Elasticity, Rubber-like'. In: Encyclopedia of Polymer Science and …nguyen.hong.hai.free.fr/EBOOKS/SCIENCE AND ENGINEERING... · 2006. 11. 9. · 218 ELASTICITY, RUBBER-LIKE Vol

ELASTICITY, RUBBER-LIKE

Introduction

Elasticity is the reversible stress–strain behavior by which a body resists andrecovers from deformation produced by a force (1). This behavior is exhibited byrubber-like materials in a unique and extremely important manner. Unlike metalsor glasses, they can undergo very large deformations without rupture (and are thussimilar 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 ofpolymers as long-chain molecules was established in the 1920s. It was observedthat a rubber gave off heat upon stretching and, when submitted to a constantload, became shorter as temperature was increased. Later, Kelvin (4) derived thethermodynamic laws of elasticity, and more careful experiments were performedby 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 changein shape or volume. Mechanical testing of rubber involves application of a forceto a specimen and measurement of the resultant deformation or application of adeformation and measurement of the required force. The results are expressed interms of stress and strain and thus are independent of specimen geometry. Stressis the force f per unit original cross-sectional area A, and strain the deformationper unit original dimension. The units of stress are Newtons per square meter(N/m2 or Pa); strain is dimensionless. The ratio of stress to strain is called the

216Encyclopedia of Polymer Science and Technology. Copyright John Wiley & Sons, Inc. All rights reserved.

Vol. 2 ELASTICITY, RUBBER-LIKE 217

A

L

f

f

L0

(a) (b)

Fig. 1. Uniaxial extension of a bar-shaped sample. (a) No load; (b) tensile load. A = area;f = force; L0 = original length; L = length.

modulus. A material is Hookean when its modulus is independent of strain (typicalof a metal spring or wire); elastomers are Hookean only in the range of very smalldeformations.

The simplest deformations are isotropic compression under hydrostatic pres-sure, uniaxial extension (or compression), and simple shear. They are used todetermine the bulk modulus K, Young’s modulus E, and shear modulus G, respec-tively (37–44). Elastomer testing for commercial applications is highly dependenton the method, the conditions (eg, strain rate, temperature), and the shape of thesamples. Therefore, mechanical tests have been standardized for uniformity andsimplicity by the American Society for Testing and Materials (ASTM) (45).

Isotropic Compression. The volume of a sample decreases from V0 to Vwhen submitted to a hydrostatic pressure p. The bulk modulus is the ratio of thispressure to the volume change per unit volume �V/V0:

K = p/(�V/V0) (1)

where �V = V0 − V; K is the reciprocal of the compressibility and can be deter-mined by direct measurement of compressibility (46) or velocity of longitudinalelastic waves (sound) or by theoretical calculations (39).

Uniaxial Extension. A rubber strip of original length L0 is stretched uni-axially to a length L, as illustrated in Figure 1. The stretch and elongation are�L= L− L0 and λ = L/L0, respectively. The strain � (also known as the relativedeformation, linear dilation, or extension) and the elongation or extension ratio λare related by

� = �L/L0 = λ − 1 (2)

218 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 w0 to the change in length perunit length is called Poisson’s ratio (47):

νp = (�w/w0)/(�L/L0) (3)

Ideal rubbers and liquids deform at constant volume, for which νp is equal to 0.5.Poisson’s ratio for real elastomers may be experimentally determined by applyingextensometers in the transverse and axial directions of a sample. Approximatevalues of νp thus determined are 0.33 for glassy polymers, 0.4 for semicrystallinepolymers, and 0.49 for elastomers. Young’s modulus E is the ratio of normal stressf /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 ofthe stress–strain curve at the origin (initial tangent modulus). It can be measuredin flexure or by tensile experiments (48).

Tension testing of a vulcanized elastomer also permits the determinationof the tensile strength, which is the maximum stress applied during stretchinga specimen to rupture; the corresponding rupture strain is called the maximumextensibility (49–52). Typical values are given in Table 1 (14). Dumbbell and ringspecimens can be used.

6

6

5

5

4

4

3

3

2

f/A, N

/mm

2

2

1

10

7 8�

�

Fig. 2. Stress–strain curve of a non-Hookean solid; typical curve for an elastomer inuniaxial extension.


Table 1. Properties of a Typical Rubber, Metal, and Glassa

Property Rubber Metal Glass

Breaking extension, % 500 Large, plastic 3Elastic limit, % 500 2 3Tensile strength, MN/m2b 10c 5 × 104 5 × 103aRef. 14.bTo convert MN/m2 (MPa) to psi, multiply by 145.c>103 if based on area at break.

Forcegauge

Recorder

CathetometerConstant T

N2

Fig. 3. Schematic diagram of a typical apparatus used to measure elastomer stress as afunction of strain (21). T = Temperature.

A typical apparatus used to measure the equilibrium stress of an elongatednetwork as a function of strain and temperature is shown in Figure 3 (21). Therubber 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 testsection of the sample is delineated by ink marks applied before loading). Valuesof the force are obtained from a calibrated stress gauge, the output of which isdisplayed as a function of time on a standard recorder. Measurements are madeat elastic equilibrium; the influence of temperature can also be studied. Anotherexample of a stretching device is an automatic stress relaxometer (53).

Simple Shear. Simple shear, illustrated in Figure 4, is a deformation inwhich the height H and surface A of the sample are held constant. The shear


A

H

y

xf

�

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’smodulus, the shear modulus (41,42) is a material constant if stress and strain aredirectly proportional. If they are not, the shear modulus at small deformation isemployed; it is defined as the slope at the origin of the stress–strain curve. Theshear 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 eventhose 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 ofthe theory of elasticity of isotropic solids, the moduli and Poisson’s ratio are inter-related (54) as follows:

E= 2G(1 + νp) = 3K(1 − 2νp) (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 chainswhich are able to alter their arrangements and extensions in space in response toan imposed stress. Only long polymeric molecules have the required exceedinglylarge number of available configurations.

It is necessary that all the configurations are accessible; this means thatrotation must be relatively free about a significant number of the bonds joiningneighboring skeletal atoms.

Table 2. Poisson’s Ratio, Moduli, and Density of Metals, Ceramics, and Polymersa

Specific properties

Young’s modulus E, Shear modulus G, Bulk modulus K, Density ρ, E/ρ, 106 G/ρ, 106 K/ρ, 106

Material νp 109 N/m2b 109 N/m2b 109 N/m2b g/cm3 m2/s2 m2/s2 m2/s2

MetalsCast iron 0.27 90 35 66 7.5 12.0 4.7 8.8Steel (mild) 0.28 220 86 166 7.8 28.0 11.0 21.0Aluminum 0.33 70 26 70 2.7 26.0 9.6 26.0Copper 0.35 120 44.5 134 8.9 13.5 4.5 15.0Lead 0.43 15 5.3 36 11.0 13.6 4.8 33.0Mercury 0.5 0 0 25 13.55 0.0 0.0 1.85

InorganicsQuartz 0.07 100 47 39 2.65 38.0 17.8 14.7Vitreous silica 0.14 70 30.5 32.5 2.20 32.0 14.0 14.7Glass 0.23 60 24.5 37 2.5 24.0 9.8 14.9Granite 0.30 30 11.5 25 2.7 11.1 4.3 9.2

WhiskersAlumina 2000 1000 667 3.96 510 253 225Carborundum 1000 500 333 3.15 315 160 106Graphite 1000 500 333 2.25 440 220 150

PolymersPolystyrene 0.33 3.2 1.2 3.0 1.05 3.05 1.15 2.85Poly(methyl Methacrylate) 0.33 4.15 1.55 4.1 1.17 3.55 1.33 3.5Nylon-6,6 0.33 2.35 0.85 3.3 1.08 2.21 0.79 2.3Polyethylene (low density) 0.45 1.0 0.35 3.33 0.91 1.1 0.385 3.7Ebonite 0.39 2.7 0.97 4.1 1.15 2.35 0.86 3.6Rubber 0.49 0.002 0.0007 0.033 0.91 0.002 0.00075 0.04

LiquidsWater 0.5 0.0 0.0 2.0 1.0 0.0 0.0 2.0Organic liquids 0.5 0.0 0.0 1.33 0.9 0.0 0.0 1.5

aRef. 39. Courtesy of Elsevier.bN/m2 = Pa. To convert Pa to psi, multiply by 0.000145.

221


Fig. 5. Segment of an elastomeric network.

Network Structure. Chains must be joined by permanent bonds calledcross-links, as illustrated in Figure 5. The network structure thus obtained isessential so as to avoid chains permanently slipping by one another, which wouldresult in flow and thus irreversibility, ie, loss of recovery. These cross-links maybe chemical bonds or physical aggregates, eg, glassy domains in multiphase blockcopolymers (55,56).

At the end of the cross-linking process, the topology of the mesh is composedof the different entities represented in Figure 6 (16,57–59). An elastically activejunction 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-likeelasticity is due to elastically active chains and junctions. Specifically, upon de-formation the number of configurations available to a chain decreases and theresulting 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 existbetween all molecules, must be small. Specifically, extensive crystallization shouldnot 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-dimensionalcrystalline lattice, joined by powerful valence forces operating at relatively short


(a) (b)

(d)(c)

Fig. 6. Structural features of a network. (a) Elastically active chain; (b) loop; (c) trappedentanglement; (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 veryhigh. After a small deformation, slippage between adjacent crystals occurs at theyield point, and the deformation increases much more rapidly than the stress andbecomes irreversible or plastic. The primary effect of stretching a metal short ofthis yield point is the increase �Em in energy caused by changing the distanced of separation between metal atoms. The sample recovers its original lengthwhen the force is removed, since this process corresponds to a decrease in energy.Heating increases oscillations about the minimum of the asymmetric potentialenergy 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. Thenumber of available configurations for a network chain is reduced in the defor-mation process. After suppression of the stress, the specimen recovers its originalshape, since this corresponds to the most disordered state. Thus the retractiveforce arises primarily from the tendency of the system to increase its entropytoward 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 chainsreach the limits of their extensibility. For polymers with regular structures, crys-tallization may also be induced. Further elongation may then require deformationof 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.


Extension of Elastomers and Compression of Gases

The extension of elastomers and the compression of gases are associated with adecrease in entropy. Thus, similar behavior is expected in regard to (adiabatic)deformation and heating. The work of deformation of a gas is dW = −pdV, wherep is the pressure and V the volume. For an elastomer it is dW = fdL, where f is theforce and L the length; the term −pdV is not taken into account, since there is onlya negligible change in rubber volume during stretching. These observations can beused to explain Gough’s experiments by making use of classical thermodynamicprinciples (64).

Statistical Distribution of End-To-End Dimensions of a Polymer Chain

Before treating the statistical properties of a network, the statistics of a singlechain must be considered, mainly to establish the relationship between the num-ber of configurations and the deformation (12,16,65–72). A polymeric chain isconstantly changing its configuration by Brownian motion. Statistical methodsand idealized models permit calculation of the average properties of such a chain.

The Freely Jointed Chain. This type of idealized chain consisting of nlinks 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 manyconfigurations, each associated with an end-to-end distance ri. In statistics, it isequivalent to consider a molecule at different times or an assembly of N moleculesat the same time. An average quantity describing the assembly is the mean-squareend-to-end distance r2 defined by

r2 = 1N

N∑i = 1

r2i (7)

where ri is the end-to-end distance of the ith chain. The vector ri is the sum of thelink vectors Ij:

ri =n∑

j = 1Ij (8)

Ij

r

Fig. 7. Ideal chain formed with n links of length l. r = End-to-end distance, Ij = linkvector.


and r2i is the scalar product of ri with itself:

r2i =n∑

j = 1I2j + 2

∑k< j

Ik·Ij (9)

If the chain is assumed to be freely jointed and volumeless, any two linkscan assume any orientation with respect to each other. Therefore the second termof equation (9) is zero and

r2i = nl2 (10)

The mean-square end-to-end distance of a freely orienting chain is deducedfrom equations (7) and (10) (73–76) as follows:

r2 = nl2 (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/π1/2)3exp( − b2r2)dxdydz (12)

where r2 = x2 + y2 + z2, and b2 = 3/2r2. This generalization is valid only for smallextensions of a relatively long chain (82). Equation (12) gives the probability thatif one extremity of the vector r is fixed at the origin of the coordinates, the other liesin the volume dx dy dz centered around the point (x, y, z). What is more interestingis the probability for a chain to have its end in a spherical shell of radius r andthickness dr centered at the origin, irrespective of direction. This is the radialdistribution

W(r)dr = (b/π1/2)3exp( − b2r2)4πr2dr (13)

which is illustrated in Figure 8 (3). The maximum occurs at r = (2nl2/3) 12 . Themean-square end-to-end distance is the second moment of the radial distributionfunction

r2 =∫ ∞

0r2W(r)dr/

∫ ∞0

W(r)dr (14)

which yields the result of equation (11)

r2 = 3/2b2 = nl2 (15)

The Gaussian distribution (eq. (12)) was obtained in the aforementionedtreatment 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


0.04

0.03

0.02

0.01

00 10 20 30 40 50

r, nm

W(r

), n

m− 1

Fig. 8. Radial distribution function W(r) of the chain displacement vectors for chainmolecules consisting of 104 freely jointed segments, each of length l = 0.25 nm; W(r) isexpressed in nm− 1 and r in nm (3). Courtesy of Cornell University Press.

take into account the significant geometric and conformational differences knownto exist among different types of polymer chains (83). The distribution obtainedfrom a more general treatment is of the more complicated form (84–86) shown asfollows:

W(r)dr = const.×exp(

−∫ r

0L− 1(r/nl)dr/l

)4πr2dr (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 + · · ·]}4πr2dr(18)

and the Gaussian distribution (eq. (13)) recovered for r � nl. A comparison of theGaussian 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 hasfixed bond angles θ . Therefore, the second term in equation (9) is no longer zero.The scalar product of two vectors Ii·Ij is l2 cos|j − i|θ . It can be shown (9,73,74) that

r2 � nl2(1 − cos θ )/(1 + cos θ) (19)


0.4

0.3

0.2

0.1

0.00 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◦ andr2 = 2nl2, twice the value for a freely jointed chain. Thus, a real chain is quitedifferent from this ideal representation. Nevertheless, any flexible real chain canbe represented by a simple model which is the statistical equivalent of a freelyjointed chain (87,88). The two conditions are that the real and freely jointed chainshave the same mean-square end-to-end distance and the same length at completeextension:

r2 = r2e = nel2e (20)

and

R= Re = nele (21)

Thus, only one model chain is equivalent to the real one, obeying the sameGaussian distribution function and composed of ne segments of length le givenby

ne = R2/r2 (22)

le = r2/R (23)


Chain with Bond-Angle Restrictions and Hindered Rotations. Theangle of rotation φ of a single bond around the axis formed by the preceding onemay be restricted by steric interferences between atoms. When n is large and theaverage value of cos φ, ie, cos φ, is not too close to unity, the following relationshipcan be established (89,90):

r2 = nl2(

1 − cos θ1 + cos θ

)(1 + cos φ1 − cos φ

)(24)

When the rotation is not hindered, ie, when cos φ = 0, equation (24) is equivalentto equation (19).

The conformations of polymer chains may be generated by a Monte Carlosimulation 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 arereported most comprehensively in the Polymer Handbook (91,92). A powerful toolto investigate chain conformations in unswollen and swollen melts and networksis small-angle neutron scattering (sans) (67,93–97).

The Equation of State for a Single Polymer Chain. The variation inthe Helmholtz free energy is the negative of the work of deformation in isothermalelongation

dA= −dW = f dr (25)

with

dA= 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 configurationsavailable to the system, is proportional to W(r) (eq. (13)). Calculation of the forcegives

f = 2kTb2r (30)


f = 2kTr/r2 (31)

Thus, f is directly proportional to the absolute temperature and to r, which meansthe chain acts as a Hookean spring with modulus 3kT/r2. 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, andforce, usually the change in force with temperature at constant length is recorded(53,98–101). It is therefore necessary to extend the thermodynamic treatment ofthe elasticity. Moreover, the force is not purely entropic, and the energetic contribu-tion carries useful information on the dependence on temperature of the averageend-to-end distance of the network chains in the unstrained state (21,102). It istherefore important to know how to deduce these quantities from a thermoelasticexperiment.

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 ofwork 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


20 40 60 800

0.1

0.2

0.3

0.4

0.5

Str

ess,

N/m

m2

60

50

40

30

20

15

10

531

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, theslopes can be negative. This phenomenon, called the thermoelastic inversion, isdue to the volume expansion occurring in any elastomer. The condition of constantlength does not correspond to constant elongation as well, since the sample’s un-strained reference length changes with temperature. The inversion is suppressedby correction of the original length (11).

The energetic and entropic contributions to the force in the intramolecularprocess of stretching the chains can be obtained in experiments where there is noother energetic contribution resulting from changes in (intermolecular) van derWaals forces. Therefore, these experiments must be performed at constant volume.A basic postulate of the elasticity of amorphous polymer networks is that the stressexhibited by a strained polymer network is assumed to be entirely intramolecularin origin. That is, intermolecular interactions play no role in deformations atconstant volume and composition.

An equation similar to equation (38) is obtained for the elastic force measuredat constant volume:

f =(

∂U∂L

)T,V

− T(

∂S∂L

)T,V

(39)


The variation in the Helmholtz free energy has the following expression:

dA= −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:

fe =(

∂U∂L

)T,V

= f − T(

∂ f∂T

)V,L

(43)

fs = −T(

∂S∂L

)T,V

= T(

∂ f∂T

)V,L

(44)

An example of thermoelastic data is given in Figure 11 (99). The change inentropy with elongation up to 350% is responsible for more than 90% of the totalstress at room temperature, whereas the contribution of internal energy is lessthan 10%. Thus, the restoring force is due almost entirely to the tendency of theextended rubber molecules to return to their unperturbed, random conformationalstates. Above 350%, crystallization appears. This is a specific feature of stereo-regular rubbers, such as natural rubber which is capable of crystallization. Dataconcerning most of the polymers studied in this manner are reviewed in References21 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, andvolume, but does not give the relationship between the molecular structure of thenetwork and elastic quantities such as the moduli. Therefore, statistical mechan-ics was introduced in the 1940s (16,86,87,107–109), and its theoretical predictionswere 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


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

Str

ess,

N/m

m2

f

fe

fs

Fig. 11. Plot of the energetic (f e) and entropic (f s) contributions to the stress at 20◦Cfor sulfur-vulcanized rubber (99). N/mm2 = 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 ofthe macroscopic strain are transmitted to the chains through these junctions atwhich the chains are multiply joined, with the internal state of the network systemspecified in terms of the positions of the cross-linkages. Deformation transformsthe arrangements of these points. The system is represented by the vectors ri,each of which connects the two ends of a chain. For a given end-to-end vector ri,the number of available configurations is directly proportional to the probabilityW(ri) or relative number of configurations. Representation of W(r) by a Gaussianfunction according to equation (13) fails as r approaches the maximum extensionrmax. Experimental evidence of deviations from the Gaussian theory have beenreported (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 havethe disadvantage of containing parameters that can be determined only by com-parisons between theory and experiments, specifically ν, the number density ofchains in the network, and n, the number of statistical links.

The tensile force is expressed by

f/A0 = (1/3)vkTn1/2[L− 1(λn− 1/2) − λ− 3/2L− 1(λ− 1/2n− 1/2)] (45)


where

L− 1(t) = 3t + (9/5)t3 + (297/175)t5 + · · ·

and A0 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, andfrom this result, an estimation of the number of monomer units in the equivalent-random link (of the Kuhn–Grün theory) of the polymeric chain (120). Anotherapproach to a non-Gaussian theory utilizes information provided by rotational iso-meric state theory on the spatial configurations of chain molecules (83), includingmost 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-endseparation 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, theprobability of an internal state of the system is the product of the probabilitiesW(ri) for each chain. The entropy is deduced by the Boltzmann equation, and thefree energy by equation (26). The three main assumptions introduced in the treat-ment of elasticity of rubber-like materials are that the intermolecular interactionsbetween chains are independent of the configurations of these chains and thus ofthe extent of deformation (125,126); the chains are Gaussian, freely jointed, andvolumeless; and the total number of configurations of an isotropic network is theproduct of the number of configurations of the individual chains.

Affine Networks. Diffusion of the junctions about their mean positionsmay be severely restricted by neighboring chains sharing the same region ofspace. The extreme case is the affine network where fluctuations are completelysuppressed, and the instantaneous distribution of chain vectors is affine in thestrain. The elastic free energy of deformation is then given by

�Ael(aff) = (1/2)vkT(I1 − 3) − (v − ξ )kT ln(V/V0) (46)

where I1 is the first invariant of the tensor of deformation

I1 = λ2x + λ2y + λ2z (47)

The quantities λx, λy, and λz are the principal extension ratios, which specify thestrain relative to an isotropic state of reference having volume V0; V is the volumeof the deformed specimen, and ν is the number of linear chains whose ends arejoined to multifunctional junctions of any functionality φ > 2. (The functionalityof 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 networkto an acyclic structure or tree (127). The cycle rank is the difference between thenumber of effective chains ν and effective junctions µ of functionality φ > 2:

ξ = ν − µ (48)


Isotropic state Deformed state

V

L

V0

L0

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 insample volume, as illustrated in Figure 12. The strain along the direction ofstretching is given by λx as

λ = λx = L/L0 (50)

Since the volume changes from V0 to V,

λxλyλz = V/V0 (51)

Because there is symmetry about the x axis, λy = λz. Combining equations (50)and (51) leads to

λy = λz = (V/λV0)1/2 (52)

The force f is the derivative of the free energy with respect to length; specifically,

f =(

∂�A∂L

)T,V

= (1/L0)(

∂�A∂λ

)T,V

(53)

Using equation (46),

f = (vkT/L0)[λ − V/(V0λ2)] (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 Lvi of the unstretched sample when itsvolume is fixed at the same volume V as occurs in the stretched state:

α = L/Liv = λL0/Liv (55)

and

L0 = Liv(V/V0)− 1/3 (56)


Equation (54) is then transformed to

f = (vkT/Liv)(V/V0)2/3(α − 1/α2). (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

fe = f − T(

∂ f∂T

)V,L

= − f T(

∂ln( f/T)∂T

)V,L

(58)

or

fef

= −T(

∂ln( f/T)∂T

)V,L

(59)

The derivative of f at constant volume and length, thus at constant α, can beobtained from equation (57), with the result

fe/ f = (2/3)T(dln V0/dT) = T(dln r20

/dT

)(60)

since V0 is the volume of the isotropic state so defined that the mean square ofthe magnitude of the chain vectors equals r20, the value for the free unperturbedchains.

The intramolecular energy changes arising from transitions of chains fromone spatial configuration to another are, by equation (60), directly related tothe temperature coefficient of the unperturbed dimensions. It is interesting tocompare the thermoelastic results for polyethylene (128), f e/f = −0.45, andpoly(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 CH2 groups (83). Since thisconformation has the highest possible spatial extension, stretching a polyethylenechain requires switching some of the gauche states, which are, of course, present inthe randomly coiled form, to the alternative trans states (106,130). These changesdecrease 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 alsoexplains the decrease in unperturbed dimensions upon increase in temperature.The additional energy causes an increase in the number of the higher energygauche 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 formis 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 thetrans conformation places CH3 side groups at distances of separation where theyare strongly attractive (83,129,131). Because of the inequality of the Si O Siand O Si O bond angles, however, this conformation is of very low spatial ex-tension. Stretching a poly(dimethylsiloxane) chain therefore requires an increasein the number of gauche states. Since these are of higher energy, this explainsthe fact that deviations from ideality for these networks are found to be positive(106,129,130).


Isotropic drystate 1 Isotropic swollenstate 2

Deformed swollenstate 3

V

L

V

L0

V0

Li�

Fig. 13. Simple extension of a swollen network.

Simple Extension of Swollen Networks. The force required to deform anelastomeric sample, from state 2 to state 3 in Figure 13, is given by equation (57),in which α = L/Liv is the extension ratio for the isotropic swollen state relative tothe deformed swollen state. The ratio V0/V is the volume fraction v2 of polymerin the swollen system, and A0 is the cross-sectional area of the dry sample, whichis generally measured before the experiment. The area of the swollen sample isgiven by

Aiv = A0(V/V0)2/3 (61)

Equation (56), of course, still holds, and L0A0 = V0. In an experiment, the force fis measured as a function of elongation. Under these conditions,

f/A0 = (νkT/L0 A0)v− 1/32 (α − 1/α2) (62)

Hence, the quantity [ f ∗]≡v1/32 /A0(α − 1/α2) should be a constant, independent ofthe degree of swelling, and be equal to vkT/V0. As shown later [f∗] is commonlyplotted versus 1/α to determine deviations from theory (112,132,133).

Simple Extension of Networks Cross-linked in the Diluted State. Thepolymer is dissolved, cross-linked, and dried. The stress–strain measurementsare carried out on the dry network, as illustrated in Figure 14. Equation (57)also holds for this case. If Aiv is the cross-sectional area of the undeformed dryspecimen, the force per unit undeformed area is

f/Aiv = (νkT/V)(V/V0)2/3(α − 1/α2) (63)

Reference statefor cross-linking

process

Dry, undeformedstate

Dry, deformedstate

L0

V0 V V

LLi�

Fig. 14. Simple extension of an unswollen network cross-linked in the diluted state(V0 > V).


which is (V/V0)2/3 times the tension of a network cross-linked in the dry state(111,134).

The Affine Shear Modulus. For V = V0, equation (54) becomes

f/A0 = (νkT/V0)(λ − 1/λ2) (64)

Making use of equation (2) gives λ = 1 + �. For small values of �, � � 1, equation(64) may be written as

f/A0 = 3(νkT/V0)� (65)

The tensile modulus Eaff is three times the shear modulus Gaff, since Poisson’sratio for elastomers is close to 0.5. Specifically,

Eaff = 3vkT/V0 = 3Gaff (66)

and hence,

Gaff = vkT/V0 (67)

If v/V0 is the molar number density of chains [equal to ρ/Mc for a perfect network,where ρ is the elastomer density and Mc is the molecular weight between cross-link points (in g/mol)], then

Gaff = vRT/V0 (68)

where R is the gas constant. In the remaining material, v is a molar quantityunless it is followed by the Boltzmann constant k.

For an affine perfect network,

Gaff = ρRT/Mc (69)

As a numerical example, the affine shear modulus of a perfect poly(dimethylsiloxane) tetrafunctional network of density ρ = 0.97 g/cm3 and Mc =11,300 g/mol is Gaff = 0.212 × 106 N/m2. However, imperfections such as chainends 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-linkingreaction, it is assumed that vm chains of length Mn are present in the melt, withvm = ρ/Mn. Each chain has two ends, thus there are 2ρ/Mn chain ends. After thecross-linking process, there are v0 chains of length Mc, v0 = ρ/Mc, along with the2ρ/Mn chain ends. Therefore the number of chains that are elastically effective is

v = ρ/Mc − 2ρ/Mn = (ρ/Mc)(1 − 2Mc/Mn) (70)

This correction is small for Mc � Mn, which is frequently the case.


Phantom Networks. In this idealized model, chains may move freelythrough 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 �r2 of the end-to-end distance r are relatedto the mean square of the end-to-end separation of the free unperturbed chainsr2(102,136,137) by

�r2/r20 = 2/φ (71)

The fluctuation range (�r2 = r20/2 for a tetrafunctional network) is generally quitelarge and of considerable importance. The instantaneous distribution of chainvectors r is not affine in the strain because it is the convolution of the distributionof the affine mean vector r with the distribution of the fluctuations �r, which areindependent of the strain. The elastic free energy of such a network is

�Ael(ph) = (1/2)ξkT(I1 − 3) = (1/2)(ν − µ)kT(I1 − 3) (72)

In simple extension, the equivalent of equation (62) is now

f/A0 = (ν − µ)kTV − 10 v− 1/32 (α − 1/α2) (73)

and the shear modulus is given by

Gph = (v − µ)RT/V0 (74)

For a perfect network of functionality φ, equation (49) holds, and therefore,

Gph = (1 − 2/φ)vRT/V0 (75)

Gph = (1 − 2/φ)ρRT/Mc (76)

The relationship between affine and phantom moduli is then

Gph = (1 − 2/φ)Gaff (77)

For example, Gaff is twice Gph for a perfect tetrafunctional network.Comparisons With Experimental Results. Stress–strain measure-

ments in uniaxial extension can be compared with the prediction of an affineGaussian 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 isobtained using Gaff = 0.39 MN/m2 (56.6 psi) (7); deviations occur when λ > 1.5.The experimental curve may nevertheless be well represented by adjusting theparameters 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 ofrubber-like elasticity.


Extension ratio

Tens

ile fo

rce

per

unit

unst

rain

ed a

rea,

N/m

m2

Theoretical

1 2 3 4 5 6 7 80.0

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/mm2 to psi, multiplyby 145. Courtesy of The Royal Society of Chemistry.

Phenomenological Theory. Continuum mechanics is used to describemathematically the stress–strain relations of elastomers over a wide range instrain. This phenomenological treatment is not based on molecular concepts, buton representations of observed behavior (132,139–143). The main goal is to find anexpression for the elastic energy W stored in the system (assumed to be perfectlyelastic, isotropic in its undeformed state, and incompressible), analogous to thefree energy of the statistical treatment. The condition of isotropy in the unstrainedstate requires that W be symmetrical with respect to the three principal extensionratios λx, λy, λz. A rotation of the material through 180◦, ie, a change of sign oftwo of the λi (i = x, y, z), does not alter W (144). The three simplest even-poweredfunctions satisfying these conditions are the strain invariants I1, I2, and I3 definedas

I1 = λ2x + λ2y + λ2z

I2 = λ2xλ2y + λ2yλ2z + λ2zλ2x

I3 = λ2xλ2yλ2z

(78)

The most general form of the strain–energy function, which vanishes at zerostrain, for an isotropic material is the power series


W =∞∑

i, j,k= 0Cijk(I1 − 3)i(I2 − 3) j(I3 − 1)k (79)

In equation (79), the experimental results obtained at sufficiently small strainsare well represented with two nonzero terms, C100 and C010:

W = C100(I1 − 3) + C010(I2 − 3) (80)In uniaxial extension, the so-called Mooney–Rivlin equation is obtained (17):

f υ1/32 /A0 = 2(C1 + C2/α)(α − 1/α2) (81)A convenient and standard way to treat stress–strain data is to plot the reducedforce

[ f ∗]≡ f ∗υ1/32 /(α − 1/α2) (82)versus 1/α, where f∗ is the nominal stress f /A0 and υ2 the volume fraction ofpolymer 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/V0 and [f∗] =(v − µ)RT/V0, respectively. It has been reported that C2 decreases as υ2 decreases,whereas C1 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 1.00.125

0.150

0.175

0.200

0.225

1/�

[f∗ ]

, N/m

m2

�2 � 1.00

Fig. 16. Plot of [f∗] versus 1/α for a natural rubber vulcanizate swollen in benzene todemonstrate the influence of v2 on C2 (114). To convert N/mm2 to psi, multiply by 145.Courtesy of The Royal Society of Chemistry.


of experimental results on swollen networks (69,114,145,146), the following formof the strain energy per unit volume of dry network is to be preferred:

W/V0 = C1(I1 − 3)+C2(I2 I − 1+m/23 − 3) (83)

The reduced force can be deduced (145) as [ f ∗] = 2C1+2C2v(4/3) − m2 α − 1. The pa-rameter C1 of the swollen network is equal to C1 of the dry network, whereasC2 (swollen) is equal to C2 (dry) v

(4/3) − m2 . Experimentally, m was found to be 0 or

12 (145). The constant C1 increases with the cross-linking density of the rubber(114,147), and 2C1 + 2C2 is approximately the shear modulus at small defor-mation (148). Although there have been many attempts (149–155), a molecularexplanation of C1 and C2 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 areextreme limits. Stress–strain measurements in uniaxial extension have revealedthat the behavior of real networks is between these limits. A theoretical attempthas been made to account for this dependence of [f∗] on 1/α in terms of a gradualtransition between the affine and phantom deformations (102,157), and a molec-ular theory has been formulated (102,158–161). In this model, the restrictions ofjunction fluctuations due to neighboring chains are represented by domains of con-straints. At small deformations, the stress is enhanced relative to that exhibitedby a phantom network. At large strains, or high dilation, the effects of restrictionson fluctuations vanish and the relationship of stress to strain converges to that fora phantom network. In a later theoretical refinement, the behavior of the networkis taken to depend on two parameters. The most important is κ which measuresthe severity of entanglement constraints relative to those imposed by the phantomnetwork. Another parameter ζ takes into account the nonaffine transformation ofthe domains of constraints with strain. Topological and mathematical treatmentleads to the expression

[ f ∗] = fph(1 + fc/ fph) (84)

where f ph is the usual phantom modulus and f c/f ph is the ratio of the force due toentanglement constraints to that for the phantom network. A specific expressionfor f c/f ph in uniaxial extension is

fc/ fph = (µ/ξ )[αK

(α21

)− α − 2K(α22)]/(α − 1/α2) (85)

where α1 = αv− 1/32 ,α2 = α − 1/2v− 1/32 , and µ/ξ is the ratio of the number of effectivejunctions to the cycle rank. For a perfect network, µ/ξ=2/(φ − 2). The functionK(x2) is given by

K(x2) = B[.B (B+ 1)− 1 + g( .g B+ g

.B )(gB+ 1)− 1] (86)


with

g = x2[1/κ + ζ (x − 1)].g = 1/κ − ζ (1 − 3x/2)

B= (x − 1)(1 + x − ζ x2)/(1 + g)2.B = B[(2x(x − 1))− 1 − 2 .g (1 + g)− 1 + (1 − 2ζ x)[2x(1 + x − ζ x2)]− 1]

It is interesting to note that junction fluctuations increase in the direction ofstretching 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 morenearly affine. The curve of [f∗] versus 1/α is sigmoidal. The parameters κ and ζ ofpoly(dimethylsiloxane) networks are determined in Figure 17 (155); the interceptof the sigmoidal curves is the phantom modulus. This Flory–Erman theory hasbeen compared successfully with such experiments in elongation and compression(155,162,162–166). It has not yet been extended to take account of limited chainextensibility or strain-induced crystallization (167).

0.20

0.12

0.08

0.04

0 0.2 0.4 0.6 0.8 1

Mn � 4000 � � 0.1 � � 0

Mn � 9500 � � 3.4 � � 0 Mn � 18500 � � 30 � � 0.01

Mn � 25600 � � 24.5 � � 0

Mn � 32900 � � 16.2 � � 0

� � 31.5 � � 0.01

� � 27.5 � � 0� � 23 � � 0

� � 21.5 � � 0

1/�

[f*]

, N/m

m2

Fig. 17. Determination of the parameters κ and ζ of the Flory–Erman theory for perfecttrifunctional poly(dimethylsiloxane) networks (155). To convert N/mm2 to psi, multiply by145. Courtesy of Springer-Verlag.


Predictions for the Parameters κ and ζ . The parameter ζ is not far fromzero, 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 κis defined as the ratio of the mean-square junction fluctuations in the equivalentphantom network, ie, in the absence of constraints, to the mean-square junctionfluctuations about the centers of domains of entanglement constraints (in theabsence of the network) in the isotropic state. Thus in a phantom network, theabsence of constraints leads to κ = 0. In an affine one, the complete suppressionof 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 anincreasing number of junctions in a volume pervaded by a chain leads to strongerconstraints on these junctions, κ was taken to be

κ = I(r20)3/2(µ/V0) (87)where I is a constant of proportionality, (r20)

3/2 is assumed to be proportional tothe volume occupied by a chain, and µ/V0 is the number of junctions per unitvolume. The mean-square unperturbed dimension r20 can be taken proportional toMc (95), the molecular weight between cross-links, and µ/V0 to M− 1c for a perfecttetrafunctional network; therefore κ � M1/2c (93,155,165).

Swelling Equilibrium. The isotropic swelling of a cross-linked elastomerby a liquid has two important opposing effects: the increase in mixing entropy ofthe system because of the presence of the small molecules, and the decrease in con-figurational entropy of the network chains by dilation. Therefore, an equilibriumdegree 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 assumedto be separable into the free energy of mixing, �Am, and the elastic free energy�Ael,

�A= �Am + �Ael (88)

although questions have been raised with regard to this separability (171,172).The contribution �Am has been calculated with the help of a lattice model (173,174). The other contribution �Ael is given in the later Flory–Erman theory (161)by

�Ael = �Ael(ph) + �Ac (89)

where �Ael(ph) is given by equation (72); �Ac is the additional term which ac-counts for the constraints and is given by

�Ac = µkT/2∑

i=x,y,z{[1 + g(λi)]B(λi) − ln[(B(λi) + 1)(g(λi)B(λi) + 1)]}. (90)

The λi are the principal extension ratios, and g and B have been defined in equation(86).


The chemical potential of the solvent in the swollen network is

µ1 − µ01 = N(

∂�Am∂n1

)T,p

+ N(

∂�Ael∂λ

)T,p

(∂λ

∂n1

)T,p

(91)

where n1 is the number of moles of solvent, and the isotropic extension ratio λ is

λ = λx = λy = λz = [(n1V1 + V0)/V0]1/3 = v− 1/32 (92)where V1 is the molar volume of the solvent and V0 the volume of the dry network.At swelling equilibrium, µ1 = µ01. Hence, the standard expression for �Am (3) leadsto

ln(1 − υ2m) + υ2m + χυ22m = −(

∂�Ael∂λ

)T,p

(∂λ

∂n1

)T,p

(RT)− 1 (93)

where v2m is the volume fraction of polymer at swelling equilibrium. The inter-action parameter χ1 may be determined, for example, by vapor pressure or byosmometry measurements (145,175–178). It depends on the concentration of poly-mer in the polymer–solvent system (175). Using the Flory–Erman expression forthe 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/V0)V1v1/32m[1 + (µ/ξ )K(v− 2/32m )] (94)

The molecular weight Mc between cross-links of a perfect network is then obtainedby combination of equations (48),(49), and (94), with v/V0 = ρ/Mc:

Mcr = (2/φ − 1)ρV1υ1/32m[1 + (ϕ/2 − 1)− 1K(V1υ − 2/32m )]/H (95)

where the subscript r is employed here for real networks.For a network deforming affinely, κ = ∞, K(λ2) = 1 − λ− 2, and

Mca = − ρV1υ1/32m(1 − 2υ2/32m /φ

)]/H (96)

For a phantom network, κ = 0, K(λ2) = 0, and

Mcp = (2/φ − 1)ρV1υ1/32m/

H (97)

Determination of the Degree of Cross-linking by Stress–StrainMeasurements and by Swelling Equilibrium

Characterization of network structures is often the main objective of theoreticaland experimental works in the field of rubber elasticity (179–183). Simple experi-ments such as swelling equilibrium have been extensively used. However, most ofthe experimental swelling results on cross-linked polymers have been interpretedusing the Flory–Rehner expression for an affinely deforming network (6,184–186).


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 phantommodulus thus determined leads to ν and Mc through equations (75) and (76) (189).Swelling equilibrium data are similarly analyzed through equation (94), with theparameter κ given by equation (87) (189).

If Mc and κ have been determined previously from stress–strain measure-ments, then the interaction parameter χ1 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 thatlinear polymers of high molecular weight exhibit a storage modulus G′(ω), whichremains relatively constant over a wide range of frequencies ω (205). This plateaumodulus G0N is independent of chain length for long chains and is insensitive totemperature. Since it varies with the volume fraction of polymer in concentratedsolutions, it could be due to pair-wise interactions between chains (20), and a uni-versal law has been proposed for the dependence of G0N on the chemical structureof the polymer (206). During the cross-linking process, some of such interactionsor entanglements could be trapped in the network and act as physical junctions.This conclusion has been tested by irradiation cross-linking of already deformednetworks (207–209), and then measuring the dimensional changes.

In the simplest phenomenological approach for rubber-like elasticity oftrapped entanglements at small deformation (210,211), the shear modulus istaken to be the sum of two terms:

G = Gc + GeTe (98)where Gc is the contribution of the chemical cross-links. Taking into account therestrictions of junction fluctuations, as in the Flory theory, leads to

Gc = (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 andζ (155,212). The additional contribution GeTe is said to arise from permanentlytrapped (interchain) entanglements in the network. The modulus Ge is thought tohave a value close to G0N, and Te, 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 affinenetworks (185,213). The possibility of a contribution at equilibrium caused bytrapped entanglements has been tested with model networks, ie, those preparedin such a way that the number and functionality of the cross-links are known. A


typical, highly specific reaction used for this purpose is the end-linking of function-ally terminated polymer chains. Specific examples would be hydroxyl-terminatedor vinyl-terminated chains of poly(dimethylsiloxane), [Si(CH3)2O]x , end-linkedwith an organic silicate or silane (214,215). A considerable body of published datahas, in fact, been interpreted as proving the existence of a trapped-entanglementcontribution 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 Te, and effective functionality φe, were calculated bythe branching theory after measurement of the network sol fraction. The mainassumption of this probabilistic method is that the sol fraction, after subtractionof the amount of nonreactive species as determined by gel permeation chromatog-raphy analysis (218), is composed only of primary reactive chains (223). On thecontrary, 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 ofnonreacted reagents, has to be significantly increased. As a result, many modelnetworks can be considered as nearly perfect. This casts some doubt on the resultsinterpreted 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-inducedcross-linking. Again some doubt exists on the method used to characterize thenetwork structure.

Molecular models treating entanglements as interstrand links that are freeto slip along the strand contours have been developed (226–228) and tube mod-els have been investigated (229,230). These approaches have been reviewed inReference 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 owingto the diffusion of segments and associated time-dependent changes of configura-tions. Such trapped entanglements as previously described are possibly of minorimportance in equilibrium stress measurements.

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J. P. QUESLELManufacture Michelin, CERL – GPAJ. E. MARKUniversity of Cincinnati

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