefficient, C, defined
as the ratio of the birefringence, ∆n, to the true stress,
σ, is often taken to be a constant (Brewster’s law).
Actually, C has a weak stress dependence and previous
workers have used comparisons of calculated and experimentally determined values of the birefringence to
assess elasticity theories. Experiments have been performed on natural rubber,5-7 PDMS,8,9 polybutadiene,10,11 and polyisoprene,11 but in all of these studies,
the data were restricted to tension. To date, a study of
birefringence of rubbers in compression has not been
carried out.
Abstract published in Advance ACS Abstracts, September 15,
1996.
X
In this paper, we describe measurements of both
stress and birefringence on deproteinized natural rubber
uniaxially deformed over the range 0.5 < λ| < 4.
Constitutive equations taken from the constraint theories of rubber elasticity were applied to the data.
Background
Early models of rubber elasticity describe two extremes in the relationship of stress to strainsa phantom
network, comprised of volumeless chains able to pass
through one another,12 and an affine network, whose
chains are firmly embedded in a deformable continuum.13 For an incompressible rubber, the engineering stress for the phantom network is given by14,15
(
σph ) 2 1 -
2
µRT(R - R-2)
φ
)
(1)
where φ is the cross-link functionality, µ the cross-link
density, and RT has its usual significance. The elongation, R, is related to the stretch ratio, λ|, by R ) λ|(V0/
V)1/3, where V0 and V are the initial and swollen
volumes, respectively. All experiments herein were
carried out on neat networks, whereby R ) λ|. Equation
1 assumes all chains are elastically active, which is
strictly true only if the primary molecular weight is
much larger than the molecular weight between crosslinks.16 The stress of an affine network, which does not
depend on the cross-link functionality, is twice that
given by eq 1 for φ ) 4.
The most obvious failing of the classical theories is
the deviation of experimental results from the predicted
proportionality between engineering stress, σeng, and the
stretch function, (λ| - λ|-2). This led to the development
of phenomenological alternatives, the most widely used
of which is the Mooney-Rivlin equation.14 The appeal
of the Mooney-Rivlin approach is largely due to the
ease with which the elastic constants can be determined
by a linear regression of the reduced stress, σeng/(λ| λ|-2)), versus 1/λ|. However, such plots are often not
linear, particularly for elastomers subjected to both
tension and compression.17,18
An important development in the theory of rubber
elasticity was Flory’s recognition of the role of intermolecular interactions in modifying the stress in a straindependent manner. This idea was incorporated into a
constrained-junction (CJ) model,19 in which the engineering stress is given by
[
σCJ ) σph 1 +
]
2
2
-2
2 RK(λ| ) - R K(λ⊥ )
φ-2
R - R-2
S0024-9297(96)00189-1
This article not subject to U.S. Copyright. Published 1996 by the American Chemical Society
(2)
6942 Mott and Roland
Macromolecules, Vol. 29, No. 21, 1996
where λ⊥ is the stretch ratio perpendicular to the applied
load ()λ|-1/2 for uniaxial deformation assuming incompressibility). The function K(λ2) is defined as
K(λ2) )
DḊ
BḂ
+
B+1 D+1
(3)
network
dicumyl
peroxide (phr)
µa
CC Model
µ
κ
NR-1
NR-2
1.0
2.0
33
66
65
103
3
3
ECJ Model
µ
κ
ζ
81
132
6
6
0.2
0.2
a Cross-link density (mol/m3) calculated assuming depletion of
the peroxide and tetrafunctional junctions.
where
B(λ2) )
Ḃ(λ2) ≡
∂B(λ2)
2
∂(λ )
κ2(λ2 - 1)
2
(4)
2
(λ + κ)
) B(λ2)
[
]
1
2
- 2
λ -1 λ +κ
2
(5)
λ2
D(λ ) ) B(λ2)
κ
∂D(λ2)
1
) [λ2Ḃ(λ2) + B(λ2)]
κ
∂(λ2)
(7)
The parameter κ in eqs 4-7 is a measure of the severity
of the topological constraints. Equation 3 reduces to the
phantom and affine results for κ ) 0 and κ ) ∞,
respectively.
Subsequently, Flory and Erman20 extended the CJ
model, introducing an additional parameter, ζ (g0), to
allow for departures from affine distortion, as occasioned, for example, by inhomogeneities in the network structure. A large value of ζ implies that the
constraints are alleviated by extension of the network
more rapidly than if the domains deformed affinely. In
this extended constrained junction (ECJ) model, K(λ2)
is still given by eq 3, but the functions B(λ2) and D(λ2)
become
B(λ2) )
κ2[λ2 - 1 - λ2ζ(λ - 1)]
(8)
[λ2 + κ + κλ2(λ - 1)]2
and
D(λ2) ) λ2
[1κ + ζ(λ - 1)]B(λ )
2
(9)
The derivatives Ḃ ≡ ∂B/∂λ2 and Ḋ ≡ ∂D/∂λ2 are not
shown. Substitution of these quantities into eq 2 yields
the engineering stress. The original CJ model is
recovered when ζ ) 0.
In both the CJ and ECJ models, the effect of the
constraints is assumed to be exerted directly on the
network junctions. An alternative is to locate their
effect at the chain’s center of gravity.21,22 This idea has
been generalized to allow the constraints to act all along
the chain, the so-called continuously constrained chain
(CC) model.23 The expression for the engineering stress
in the CC model is
σCC )
σph 1 +
RK(λ|2) - R-2K(λ⊥2)
∫0 W(Θ)
φ
φ-2
1
R - R-2
[
(6)
and
Ḋ(λ2) ≡
constraints is taken to be constant along the chain,
W(Θ) equals unity.23 Contrary to the earlier models,
in which κ is constant, for the CC model the constraint
parameter κ embedded in eq 10 is a function of position
along the chain,
κ(Θ) ) κ 1 +
2
[
Table 1. Rubber Networks and Fitting Parameters
]
dΘ (10)
where K(λ2) is again given by eq 3. W(Θ) is a distribution function describing the effectiveness of the constraints at each point along the chain, and Θ, which
varies from 0 to 1, is the relative distance of a point on
the chain from the junction site. If the strength of the
]
(Θ - 2)2Θ(1 - Θ)
φ-1
(11)
The constraint parameter κ(Θ) replaces the κ in eqs 4-7.
While the stress-optical coefficient remains constant
through moderate strains, deviations from Brewster’s
law occur for larger deformations. Thus, the strain
dependence of the birefringence ∆n provides a useful
quantity to independently assess the validity of a
constitutive equation. According to the ECJ model,8
1+
B(λ|2) - B(λ⊥2) + bD(λ|2) - bD(λ⊥2)
(φ/2 - 1)(V/V0)2/3(R2 - R-1)
∆n ) σC
1+
λ|2K(λ|2) - λ⊥2K(λ⊥2)
(12)
(φ/2 - 1)(V/V0)2/3(R2 - R-1)
where C is now the stress-optical coefficient measured
at infinitesimal strain, and b is an adjustable parameter
(0 e b e 1) reflecting the effect of orientation of the
domains of constraint. The functions K(λ2), B(λ2), and
D(λ2) are given by eqs 3, 8, and 9, respectively. The CC
model makes analogous predictions for ∆n.9,24,25
Experimental Section
The polymer was deproteinized natural rubber (SMR-S from
H. A. Astlett Co.), which is cis-1,4-polyisoprene. Removal of
the proteins significantly enhances transparency in comparison to conventional Hevea brasiliensis. The cross-linking
agent, dicumyl peroxide, was mixed into the rubber using a
two-roll mill (Table 1). Both films (65 × 13 × 1.6 mm) and
cylinders (12.2 mm diameter × 17.8 mm high) were prepared
by compression molding for 30 min at 160 °C. The cross-link
density calculated for the two networks, assuming depletion
of the peroxide, is listed in Table 1.
The apparatus used to obtain compression data on the
rubber cylinders is depicted in Figure 1. The cylinder ends
were bonded to aluminum plates with cyanoacrylate adhesive.
Calibrated weights were applied, and the change in sample
length measured using a linear voltage differential transducer.
Although bonding of the cylinder ends to the confining plates
allows both tension and compression data to be obtained, it
also gives rise to additional stress, due to shear forces acting
on the cylinder ends.26 Correcting for the resulting barreling
(or hourglass shape if R > 1) gives27
σeng )
σapp
1+
R02
(13)
2H02
where R0 and H0 are the initial sample radius and height,
respectively, and σapp, the apparent stress, is the ratio of the
applied force to the initial cross-sectional area of the cylinder.
The corrected engineering stress σeng given by eq 13 corresponds to the engineering stress that would be observed in a
homogeneously-strained cylinder, deformed uniaxially. A
Macromolecules, Vol. 29, No. 21, 1996
Elasticity of Natural Rubber Networks 6943
Figure 1. Schematic of the apparatus used for simultaneous
measurement of the force and birefringence of rubber cylinders
in uniaxial deformation.
comprehensive experimental assessment of eq 13 was recently
carried out,28 involving a comparison of stress, strain, and
birefringence data obtained from bonded cylinders of varying
aspect ratio as well as from nonbonded samples. This investigation demonstrated that eq 13 corrects the experimental
data to an accuracy of better than 5% over substantial ranges
of strain.28 Thus, any residual error due to bonding of the
cylinder ends is less than the cumulative error intrinsic to
mechanical experiments (e.g., reproducing the cross-link density, attaining mechanical equilibrium, etc.). This cumulative
error we judge to be no more than 10%.
Extension measurements were carried out by two methods.
To ensure an absence of systematic error between the compression and tension experiments, the cylindrical samples were
extended while in the compression geometry by means of a
screw pulling on the loading stage; this yielded tension data
through λ ) 1.8. Additional experiments to much higher
extension were performed using films. Weights were attached
to the samples, and the resulting strain determined from
displacement of fiducial marks, measured using a cathetometer.
The birefringence was determined from the rotation, θ, of
an analyzer polarizing filter, as shown in Figure 1. The
birefringence is given by 29
∆n )
θΛ
π l
(14)
where Λ ) 632.8 nm for He-Ne laser radiation. The rotation
of the analyzer reached values as high as 40π for the highest
compressive strains. The path length of the beam through the
sample, l, was deduced for the extended strips by assuming
incompressibility. For the cylinders, l corresponds to the
maximum (compression) or minimum (extension) diameter of
the deformed cylinder. This was measured by translating the
deformed cylinder horizontally and identifying the outer
surfaces with the laser beam. Parenthetically, we note that
the usual assumption that compressed cylinders exhibit a
parabolic profile is incorrect.28
All experimental results herein correspond to room temperature and mechanical equilibrium. A minimum of 6 h
elapsed between applying the load and measuring the strain
and the birefringence; thus, no more than two measurements
could be obtained per day. Furthermore, to verify the absence
of hysteresis, alternately increasing and decreasing changes
in the load were made.
Results and Discussion
In Figure 2 the engineering stress versus stretch ratio
for the natural rubber networks is displayed, after
correction for the effect of inhomogeneous strain in the
bonded cylinders (eq 13). Concerning the shape of the
stress-strain curves in Figure 2, three observations can
be made, all having been previously established for
rubbery networks. First, the slopes of the curves
Figure 2. Engineering stress versus stretch ratio for the
peroxide-cross-linked, deproteinized natural rubbers.
Figure 3. Optical birefringence as a function of true stress
for NR-1 (() and NR-2 (b) over the range 0.8 e λ| e 1.6. Linear
regression yields the indicated value for the stress optical
coefficient C.
decrease as the material is taken from compression into
tension. This lower modulus, as well as the reduced
strain dependency of the modulus in compression,
represents a classic failing of the most popular equation
of state for elastomers, the Mooney-Rivlin expression.17,18 Second, the data in Figure 2 are continuous
through λ ) 1. Although unsurprising, experimental
data demonstrating this fact are relatively scarce.2,4,30,31
Third, at the highest tensile strains, the stress for the
more cross-linked NR-2 shows an upturn, reflecting
strain-induced crystallization.32-35 Natural rubber has
an unusual capacity for orientational crystallization,
which underlies much of its commercial utility.36 Of
course, once crystallization commences, rubber elasticity
theories are no longer applicable.
In Figure 3 we plot the birefringence measured at the
center of the cylinders versus the true stress for 0.8 e
λ| e 1.6. Over this limited range of strains, the stressoptical coefficient is constant to within the experimental
error. Neglecting any weak dependence of C on crosslink density,14,28 we combine the data for NR-1 and NR-2
to obtain C ) 2.2 GPa-1. This result is about 5% higher
than the value reported for a natural rubber network
of comparable cross-link density, but from which the
proteins had not been removed.6
To compare the experimental stress-strain data to
the constitutive models described above, the reduced
stress, σeng/(λ| - λ|-2), is plotted against 1/λ|. The data
6944 Mott and Roland
Figure 4. Reduced force versus reciprocal of the stretch ratio
for the two natural rubber networks, along with the best fits
to the ECJ theory (eqs 2, 8, and 9) and to the CC theory (eqs
10 and 11). The inset shows the experimental uncertainty in
the vicinity of λ|-1 ) 1.
from Figure 2 are displayed in this “Mooney” form in
Figure 4. For values near λ|-1 ) 1, the reduced stress
is extremely sensitive to errors in length. An error
analysis37 carried out to determine this uncertainty is
shown as an inset to Figure 4. Thus, extreme scatter
of the data in the range 0.987 e λ|-1 e 1.02 is expected.
Interestingly, these deviations are positive for tension
and negative for compression. This can perhaps be
ascribed to systematic error (e.g., friction from the
bearings in the support stage in Figure 1). However,
more accurate, small-strain experiments carried out on
natural rubber in both tension and compression revealed very similar deviations in the reduced stress,
which the authors did not attribute to experimental
error.30 Such behavior is not anticipated by any rubber
elasticity theory, and perhaps deserves more careful
study.
At moderate strains, the data in Figure 4 exhibit a
maximum, roughly in the vicinity of λ|-1 ) 1.2, in accord
with previous results on PDMS.2,4 At higher strains,
particularly for the more cross-linked NR-2, straininduced crystallization35 causes a marked curvature in
the experimental points. The curvature is the result of
two competing effects: at the onset of crystallization,
some reduction in the reduced stress occurs, since the
formation of crystals reduces the number of elastically
active chains. Eventually at higher strains, however,
crystallinity raises the stress, because the crystallites
serve to increase the effective cross-link density of the
network.33
We can compare the present results to the network
theories described above. For the continuously constrained-chain theory (eqs 10 and 11), the distribution
of constraints along the chain is taken to be uniform
(i.e., W(Θ) ) 1), with the cross-links assumed to be
tetrafunctional (φ ) 4), as reputed for peroxide-crosslinked natural rubber.38 Thus, the only adjustable
parameters are the cross-link density µ and the constraint parameter κ. In fitting the CC model to the
NR-2 data, we ignored the highest strains, since these
points were influenced by crystallization. The solid
lines in Figure 4 represent fitting to the CC model, with
the values for the parameters µ and κ listed in Table 1.
The figure shows that in extension (λ|-1 < 1), the model
describes the experimental data for both networks quite
well; the difference between the calculated and measured results is less than the experimental error.
Macromolecules, Vol. 29, No. 21, 1996
Figure 5. Reduced force versus inverse stretch ratio for
sulfur-cross-linked natural rubber. The solid line was calculated from the CC theory (eqs 10 and 11).
However, the CC theory does not give good agreement
with the data for either network in compression (λ|-1 >
1); the reduced stress (modulus) decreases more strongly
with compressive strain than predicted. Any modification of κ to improve the agreement in compression
causes the fitting for the tension data to deteriorate.
This lack of a good fit over the entire range of deformation introduces some uncertainty in the parameters µ
and κ (Table 1), and their interpretation in terms of the
details of the network structure is problematic.
An alternative to the CC model is the extended
constrained-junction model (eqs 2, 8, and 9), in which
the constraints on the chain fluctuations act only on the
cross-links. Our interest in applying the ECJ model is
whether the extra parameter, ζ, provides for improved
agreement with experiment. The dashed lines in Figure
4 show the results of fitting the ECJ model, with the
values for the fit variables, µ, κ, and ζ, listed in Table
1. Again, the imperfect quality of the fit confers some
uncertainty to the value of the fitting parameters. Note
that differences between the CC and ECJ models are
minor in comparison to their deviation from the experimental compression data.
The limited success of the models in describing both
tension and compression data can be compared to the
results of fitting previously published measurements on
sulfur-vulcanized natural rubber.1 Unfortunately, there
are some problems with this data. Unless the crosslinks are monosulfidic, sulfur vulcanization produces
networks which are mechanically labile. In fact, a hint
of such is evident in the authors’ comment1 that
consistent results (without hysteresis) required that
only load-increasing experiments be conducted. This
study also relied on two different experimental techniquessextension of strips for R > 1 and inflation of
sheets for R < 1swhich can be a source of error,
particularly near R ) 1.
Shown in Figure 5 are the results of fitting the data
for the sulfur-cured networks1 to the CC model (eqs 10
and 11). Agreement between experiment and theory is
again found only for extension. We also note that,
consistent with our findings, this data was previously
shown to conform to the original constrained junction
model (i.e., ζ ) 0) only for tension.2
While in Figure 3 the stress-optical coefficient C was
determined from the proportionality between the birefringence ∆n and the true stress σ, deviations from
linearity were notable at higher strains. In Figure 6,
the birefringence ∆n from the two networks is plotted
Macromolecules, Vol. 29, No. 21, 1996
Figure 6. Optical birefringence as a function of strain for the
two natural rubber networks, along with the curve calculated
using eq 12. The deviations at high extension are due, at least
in part, to orientational crystallization.
against λ|, along with the best fit of the data to the ECJ
model (eq 12), with b ) 0. Given that the stress is only
roughly described by the elasticity models for compression (λ| < 1), we cannot expect better success with
regard to the birefringence. There is an additional
complication from crystallization, apparent in Figure 4,
but more evident in the birefringence data. Contrary
to its effect on stress, crystallinity always augments the
birefringence,31 and thus it is a sensitive means to detect
its onset. This effect is seen in Figure 6 in the
underestimation of ∆n by the fitted ECJ model, beginning at moderate extensions (λ| > 2). At the highest
extensions (λ| > 3.5), Figure 6 shows the birefringence
increasing dramatically, corresponding to an order of
magnitude increase in the stress-optical coefficient C.
Summary
The elasticity models of Flory and their extensions
by various workers were assessed over a range of
strains, including both compression and tension. We
find that the constraint models describe stress, strain,
and birefringence data on natural rubber networks quite
well in tension; however, the agreement is unsatisfactory when both compression and tension data are
considered together. The mediocre fit of theory to
experiment was most evident when the reduced stress
was analyzed. If consideration is restricted to the more
conventional engineering stress-strain, the constraint
models would suffice for many practical applications.
There are, of course, other rubber elasticity models,
which have not been examined herein. Some39-42 are
in many respects similar to the constraint models.
Others have a more phenomenological basis1,31,43-46 and
thus lack the intuitive clarity of the constraint theories.
Our focus on the constraint theories reflects their
widespread usage. Be that as it may, it is evident that
further developments are required for a quantitative
characterization of the behavior of rubbery networks.
It is likely that contributions in this regard can be
realized from investigations of network dynamics.41,42,47-54
Acknowledgment. This work was supported by the
Office of Naval Research. P.H.M. expresses his gratitude for a National Research Council/Naval Research
Laboratory postdoctoral fellowship.
References and Notes
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Elasticity of Natural Rubber Networks 6945
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