ELASTICITY, RUBBER-LIKE
Introduction
Elasticity is the reversible stress–strain behavior by which a body resists and
recovers from deformation produced by a force (1). This behavior is exhibited by
rubber-like materials in a unique and extremely important manner. Unlike metals
or glasses, they can undergo very large deformations without rupture (and are thus
similar to liquids) and then can come back to their original shape (as do solids).
This exceptional ability was investigated in 1805 (2,3), well before the concept of
polymers as long-chain molecules was established in the 1920s. It was observed
that a rubber gave off heat upon stretching and, when submitted to a constant
load, became shorter as temperature was increased. Later, Kelvin (4) derived the
thermodynamic laws of elasticity, and more careful experiments were performed
by Joule (5). Many books and review papers have been published on the subject
(6–36).
Deformation and Mechanical Testing
When a body is submitted to external stress or pressure, it undergoes a change
in shape or volume. Mechanical testing of rubber involves application of a force
to a specimen and measurement of the resultant deformation or application of a
deformation and measurement of the required force. The results are expressed in
terms of stress and strain and thus are independent of specimen geometry. Stress
is the force f per unit original cross-sectional area A, and strain the deformation
per unit original dimension. The units of stress are Newtons per square meter
(N/m
2
or Pa); strain is dimensionless. The ratio of stress to strain is called the
216
Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.
Vol. 2
ELASTICITY, RUBBER-LIKE
217
A
L
f
f
L
0
(a)
(b)
Fig. 1.
Uniaxial extension of a bar-shaped sample. (a) No load; (b) tensile load. A
= area;
f
= force; L
0
= original length; L = length.
modulus. A material is Hookean when its modulus is independent of strain (typical
of a metal spring or wire); elastomers are Hookean only in the range of very small
deformations.
The simplest deformations are isotropic compression under hydrostatic pres-
sure, uniaxial extension (or compression), and simple shear. They are used to
determine the bulk modulus K, Young’s modulus E, and shear modulus G, respec-
tively (37–44). Elastomer testing for commercial applications is highly dependent
on the method, the conditions (eg, strain rate, temperature), and the shape of the
samples. Therefore, mechanical tests have been standardized for uniformity and
simplicity by the American Society for Testing and Materials (ASTM) (45).
Isotropic Compression.
The volume of a sample decreases from V
0
to V
when submitted to a hydrostatic pressure p. The bulk modulus is the ratio of this
pressure to the volume change per unit volume
V/V
0
:
K
= p/(V/V
0
)
(1)
where
V = V
0
− V; K is the reciprocal of the compressibility and can be deter-
mined by direct measurement of compressibility (46) or velocity of longitudinal
elastic waves (sound) or by theoretical calculations (39).
Uniaxial Extension.
A rubber strip of original length L
0
is stretched uni-
axially to a length L, as illustrated in Figure 1. The stretch and elongation are
L = L − L
0
and
λ = L/L
0
, respectively. The strain
(also known as the relative
deformation, linear dilation, or extension) and the elongation or extension ratio
λ
are related by
= L/L
0
= λ − 1
(2)
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ELASTICITY, RUBBER-LIKE
Vol. 2
The lengthwise extension is in general accompanied by a transverse contraction.
The ratio of the change
w in width per unit width w
0
to the change in length per
unit length is called Poisson’s ratio (47):
ν
p
= (w/w
0
)
/(L/L
0
)
(3)
Ideal rubbers and liquids deform at constant volume, for which
ν
p
is equal to 0.5.
Poisson’s ratio for real elastomers may be experimentally determined by applying
extensometers in the transverse and axial directions of a sample. Approximate
values of
ν
p
thus determined are 0.33 for glassy polymers, 0.4 for semicrystalline
polymers, and 0.49 for elastomers. Young’s modulus E is the ratio of normal stress
f /A to corresponding strain
:
E
= ( f/A)/
(4)
Generally, stress–strain curves deviate markedly from a straight line, as illus-
trated in Figure 2. The Young’s modulus at small deformation is the slope tan
θ of
the stress–strain curve at the origin (initial tangent modulus). It can be measured
in flexure or by tensile experiments (48).
Tension testing of a vulcanized elastomer also permits the determination
of the tensile strength, which is the maximum stress applied during stretching
a specimen to rupture; the corresponding rupture strain is called the maximum
extensibility (49–52). Typical values are given in Table 1 (14). Dumbbell and ring
specimens can be used.
6
6
5
5
4
4
3
3
2
f/A
, N/mm
2
2
1
1
0
7
8
Fig. 2.
Stress–strain curve of a non-Hookean solid; typical curve for an elastomer in
uniaxial extension.
Vol. 2
ELASTICITY, RUBBER-LIKE
219
Table 1. Properties of a Typical Rubber, Metal, and Glass
a
Property
Rubber
Metal
Glass
Breaking extension, %
500
Large, plastic
3
Elastic limit, %
500
2
3
Tensile strength, MN/m
2b
10
c
5
× 10
4
5
× 10
3
a
Ref. 14.
b
To convert MN/m
2
(MPa) to psi, multiply by 145.
c
>10
3
if based on area at break.
Force
gauge
Recorder
Cathetometer
Constant
T
N
2
Fig. 3.
Schematic diagram of a typical apparatus used to measure elastomer stress as a
function of strain (21). T
= Temperature.
A typical apparatus used to measure the equilibrium stress of an elongated
network as a function of strain and temperature is shown in Figure 3 (21). The
rubber strip is held between two clamps and maintained under a protective atmo-
sphere of nitrogen. The sample length, required to characterize its deformation,
is obtained by means of a cathetometer or traveling microscope (the central test
section of the sample is delineated by ink marks applied before loading). Values
of the force are obtained from a calibrated stress gauge, the output of which is
displayed as a function of time on a standard recorder. Measurements are made
at elastic equilibrium; the influence of temperature can also be studied. Another
example of a stretching device is an automatic stress relaxometer (53).
Simple Shear.
Simple shear, illustrated in Figure 4, is a deformation in
which the height H and surface A of the sample are held constant. The shear
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ELASTICITY, RUBBER-LIKE
Vol. 2
A
H
y
x
f
U
y
A
H
f
x
(a)
(b)
Fig. 4.
Simple shear of a rectangular sample. (a) No deformation; (b) simple shear.
A
= area; H = height; f = force; U = linear displacement.
modulus or rigidity G is expressed by the relations
G
= ( f/A)/(U/H) = ( f/A)tan γ f/γ A
(5)
where
γ is the shear strain and f/A the shear stress. As in the case of Young’s
modulus, the shear modulus (41,42) is a material constant if stress and strain are
directly proportional. If they are not, the shear modulus at small deformation is
employed; it is defined as the slope at the origin of the stress–strain curve. The
shear modulus is generally measured on a cylindrical specimen and a tension-
testing machine.
Typical values of moduli and Poisson’s ratios for metals, ceramics, and poly-
mers are given in Table 2 (39). Moduli of rubbers are strikingly low, and even
those of other types of polymers are of the order of one tenth of those of metals.
However, compared at equal weights, ie, ratio of modulus to density
ρ, polymers
(except rubbers) compare favorably.
Relationships between Moduli and Poisson’s Ratio.
On the basis of
the theory of elasticity of isotropic solids, the moduli and Poisson’s ratio are inter-
related (54) as follows:
E
= 2G(1 + ν
p
)
= 3K(1 − 2ν
p
)
(6)
For elastomers, the Poisson’s ratio is nearly 0.5, and thus E
3G.
Conditions for Rubber-like Elasticity
Long, Highly Flexible Chains.
Elastomers consist of polymeric chains
which are able to alter their arrangements and extensions in space in response to
an imposed stress. Only long polymeric molecules have the required exceedingly
large number of available configurations.
It is necessary that all the configurations are accessible; this means that
rotation must be relatively free about a significant number of the bonds joining
neighboring skeletal atoms.
Table 2. Poisson’s Ratio, Moduli, and Density of Metals, Ceramics, and Polymers
a
Specific properties
Young’s modulus E, Shear modulus G, Bulk modulus K, Density
ρ, E/ρ, 10
6
G/
ρ, 10
6
K/
ρ, 10
6
Material
ν
p
10
9
N/m
2b
10
9
N/m
2b
10
9
N/m
2b
g/cm
3
m
2
/s
2
m
2
/s
2
m
2
/s
2
Metals
Cast iron
0.27
90
35
66
7.5
12.0
4.7
8.8
Steel (mild)
0.28
220
86
166
7.8
28.0
11.0
21.0
Aluminum
0.33
70
26
70
2.7
26.0
9.6
26.0
Copper
0.35
120
44.5
134
8.9
13.5
4.5
15.0
Lead
0.43
15
5.3
36
11.0
13.6
4.8
33.0
Mercury
0.5
0
0
25
13.55
0.0
0.0
1.85
Inorganics
Quartz
0.07
100
47
39
2.65
38.0
17.8
14.7
Vitreous silica
0.14
70
30.5
32.5
2.20
32.0
14.0
14.7
Glass
0.23
60
24.5
37
2.5
24.0
9.8
14.9
Granite
0.30
30
11.5
25
2.7
11.1
4.3
9.2
Whiskers
Alumina
2000
1000
667
3.96
510
253
225
Carborundum
1000
500
333
3.15
315
160
106
Graphite
1000
500
333
2.25
440
220
150
Polymers
Polystyrene
0.33
3.2
1.2
3.0
1.05
3.05
1.15
2.85
Poly(methyl Methacrylate) 0.33
4.15
1.55
4.1
1.17
3.55
1.33
3.5
Nylon-6,6
0.33
2.35
0.85
3.3
1.08
2.21
0.79
2.3
Polyethylene (low density)
0.45
1.0
0.35
3.33
0.91
1.1
0.385
3.7
Ebonite
0.39
2.7
0.97
4.1
1.15
2.35
0.86
3.6
Rubber
0.49
0.002
0.0007
0.033
0.91
0.002
0.00075
0.04
Liquids
Water
0.5
0.0
0.0
2.0
1.0
0.0
0.0
2.0
Organic liquids
0.5
0.0
0.0
1.33
0.9
0.0
0.0
1.5
a
Ref. 39. Courtesy of Elsevier.
b
N/m
2
= Pa. To convert Pa to psi, multiply by 0.000145.
221
222
ELASTICITY, RUBBER-LIKE
Vol. 2
Fig. 5.
Segment of an elastomeric network.
Network Structure.
Chains must be joined by permanent bonds called
cross-links, as illustrated in Figure 5. The network structure thus obtained is
essential so as to avoid chains permanently slipping by one another, which would
result in flow and thus irreversibility, ie, loss of recovery. These cross-links may
be chemical bonds or physical aggregates, eg, glassy domains in multiphase block
copolymers (55,56).
At the end of the cross-linking process, the topology of the mesh is composed
of the different entities represented in Figure 6 (16,57–59). An elastically active
junction is one joined by at least three paths to the gel network (60,61). An ac-
tive chain is one terminated by an active junction at both its ends. Rubber-like
elasticity is due to elastically active chains and junctions. Specifically, upon de-
formation the number of configurations available to a chain decreases and the
resulting decrease in entropy gives rise to the retractive force.
Weak Interchain Interactions.
Apart from the effects of the cross-links,
the molecules must be free to move reversibly past one another, that is, the in-
termolecular attractions known as secondary or van der Waals forces, which exist
between all molecules, must be small. Specifically, extensive crystallization should
not be present, and the polymer should not be in the glassy state.
Differences between Elastomers and Metals
Elastomers and metals differ greatly with regard to deformation mechanisms (26,
62,63). Metals and minerals are formed of atoms arranged in a three-dimensional
crystalline lattice, joined by powerful valence forces operating at relatively short
Vol. 2
ELASTICITY, RUBBER-LIKE
223
(a)
(b)
(d)
(c)
Fig. 6.
Structural features of a network. (a) Elastically active chain; (b) loop; (c) trapped
entanglement; (d) chain end.
range. Deformation of such materials involves changes in the interatomic distance,
which requires large forces; hence the elastic modulus of these materials is very
high. After a small deformation, slippage between adjacent crystals occurs at the
yield point, and the deformation increases much more rapidly than the stress and
becomes irreversible or plastic. The primary effect of stretching a metal short of
this yield point is the increase
E
m
in energy caused by changing the distance
d of separation between metal atoms. The sample recovers its original length
when the force is removed, since this process corresponds to a decrease in energy.
Heating increases oscillations about the minimum of the asymmetric potential
energy curve and thus causes the usual volume expansion.
Stretching of elastomers does not involve any significant changes in the in-
teratomic distances, and therefore the forces required are considerably lower. The
number of available configurations for a network chain is reduced in the defor-
mation process. After suppression of the stress, the specimen recovers its original
shape, since this corresponds to the most disordered state. Thus the retractive
force arises primarily from the tendency of the system to increase its entropy
toward the maximum value it had in the undeformed state. At high elongations,
stress–strain curves turn upward, a behavior very unlike that of metals. The num-
ber of available configurations is drastically reduced in this region, and chains
reach the limits of their extensibility. For polymers with regular structures, crys-
tallization may also be induced. Further elongation may then require deformation
of bond angles and length, which requires much larger forces.
At constant force, heating increases disorder, forcing the sample in the direc-
tion of the state of disorder it had at a lower temperature and smaller deformation.
The result is therefore a decrease in length.
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ELASTICITY, RUBBER-LIKE
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Extension of Elastomers and Compression of Gases
The extension of elastomers and the compression of gases are associated with a
decrease in entropy. Thus, similar behavior is expected in regard to (adiabatic)
deformation and heating. The work of deformation of a gas is dW
= −pdV, where
p is the pressure and V the volume. For an elastomer it is dW
= f dL, where f is the
force and L the length; the term
−pdV is not taken into account, since there is only
a negligible change in rubber volume during stretching. These observations can be
used to explain Gough’s experiments by making use of classical thermodynamic
principles (64).
Statistical Distribution of End-To-End Dimensions of a Polymer Chain
Before treating the statistical properties of a network, the statistics of a single
chain must be considered, mainly to establish the relationship between the num-
ber of configurations and the deformation (12,16,65–72). A polymeric chain is
constantly changing its configuration by Brownian motion. Statistical methods
and idealized models permit calculation of the average properties of such a chain.
The Freely Jointed Chain.
This type of idealized chain consisting of n
links of length l is represented in Figure 7, where r is the end-to-end distance.
When the chain is completely extended, R
= nl. The chain may assume many
configurations, each associated with an end-to-end distance r
i
. In statistics, it is
equivalent to consider a molecule at different times or an assembly of N molecules
at the same time. An average quantity describing the assembly is the mean-square
end-to-end distance r
2
defined by
r
2
=
1
N
N
i
= 1
r
2
i
(7)
where r
i
is the end-to-end distance of the ith chain. The vector r
i
is the sum of the
link vectors I
j
:
r
i
=
n
j
= 1
I
j
(8)
I
j
r
Fig. 7.
Ideal chain formed with n links of length l. r
= End-to-end distance, I
j
= link
vector.
Vol. 2
ELASTICITY, RUBBER-LIKE
225
and r
2
i
is the scalar product of r
i
with itself:
r
2
i
=
n
j
= 1
I
2
j
+ 2
k
< j
I
k
·I
j
(9)
If the chain is assumed to be freely jointed and volumeless, any two links
can assume any orientation with respect to each other. Therefore the second term
of equation (9) is zero and
r
2
i
= nl
2
(10)
The mean-square end-to-end distance of a freely orienting chain is deduced
from equations (7) and (10) (73–76) as follows:
r
2
= nl
2
(11)
Another interesting quantity is the probability that a chain has a given end-
to-end distance (77–81). This is called the Gaussian distribution function
W(x
,y,z)dxdydz = W(x)dxW(y)dyW(z)dz = (b/π
1
/2
)
3
exp(
− b
2
r
2
)dxdydz
(12)
where r
2
= x
2
+ y
2
+ z
2
, and b
2
= 3/2r
2
. This generalization is valid only for small
extensions of a relatively long chain (82). Equation (12) gives the probability that
if one extremity of the vector r is fixed at the origin of the coordinates, the other lies
in the volume dx dy dz centered around the point (x, y, z). What is more interesting
is the probability for a chain to have its end in a spherical shell of radius r and
thickness dr centered at the origin, irrespective of direction. This is the radial
distribution
W(r)dr
= (b/π
1
/2
)
3
exp(
− b
2
r
2
)4
πr
2
dr
(13)
which is illustrated in Figure 8 (3). The maximum occurs at r
= (2nl
2
/3)
1
2
. The
mean-square end-to-end distance is the second moment of the radial distribution
function
r
2
=
∞
0
r
2
W(r)dr
/
∞
0
W(r)dr
(14)
which yields the result of equation (11)
r
2
= 3/2b
2
= nl
2
(15)
The Gaussian distribution (eq. (12)) was obtained in the aforementioned
treatment with the assumption that chains are far from their full extension. More-
over, the Gaussian distribution function predicts zero probability only for r
= ∞
instead of for all r in excess of that for full chain extension, and does not adequately
226
ELASTICITY, RUBBER-LIKE
Vol. 2
0.04
0.03
0.02
0.01
0
0
10
20
30
40
50
r, nm
W(r)
, nm
−
1
Fig. 8.
Radial distribution function W(r) of the chain displacement vectors for chain
molecules consisting of 10
4
freely jointed segments, each of length l
= 0.25 nm; W(r) is
expressed in nm
− 1
and r in nm (3). Courtesy of Cornell University Press.
take into account the significant geometric and conformational differences known
to exist among different types of polymer chains (83). The distribution obtained
from a more general treatment is of the more complicated form (84–86) shown as
follows:
W(r)dr
= const.×exp
−
r
0
L
− 1
(r
/nl)dr/l
4
πr
2
dr
(16)
where
L
− 1
is the inverse Langevin function and
L(u) = coth u− 1/u
(17)
Equation (16) can be expanded in the series
W(r)dr
= const.×exp{−n[(3/2)(r/nl)
2
+ (9/20)(r/nl)
4
+ (99/350)(r/nl)
6
+ · · ·]}4πr
2
dr
(18)
and the Gaussian distribution (eq. (13)) recovered for r
nl. A comparison of the
Gaussian and inverse Langevin distributions for n
= 6 is shown in Figure 9 (7).
Chain with Bond-Angle Restrictions.
Although the chain in the afore-
mentioned treatment was assumed to be freely jointed, a real polymer chain has
fixed bond angles
θ. Therefore, the second term in equation (9) is no longer zero.
The scalar product of two vectors I
i
·I
j
is l
2
cos
|j − i|
θ. It can be shown (9,73,74) that
r
2
nl
2
(1
− cos θ)/(1 + cos θ)
(19)
Vol. 2
ELASTICITY, RUBBER-LIKE
227
0.4
0.3
0.2
0.1
0.0
0
1
2
3
4
5
6
W(r)
A
B
Fig. 9.
Distribution function W(r) for a six-link random chain, with length l
= 1 in arbi-
trary units: A, Gaussian limit; B, inverse Langevin function (7). Courtesy of Oxford Uni-
versity Press.
In the particular case of a tetrahedrally bonded chain,
θ = 109.5
◦
and
r
2
= 2nl
2
, twice the value for a freely jointed chain. Thus, a real chain is quite
different from this ideal representation. Nevertheless, any flexible real chain can
be represented by a simple model which is the statistical equivalent of a freely
jointed chain (87,88). The two conditions are that the real and freely jointed chains
have the same mean-square end-to-end distance and the same length at complete
extension:
r
2
= r
2
e
= n
e
l
2
e
(20)
and
R
= R
e
= n
e
l
e
(21)
Thus, only one model chain is equivalent to the real one, obeying the same
Gaussian distribution function and composed of n
e
segments of length l
e
given
by
n
e
= R
2
/r
2
(22)
l
e
= r
2
/R
(23)
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ELASTICITY, RUBBER-LIKE
Vol. 2
Chain with Bond-Angle Restrictions and Hindered Rotations.
The
angle of rotation
φ of a single bond around the axis formed by the preceding one
may be restricted by steric interferences between atoms. When n is large and the
average value of cos
φ, ie, cos φ, is not too close to unity, the following relationship
can be established (89,90):
r
2
= nl
2
1
− cos θ
1
+ cos θ
1
+ cos φ
1
− cos φ
(24)
When the rotation is not hindered, ie, when cos
φ = 0, equation (24) is equivalent
to equation (19).
The conformations of polymer chains may be generated by a Monte Carlo
simulation method. The dimensions of linear chains, unperturbed by excluded-
volume interactions, have been measured in various solvents by light scattering, x-
ray small-angle scattering, and dilute-solution viscosity measurements. They are
reported most comprehensively in the Polymer Handbook (91,92). A powerful tool
to investigate chain conformations in unswollen and swollen melts and networks
is small-angle neutron scattering (sans) (67,93–97).
The Equation of State for a Single Polymer Chain.
The variation in
the Helmholtz free energy is the negative of the work of deformation in isothermal
elongation
d A
= −dW = f dr
(25)
with
d A
= dU − TdS
(26)
The tensile force f on a polymer chain for a given length r is
f
=
∂ A
∂r
T
=
∂U
∂r
T
− T
∂ S
∂r
T
(27)
In freely jointed and freely rotating model chains, no rotation is preferred;
therefore, the internal energy is the same for all the conformations. Then,
f
= −T
∂ S
∂r
T
(28)
The entropy is given by the Boltzmann relation
S
= k ln
(29)
where k is the Boltzmann constant, and
, the total number of configurations
available to the system, is proportional to W(r) (eq. (13)). Calculation of the force
gives
f
= 2kTb
2
r
(30)
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ELASTICITY, RUBBER-LIKE
229
f
= 2kTr/r
2
(31)
Thus, f is directly proportional to the absolute temperature and to r, which means
the chain acts as a Hookean spring with modulus 3kT
/r
2
. The use of a non-
Gaussian distribution for
leads to (71,85,86)
f
= (kT/l)L
− 1
(r
/nl)
(32)
Classical Thermodynamics of Rubber-like Elasticity
In experiments concerning the relationships between length, temperature, and
force, usually the change in force with temperature at constant length is recorded
(53,98–101). It is therefore necessary to extend the thermodynamic treatment of
the elasticity. Moreover, the force is not purely entropic, and the energetic contribu-
tion carries useful information on the dependence on temperature of the average
end-to-end distance of the network chains in the unstrained state (21,102). It is
therefore important to know how to deduce these quantities from a thermoelastic
experiment.
The change in internal energy during stretching an elastic body is
dU
= dQ − dW
(33)
where dQ is the element of heat absorbed by the system and dW the element of
work done by the system on the surroundings. In a reversible process,
dQ
= TdS
(34)
where S is the entropy of the body. The work dW can be expressed as the sum
−dW = −pdV + f dL
(35)
where p is the equilibrium external pressure, dV the volume dilation accompany-
ing the elongation of the elastomer, and f the equilibrium tension. Thus,
dU
= TdS− pdV + f dL
(36)
At constant pressure, the enthalpy change is
dH
= dU + pdV = TdS+ f dL
(37)
A deformation dL at constant pressure and temperature induces a retractive force
f
=
∂ H
∂ L
T
,p
− T
∂ S
∂ L
T
,p
(38)
Expression 38 is one of the forms of the thermodynamic elastic equation of state.
Measurements of stress at constant length as a function of temperature have been
230
ELASTICITY, RUBBER-LIKE
Vol. 2
20
40
60
80
0
0.1
0.2
0.3
0.4
0.5
Stress
, N/mm
2
60
50
40
30
20
15
10
5
3
1
Temperature,
°C
Fig. 10.
Stress–temperature curves for sulfur-vulcanized natural rubber (99,103). Cour-
tesy of the American Chemical Society.
performed (see Fig. 10) (97,103–105). All the curves appear to be straight lines;
the slope increases with increasing elongation, but for very small elongations, the
slopes can be negative. This phenomenon, called the thermoelastic inversion, is
due to the volume expansion occurring in any elastomer. The condition of constant
length does not correspond to constant elongation as well, since the sample’s un-
strained reference length changes with temperature. The inversion is suppressed
by correction of the original length (11).
The energetic and entropic contributions to the force in the intramolecular
process of stretching the chains can be obtained in experiments where there is no
other energetic contribution resulting from changes in (intermolecular) van der
Waals forces. Therefore, these experiments must be performed at constant volume.
A basic postulate of the elasticity of amorphous polymer networks is that the stress
exhibited by a strained polymer network is assumed to be entirely intramolecular
in origin. That is, intermolecular interactions play no role in deformations at
constant volume and composition.
An equation similar to equation (38) is obtained for the elastic force measured
at constant volume:
f
=
∂U
∂ L
T
,V
− T
∂ S
∂ L
T
,V
(39)
Vol. 2
ELASTICITY, RUBBER-LIKE
231
The variation in the Helmholtz free energy has the following expression:
d A
= −SdT − pdV + f dL
(40)
The second derivative obtained by differentiating (
∂ A
∂ L
)
V
,T
with respect to T at
constant V and L is identical with that obtained by differentiating (
∂ A
∂T
)
V
,L
with
respect to L at constant V and T:
−
∂ f
∂T
V
,L
=
∂ S
∂ L
V
,T
(41)
Thus, equation (39) can be written as
f
=
∂U
∂ L
T
,V
+ T
∂ f
∂T
V
,L
(42)
The energetic and entropic components of the elastic force, f
e
and f
s
, respectively,
are obtained from thermoelastic experiments using the following equations:
f
e
=
∂U
∂ L
T
,V
= f − T
∂ f
∂T
V
,L
(43)
f
s
= −T
∂ S
∂ L
T
,V
= T
∂ f
∂T
V
,L
(44)
An example of thermoelastic data is given in Figure 11 (99). The change in
entropy with elongation up to 350% is responsible for more than 90% of the total
stress at room temperature, whereas the contribution of internal energy is less
than 10%. Thus, the restoring force is due almost entirely to the tendency of the
extended rubber molecules to return to their unperturbed, random conformational
states. Above 350%, crystallization appears. This is a specific feature of stereo-
regular rubbers, such as natural rubber which is capable of crystallization. Data
concerning most of the polymers studied in this manner are reviewed in References
21 and 106.
Statistical Treatment of Rubber-like Elasticity
A network is an ensemble of macromolecules linked together, each of them re-
arranging its configurations by Brownian motion. Classical thermodynamics ex-
plains the behavior of elastomers with regard to force, temperature, pressure, and
volume, but does not give the relationship between the molecular structure of the
network and elastic quantities such as the moduli. Therefore, statistical mechan-
ics was introduced in the 1940s (16,86,87,107–109), and its theoretical predictions
were tested (110–112). Because of the complexity of network structures, two mod-
els based on affine and phantom networks were studied. The cross-linking points
232
ELASTICITY, RUBBER-LIKE
Vol. 2
Elongation,
%
0
0
0.2
−0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
50
100
150
200
250
300
350
400
Stress
, N/mm
2
f
f
e
f
s
Fig. 11.
Plot of the energetic (f
e
) and entropic (f
s
) contributions to the stress at 20
◦
C
for sulfur-vulcanized rubber (99). N/mm
2
= MPa; to convert MPa to psi, multiply by 145.
Courtesy of the American Chemical Society.
of the affine network are fixed, whereas those of the phantom network can un-
dergo fluctuations independent of their immediate surroundings. The effects of
the macroscopic strain are transmitted to the chains through these junctions at
which the chains are multiply joined, with the internal state of the network system
specified in terms of the positions of the cross-linkages. Deformation transforms
the arrangements of these points. The system is represented by the vectors r
i
,
each of which connects the two ends of a chain. For a given end-to-end vector r
i
,
the number of available configurations is directly proportional to the probability
W(r
i
) or relative number of configurations. Representation of W(r) by a Gaussian
function according to equation (13) fails as r approaches the maximum extension
r
max
. Experimental evidence of deviations from the Gaussian theory have been
reported (24,113–116) and non-Gaussian theories based on an expression for W(r)
similar to equation (16) have been developed (84,85,117–119). These theories have
the disadvantage of containing parameters that can be determined only by com-
parisons between theory and experiments, specifically
ν, the number density of
chains in the network, and n, the number of statistical links.
The tensile force is expressed by
f
/A
0
= (1/3)vkTn
1
/2
L
− 1
(
λn
− 1/2
)
− λ
− 3/2
L
− 1
(
λ
− 1/2
n
− 1/2
)
(45)
Vol. 2
ELASTICITY, RUBBER-LIKE
233
where
L
− 1
(t)
= 3t + (9/5)t
3
+ (297/175)t
5
+ · · ·
and A
0
is the cross-sectional area of the unstrained sample. The number n of ran-
dom links may be obtained by birefringence and stress–strain measurements, and
from this result, an estimation of the number of monomer units in the equivalent-
random link (of the Kuhn–Gr ¨
un theory) of the polymeric chain (120). Another
approach to a non-Gaussian theory utilizes information provided by rotational iso-
meric state theory on the spatial configurations of chain molecules (83), including
most of those used in elastomeric networks. Specifically, Monte Carlo calculations
(121–123) based on the rotational isomeric state approximation are used to sim-
ulate spatial configurations followed by distribution functions for the end-to-end
separation r of the network chains (124).
The theory in the Gaussian limit has been refined greatly to take into ac-
count the possible fluctuations of the junction points. In these approaches, the
probability of an internal state of the system is the product of the probabilities
W(r
i
) for each chain. The entropy is deduced by the Boltzmann equation, and the
free energy by equation (26). The three main assumptions introduced in the treat-
ment of elasticity of rubber-like materials are that the intermolecular interactions
between chains are independent of the configurations of these chains and thus of
the extent of deformation (125,126); the chains are Gaussian, freely jointed, and
volumeless; and the total number of configurations of an isotropic network is the
product of the number of configurations of the individual chains.
Affine Networks.
Diffusion of the junctions about their mean positions
may be severely restricted by neighboring chains sharing the same region of
space. The extreme case is the affine network where fluctuations are completely
suppressed, and the instantaneous distribution of chain vectors is affine in the
strain. The elastic free energy of deformation is then given by
A
el
(aff )
= (1/2)vkT(I
1
− 3) − (v − ξ)kT ln(V/V
0
)
(46)
where I
1
is the first invariant of the tensor of deformation
I
1
= λ
2
x
+ λ
2
y
+ λ
2
z
(47)
The quantities
λ
x
,
λ
y
, and
λ
z
are the principal extension ratios, which specify the
strain relative to an isotropic state of reference having volume V
0
; V is the volume
of the deformed specimen, and
ν is the number of linear chains whose ends are
joined to multifunctional junctions of any functionality
φ > 2. (The functionality
of a junction is defined as the number of chains connected to it.) The cycle rank
ξ
(102) represents the number of chains that have to be cut to reduce the network
to an acyclic structure or tree (127). The cycle rank is the difference between the
number of effective chains
ν and effective junctions µ of functionality φ > 2:
ξ = ν − µ
(48)
234
ELASTICITY, RUBBER-LIKE
Vol. 2
Isotropic state
Deformed state
V
L
V
0
L
0
Fig. 12.
Simple extension of an unswollen network prepared in the undiluted state (99).
Courtesy of the American Chemical Society.
For a perfect network of functionality
φ,
ξ = (1 − 2/φ)ν
(49)
Simple Extension.
The most general case is an extension with change in
sample volume, as illustrated in Figure 12. The strain along the direction of
stretching is given by
λ
x
as
λ = λ
x
= L/L
0
(50)
Since the volume changes from V
0
to V,
λ
x
λ
y
λ
z
= V/V
0
(51)
Because there is symmetry about the x axis,
λ
y
= λ
z
. Combining equations (50)
and (51) leads to
λ
y
= λ
z
= (V/λV
0
)
1
/2
(52)
The force f is the derivative of the free energy with respect to length; specifically,
f
=
∂A
∂ L
T
,V
= (1/L
0
)
∂A
∂λ
T
,V
(53)
Using equation (46),
f
= (vkT/L
0
)
λ − V/
V
0
λ
2
(54)
It is also possible to deduce the expression for the force as a function of the ex-
tension
α measured relative to the length L
v
i
of the unstretched sample when its
volume is fixed at the same volume V as occurs in the stretched state:
α = L/L
i
v
= λL
0
/L
i
v
(55)
and
L
0
= L
i
v
(V
/V
0
)
− 1/3
(56)
Vol. 2
ELASTICITY, RUBBER-LIKE
235
Equation (54) is then transformed to
f
=
vkT
/L
i
v
(V
/V
0
)
2
/3
(
α − 1/α
2
)
.
(57)
Energetic Contributions.
Equation (57) can be used for the molecular in-
terpretation of the ratio f
e
/f . Thus, equation (43) can be rewritten as
f
e
= f − T
∂ f
∂T
V
,L
= − f T
∂ln( f/T)
∂T
V
,L
(58)
or
f
e
f
= −T
∂ln( f/T)
∂T
V
,L
(59)
The derivative of f at constant volume and length, thus at constant
α, can be
obtained from equation (57), with the result
f
e
/f = (2/3)T(dln V
0
/dT) = T
dln r
2
0
dT
(60)
since V
0
is the volume of the isotropic state so defined that the mean square of
the magnitude of the chain vectors equals r
2
0
, the value for the free unperturbed
chains.
The intramolecular energy changes arising from transitions of chains from
one spatial configuration to another are, by equation (60), directly related to
the temperature coefficient of the unperturbed dimensions. It is interesting to
compare the thermoelastic results for polyethylene (128), f
e
/f
= −0.45, and
poly(dimethylsiloxane) (129), f
e
/f
= 0.25. The preferred (lowest energy) confor-
mation of the polyethylene chain is the all-trans form, since gauche states at rota-
tional angles of
±120
◦
cause steric repulsions between CH
2
groups (83). Since this
conformation has the highest possible spatial extension, stretching a polyethylene
chain requires switching some of the gauche states, which are, of course, present in
the randomly coiled form, to the alternative trans states (106,130). These changes
decrease the conformational energy and are the origin of the negative type of ide-
ality represented in the experimental value of f
e
/f . (This physical picture also
explains the decrease in unperturbed dimensions upon increase in temperature.
The additional energy causes an increase in the number of the higher energy
gauche states, which are more compact than the trans ones.) The opposite be-
havior is observed in the case of poly(dimethylsiloxane) (26). The all-trans form
is again the preferred conformation; the relatively long Si O bonds and the un-
usually large Si O Si bond angles reduce steric repulsion in general, and the
trans conformation places CH
3
side groups at distances of separation where they
are strongly attractive (83,129,131). Because of the inequality of the Si O Si
and O Si O bond angles, however, this conformation is of very low spatial ex-
tension. Stretching a poly(dimethylsiloxane) chain therefore requires an increase
in the number of gauche states. Since these are of higher energy, this explains
the fact that deviations from ideality for these networks are found to be positive
(106,129,130).
236
ELASTICITY, RUBBER-LIKE
Vol. 2
Isotropic dry
state 1
Isotropic swollen
state 2
Deformed swollen
state 3
V
L
V
L
0
V
0
L
i
Fig. 13.
Simple extension of a swollen network.
Simple Extension of Swollen Networks.
The force required to deform an
elastomeric sample, from state 2 to state 3 in Figure 13, is given by equation (57),
in which
α = L/L
i
v
is the extension ratio for the isotropic swollen state relative to
the deformed swollen state. The ratio V
0
/V is the volume fraction v
2
of polymer
in the swollen system, and A
0
is the cross-sectional area of the dry sample, which
is generally measured before the experiment. The area of the swollen sample is
given by
A
i
v
= A
0
(V
/V
0
)
2
/3
(61)
Equation (56), of course, still holds, and L
0
A
0
= V
0
. In an experiment, the force f
is measured as a function of elongation. Under these conditions,
f
/A
0
= (νkT/L
0
A
0
)v
− 1/3
2
(
α − 1/α
2
)
(62)
Hence, the quantity [ f
∗]≡v
1
/3
2
/A
0
(
α − 1/α
2
) should be a constant, independent of
the degree of swelling, and be equal to vkT/V
0
. As shown later [f
∗] is commonly
plotted versus 1/
α to determine deviations from theory (112,132,133).
Simple Extension of Networks Cross-linked in the Diluted State.
The
polymer is dissolved, cross-linked, and dried. The stress–strain measurements
are carried out on the dry network, as illustrated in Figure 14. Equation (57)
also holds for this case. If A
i
v
is the cross-sectional area of the undeformed dry
specimen, the force per unit undeformed area is
f
/A
i
v
= (νkT/V)(V/V
0
)
2
/3
(
α − 1/α
2
)
(63)
Reference state
for cross-linking
process
Dry, undeformed
state
Dry, deformed
state
L
0
V
0
V
V
L
L
i
Fig. 14.
Simple extension of an unswollen network cross-linked in the diluted state
(V
0
> V).
Vol. 2
ELASTICITY, RUBBER-LIKE
237
which is (V/V
0
)
2
/3
times the tension of a network cross-linked in the dry state
(111,134).
The Affine Shear Modulus.
For V
= V
0
, equation (54) becomes
f
/A
0
= (νkT/V
0
)(
λ − 1/λ
2
)
(64)
Making use of equation (2) gives
λ = 1 + . For small values of , 1, equation
(64) may be written as
f
/A
0
= 3(νkT/V
0
)
(65)
The tensile modulus E
aff
is three times the shear modulus G
aff
, since Poisson’s
ratio for elastomers is close to 0.5. Specifically,
E
aff
= 3vkT/V
0
= 3G
aff
(66)
and hence,
G
aff
= vkT/V
0
(67)
If v/V
0
is the molar number density of chains [equal to
ρ/M
c
for a perfect network,
where
ρ is the elastomer density and M
c
is the molecular weight between cross-
link points (in g/mol)], then
G
aff
= vRT/V
0
(68)
where R is the gas constant. In the remaining material, v is a molar quantity
unless it is followed by the Boltzmann constant k.
For an affine perfect network,
G
aff
= ρ RT/M
c
(69)
As a numerical example, the affine shear modulus of a perfect poly
(dimethylsiloxane) tetrafunctional network of density
ρ = 0.97 g/cm
3
and M
c
=
11,300 g/mol is G
aff
= 0.212 × 10
6
N/m
2
. However, imperfections such as chain
ends exist in typical networks, as illustrated in Figure 6. The following correc-
tion can be made to account for this circumstance (16). Before the cross-linking
reaction, it is assumed that v
m
chains of length M
n
are present in the melt, with
v
m
= ρ/M
n
. Each chain has two ends, thus there are 2
ρ/M
n
chain ends. After the
cross-linking process, there are v
0
chains of length M
c
, v
0
= ρ/M
c
, along with the
2
ρ/M
n
chain ends. Therefore the number of chains that are elastically effective is
v
= ρ/M
c
− 2ρ/M
n
= (ρ/M
c
)(1
− 2M
c
/M
n
)
(70)
This correction is small for M
c
M
n
, which is frequently the case.
238
ELASTICITY, RUBBER-LIKE
Vol. 2
Phantom Networks.
In this idealized model, chains may move freely
through one another (135). Junctions fluctuate around their mean positions be-
cause of Brownian motion, and these fluctuations are independent of deforma-
tion. The mean square fluctuations
r
2
of the end-to-end distance r are related
to the mean square of the end-to-end separation of the free unperturbed chains
r
2
(102,136,137) by
r
2
/r
2
0
= 2/φ
(71)
The fluctuation range (
r
2
= r
2
0
/2 for a tetrafunctional network) is generally quite
large and of considerable importance. The instantaneous distribution of chain
vectors r is not affine in the strain because it is the convolution of the distribution
of the affine mean vector r with the distribution of the fluctuations
r, which are
independent of the strain. The elastic free energy of such a network is
A
el
(ph)
= (1/2)ξkT(I
1
− 3) = (1/2)(ν − µ)kT(I
1
− 3)
(72)
In simple extension, the equivalent of equation (62) is now
f
/A
0
= (ν − µ)kTV
− 1
0
v
− 1/3
2
(
α − 1/α
2
)
(73)
and the shear modulus is given by
G
ph
= (v − µ)RT/V
0
(74)
For a perfect network of functionality
φ, equation (49) holds, and therefore,
G
ph
= (1 − 2/φ)vRT/V
0
(75)
G
ph
= (1 − 2/φ)ρ RT/M
c
(76)
The relationship between affine and phantom moduli is then
G
ph
= (1 − 2/φ)G
aff
(77)
For example, G
aff
is twice G
ph
for a perfect tetrafunctional network.
Comparisons With Experimental Results.
Stress–strain measure-
ments in uniaxial extension can be compared with the prediction of an affine
Gaussian network (eq. (64)), as illustrated in Figure 15 (113). The Gaussian re-
lationship in the affine limit is valid only at small deformations. The best fit is
obtained using G
aff
= 0.39 MN/m
2
(56.6 psi) (7); deviations occur when
λ > 1.5.
The experimental curve may nevertheless be well represented by adjusting the
parameters of the non-Gaussian stress–strain relationship (eq. (45)) (84,117,138).
These disagreements between experiments and the simple predictions of statis-
tical mechanics have led some workers to develop a phenomenological theory of
rubber-like elasticity.
Vol. 2
ELASTICITY, RUBBER-LIKE
239
Extension ratio
T
ensile f
orce per unit unstr
ained area, N/mm
2
Theoretical
1
2
3
4
5
6
7
8
0.0
1.0
2.0
3.0
4.0
5.0
6.0
7.0
Fig. 15.
Stress–strain isotherms in simple elongation; comparison of experimental curve
(open circles) with theoretical prediction (eq. (64)) (113). To convert N/mm
2
to psi, multiply
by 145. Courtesy of The Royal Society of Chemistry.
Phenomenological Theory.
Continuum mechanics is used to describe
mathematically the stress–strain relations of elastomers over a wide range in
strain. This phenomenological treatment is not based on molecular concepts, but
on representations of observed behavior (132,139–143). The main goal is to find an
expression for the elastic energy W stored in the system (assumed to be perfectly
elastic, isotropic in its undeformed state, and incompressible), analogous to the
free energy of the statistical treatment. The condition of isotropy in the unstrained
state requires that W be symmetrical with respect to the three principal extension
ratios
λ
x
,
λ
y
,
λ
z
. A rotation of the material through 180
◦
, ie, a change of sign of
two of the
λ
i
(i
= x, y, z), does not alter W (144). The three simplest even-powered
functions satisfying these conditions are the strain invariants I
1
, I
2
, and I
3
defined
as
I
1
= λ
2
x
+ λ
2
y
+ λ
2
z
I
2
= λ
2
x
λ
2
y
+ λ
2
y
λ
2
z
+ λ
2
z
λ
2
x
I
3
= λ
2
x
λ
2
y
λ
2
z
(78)
The most general form of the strain–energy function, which vanishes at zero
strain, for an isotropic material is the power series
240
ELASTICITY, RUBBER-LIKE
Vol. 2
W
=
∞
i
, j,k= 0
C
i jk
(I
1
− 3)
i
(I
2
− 3)
j
(I
3
− 1)
k
(79)
In equation (79), the experimental results obtained at sufficiently small strains
are well represented with two nonzero terms, C
100
and C
010
:
W
= C
100
(I
1
− 3) + C
010
(I
2
− 3)
(80)
In uniaxial extension, the so-called Mooney–Rivlin equation is obtained (17):
f
υ
1
/3
2
/A
0
= 2(C
1
+ C
2
/α)(α − 1/α
2
)
(81)
A convenient and standard way to treat stress–strain data is to plot the reduced
force
[ f
∗]≡ f
∗
υ
1
/3
2
/(α − 1/α
2
)
(82)
versus 1/
α, where f ∗ is the nominal stress f /A
0
and
υ
2
the volume fraction of
polymer in the network, if swollen (Fig. 16).
In this scheme, the affine and phantom networks are represented by hor-
izontal lines (eqs. (62) and (73)) from equations [f
∗] = vRT/V
0
and [f
∗] =
(v
− µ)RT/V
0
, respectively. It has been reported that C
2
decreases as
υ
2
decreases,
whereas C
1
is approximately constant, as illustrated in Figure 16 (112). In view
0.20
0.29
0.40
0.55
0.74
0.4
0.6
0.8
1.0
0.125
0.150
0.175
0.200
0.225
1/
[
f∗
], N/mm
2
2
1.00
Fig. 16.
Plot of [f
∗] versus 1/α for a natural rubber vulcanizate swollen in benzene to
demonstrate the influence of v
2
on C
2
(114). To convert N/mm
2
to psi, multiply by 145.
Courtesy of The Royal Society of Chemistry.
Vol. 2
ELASTICITY, RUBBER-LIKE
241
of experimental results on swollen networks (69,114,145,146), the following form
of the strain energy per unit volume of dry network is to be preferred:
W
/V
0
= C
1
(I
1
− 3)+C
2
(I
2
I
− 1+m/2
3
− 3)
(83)
The reduced force can be deduced (145) as [ f
∗] = 2C
1
+2C
2
v
(4
/3) − m
2
α
− 1
. The pa-
rameter C
1
of the swollen network is equal to C
1
of the dry network, whereas
C
2
(swollen) is equal to C
2
(dry) v
(4
/3) − m
2
. Experimentally, m was found to be 0 or
1
2
(145). The constant C
1
increases with the cross-linking density of the rubber
(114,147), and 2C
1
+ 2C
2
is approximately the shear modulus at small defor-
mation (148). Although there have been many attempts (149–155), a molecular
explanation of C
1
and C
2
has been achieved only recently, as is described later.
The experimental error range is of great importance. A 1% error in the de-
termination of
λ has a tremendous effect on the Mooney curve when 1/α > 0.9;
this part of the curve is therefore highly unreliable (156).
Statistical Theory of Real Networks.
Affine and phantom networks are
extreme limits. Stress–strain measurements in uniaxial extension have revealed
that the behavior of real networks is between these limits. A theoretical attempt
has been made to account for this dependence of [f
∗] on 1/α in terms of a gradual
transition between the affine and phantom deformations (102,157), and a molec-
ular theory has been formulated (102,158–161). In this model, the restrictions of
junction fluctuations due to neighboring chains are represented by domains of con-
straints. At small deformations, the stress is enhanced relative to that exhibited
by a phantom network. At large strains, or high dilation, the effects of restrictions
on fluctuations vanish and the relationship of stress to strain converges to that for
a phantom network. In a later theoretical refinement, the behavior of the network
is taken to depend on two parameters. The most important is
κ which measures
the severity of entanglement constraints relative to those imposed by the phantom
network. Another parameter
ζ takes into account the nonaffine transformation of
the domains of constraints with strain. Topological and mathematical treatment
leads to the expression
[ f
∗] = f
ph
(1
+ f
c
/f
ph
)
(84)
where f
ph
is the usual phantom modulus and f
c
/f
ph
is the ratio of the force due to
entanglement constraints to that for the phantom network. A specific expression
for f
c
/f
ph
in uniaxial extension is
f
c
/f
ph
= (µ/ξ)
αK
α
2
1
− α
− 2
K
α
2
2
/(α − 1/α
2
)
(85)
where
α
1
= αv
− 1/3
2
,α
2
= α
− 1/2
v
− 1/3
2
, and
µ/ξ is the ratio of the number of effective
junctions to the cycle rank. For a perfect network,
µ/ξ=2/(φ − 2). The function
K(x
2
) is given by
K(x
2
)
= B[
.
B (B
+ 1)
− 1
+ g(
.
g B
+ g
.
B )(gB
+ 1)
− 1
]
(86)
242
ELASTICITY, RUBBER-LIKE
Vol. 2
with
g
= x
2
[1
/κ + ζ (x − 1)]
.
g
= 1/κ − ζ(1 − 3x/2)
B
= (x − 1)(1 + x − ζ x
2
)
/(1 + g)
2
.
B
= B
(2x(x
− 1))
− 1
− 2
.
g (1
+ g)
− 1
+ (1 − 2ζ x)[2x(1 + x − ζ x
2
)]
− 1
It is interesting to note that junction fluctuations increase in the direction of
stretching but decrease in the direction perpendicular to it. Therefore the modu-
lus decreases in the direction of stretching, but increases in the normal direc-
tion since the state of the network probed in this direction tends to be more
nearly affine. The curve of [f
∗] versus 1/α is sigmoidal. The parameters κ and ζ of
poly(dimethylsiloxane) networks are determined in Figure 17 (155); the intercept
of the sigmoidal curves is the phantom modulus. This Flory–Erman theory has
been compared successfully with such experiments in elongation and compression
(155,162,162–166). It has not yet been extended to take account of limited chain
extensibility or strain-induced crystallization (167).
0.20
0.12
0.08
0.04
0
0.2
0.4
0.6
0.8
1
M
n
4000 0.1 0
M
n
9500 3.4 0
M
n
18500 30 0.01
M
n
25600 24.5 0
M
n
32900 16.2 0
31.5 0.01
27.5 0
23 0
21.5 0
1/
[
f*], N/mm
2
Fig. 17.
Determination of the parameters
κ and ζ of the Flory–Erman theory for perfect
trifunctional poly(dimethylsiloxane) networks (155). To convert N/mm
2
to psi, multiply by
145. Courtesy of Springer-Verlag.
Vol. 2
ELASTICITY, RUBBER-LIKE
243
Predictions for the Parameters
κ and ζ.
The parameter
ζ is not far from
zero, which is to be expected since the surroundings of junctions cause their defor-
mation to be nearly affine with the macroscopic strain. The primary parameter
κ
is defined as the ratio of the mean-square junction fluctuations in the equivalent
phantom network, ie, in the absence of constraints, to the mean-square junction
fluctuations about the centers of domains of entanglement constraints (in the
absence of the network) in the isotropic state. Thus in a phantom network, the
absence of constraints leads to
κ = 0. In an affine one, the complete suppression
of fluctuations is equivalent to
κ = ∞. It has been proposed that κ should be pro-
portional to the degree of interpenetration of chains and junctions (165). Since an
increasing number of junctions in a volume pervaded by a chain leads to stronger
constraints on these junctions,
κ was taken to be
κ = I
r
2
0
3
/2
(
µ/V
0
)
(87)
where I is a constant of proportionality, (r
2
0
)
3
/2
is assumed to be proportional to
the volume occupied by a chain, and
µ/V
0
is the number of junctions per unit
volume. The mean-square unperturbed dimension r
2
0
can be taken proportional to
M
c
(95), the molecular weight between cross-links, and
µ/V
0
to M
− 1
c
for a perfect
tetrafunctional network; therefore
κ M
1
/2
c
(93,155,165).
Swelling Equilibrium.
The isotropic swelling of a cross-linked elastomer
by a liquid has two important opposing effects: the increase in mixing entropy of
the system because of the presence of the small molecules, and the decrease in con-
figurational entropy of the network chains by dilation. Therefore, an equilibrium
degree of swelling is established, which increases as the cross-linking density de-
creases (168–170). The free energy change
A for this process is usually assumed
to be separable into the free energy of mixing,
A
m
, and the elastic free energy
A
el
,
A= A
m
+ A
el
(88)
although questions have been raised with regard to this separability (171,172).
The contribution
A
m
has been calculated with the help of a lattice model (173,
174). The other contribution
A
el
is given in the later Flory–Erman theory (161)
by
A
el
= A
el
(ph)
+ A
c
(89)
where
A
el
(ph) is given by equation (72);
A
c
is the additional term which ac-
counts for the constraints and is given by
A
c
= µkT/2
i
=x,y,z
{[1 + g(λ
i
)]B(
λ
i
)
− ln[(B(λ
i
)
+ 1)(g(λ
i
)B(
λ
i
)
+ 1)]}.
(90)
The
λ
i
are the principal extension ratios, and g and B have been defined in equation
(86).
244
ELASTICITY, RUBBER-LIKE
Vol. 2
The chemical potential of the solvent in the swollen network is
µ
1
− µ
0
1
= N
∂A
m
∂n
1
T
,p
+ N
∂A
el
∂λ
T
,p
∂λ
∂n
1
T
,p
(91)
where n
1
is the number of moles of solvent, and the isotropic extension ratio
λ is
λ = λ
x
= λ
y
= λ
z
= [(n
1
V
1
+ V
0
)
/V
0
]
1
/3
= v
− 1/3
2
(92)
where V
1
is the molar volume of the solvent and V
0
the volume of the dry network.
At swelling equilibrium,
µ
1
= µ
0
1
. Hence, the standard expression for
A
m
(3) leads
to
ln(1
− υ
2m
)
+ υ
2m
+ χυ
2
2m
= −
∂A
el
∂λ
T
,p
∂λ
∂n
1
T
,p
(RT)
− 1
(93)
where v
2m
is the volume fraction of polymer at swelling equilibrium. The inter-
action parameter
χ
1
may be determined, for example, by vapor pressure or by
osmometry measurements (145,175–178). It depends on the concentration of poly-
mer in the polymer–solvent system (175). Using the Flory–Erman expression for
the elastic energy and assuming the parameter
κ to be independent of swelling
(isotropic swelling does not change the relative topology of the network), equation
(93) (with the left-hand side abbreviated as H) becomes
H
= − (ξ/V
0
)V
1
v
1
/3
2m
1
+ (µ/ξ)K
v
− 2/3
2m
(94)
The molecular weight M
c
between cross-links of a perfect network is then obtained
by combination of equations (48),(49), and (94), with v/V
0
= ρ/M
c
:
M
cr
= (2/φ − 1)ρV
1
υ
1
/3
2m
1
+ (ϕ/2 − 1)
− 1
K
V
1
υ
− 2/3
2m
H
(95)
where the subscript r is employed here for real networks.
For a network deforming affinely,
κ = ∞, K(λ
2
)
= 1 − λ
− 2
, and
M
ca
= − ρV
1
υ
1
/3
2m
1
− 2υ
2
/3
2m
/φ
H
(96)
For a phantom network,
κ = 0, K(λ
2
)
= 0, and
M
cp
= (2/φ − 1)ρV
1
υ
1
/3
2m
H
(97)
Determination of the Degree of Cross-linking by Stress–Strain
Measurements and by Swelling Equilibrium
Characterization of network structures is often the main objective of theoretical
and experimental works in the field of rubber elasticity (179–183). Simple experi-
ments such as swelling equilibrium have been extensively used. However, most of
the experimental swelling results on cross-linked polymers have been interpreted
using the Flory–Rehner expression for an affinely deforming network (6,184–186).
Vol. 2
ELASTICITY, RUBBER-LIKE
245
The modern theory of real networks now permits a more accurate determina-
tion of network structures through use of equations (84),(87), and (95) (187–204).
Stress–strain measurements can be analyzed as shown in Figure 17. The phantom
modulus thus determined leads to
ν and M
c
through equations (75) and (76) (189).
Swelling equilibrium data are similarly analyzed through equation (94), with the
parameter
κ given by equation (87) (189).
If M
c
and
κ have been determined previously from stress–strain measure-
ments, then the interaction parameter
χ
1
may be calculated through equation
(95) (187,190).
Entanglements
Entanglements have been introduced in the later Flory–Erman theory as con-
straints that restrict junction fluctuations. Another viewpoint considers entangle-
ments to act as physical cross-links, being based, in part, on the observation that
linear polymers of high molecular weight exhibit a storage modulus G
(
ω), which
remains relatively constant over a wide range of frequencies
ω (205). This plateau
modulus G
0
N
is independent of chain length for long chains and is insensitive to
temperature. Since it varies with the volume fraction of polymer in concentrated
solutions, it could be due to pair-wise interactions between chains (20), and a uni-
versal law has been proposed for the dependence of G
0
N
on the chemical structure
of the polymer (206). During the cross-linking process, some of such interactions
or entanglements could be trapped in the network and act as physical junctions.
This conclusion has been tested by irradiation cross-linking of already deformed
networks (207–209), and then measuring the dimensional changes.
In the simplest phenomenological approach for rubber-like elasticity of
trapped entanglements at small deformation (210,211), the shear modulus is
taken to be the sum of two terms:
G
= G
c
+ G
e
T
e
(98)
where G
c
is the contribution of the chemical cross-links. Taking into account the
restrictions of junction fluctuations, as in the Flory theory, leads to
G
c
= (v − hµ)RT
(99)
in which the empirical parameter h was introduced (153). Its value, between 0
(affine) and 1 (phantom), characterizes the nature of the networks at small de-
formation; h can also be expressed as a function of the Flory parameters
κ and
ζ (155,212). The additional contribution G
e
T
e
is said to arise from permanently
trapped (interchain) entanglements in the network. The modulus G
e
is thought to
have a value close to G
0
N
, and T
e
, the “trapped entanglement factor,” is the proba-
bility that all of the four directions from two randomly chosen points in the system,
which may potentially contribute an entanglement, lead to the gel fraction.
The idea of entanglements acting as physical junctions was originally de-
veloped in the literature to explain deviations from the predictions for affine
networks (185,213). The possibility of a contribution at equilibrium caused by
trapped entanglements has been tested with model networks, ie, those prepared
in such a way that the number and functionality of the cross-links are known. A
246
ELASTICITY, RUBBER-LIKE
Vol. 2
typical, highly specific reaction used for this purpose is the end-linking of function-
ally terminated polymer chains. Specific examples would be hydroxyl-terminated
or vinyl-terminated chains of poly(dimethylsiloxane), [Si(CH
3
)
2
O]
x
, end-linked
with an organic silicate or silane (214,215). A considerable body of published data
has, in fact, been interpreted as proving the existence of a trapped-entanglement
contribution in this kind of network (216–222). Typically, the network charac-
teristics, eg, number of chains and junctions, extent of cross-linking reaction p,
trapped entanglement factor T
e
, and effective functionality
φ
e
, were calculated by
the branching theory after measurement of the network sol fraction. The main
assumption of this probabilistic method is that the sol fraction, after subtraction
of the amount of nonreactive species as determined by gel permeation chromatog-
raphy analysis (218), is composed only of primary reactive chains (223). On the
contrary, however, the sol fraction is a complicated mixture of reactive, unreactive,
reacted, and unreacted molecules. For example, simulation of nonlinear polymer-
izations has shown that about half of the sol molecules are (reacted) cyclics (224).
Their presence indicates that the value of the extent of the cross-linking reac-
tion, formerly calculated with the assumption that the sol fraction is composed of
nonreacted reagents, has to be significantly increased. As a result, many model
networks can be considered as nearly perfect. This casts some doubt on the results
interpreted as showing an entanglement contribution at equilibrium. If these net-
works are considered as perfect, such a contribution does not seem to be important
(155,225). Entanglement contributions have been reported for polybutadiene (153)
and ethylene–propylene copolymer (154) networks prepared by radiation-induced
cross-linking. Again some doubt exists on the method used to characterize the
network structure.
Molecular models treating entanglements as interstrand links that are free
to slip along the strand contours have been developed (226–228) and tube mod-
els have been investigated (229,230). These approaches have been reviewed in
Reference 27.
The question of entanglements is still controversial. It has been postulated
(160) that an entanglement cannot be equivalent to a chemical cross-link. Con-
tacts between a pair of entangled chains are transitory and of short duration owing
to the diffusion of segments and associated time-dependent changes of configura-
tions. Such trapped entanglements as previously described are possibly of minor
importance in equilibrium stress measurements.
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J. P. Q
UESLEL
Manufacture Michelin, CERL – GPA
J. E. M
ARK
University of Cincinnati
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