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/m2 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.

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f

A

L0

L

f (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 (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 pressure, uniaxial extension (or compression), and simple shear. They are used to determine the bulk modulus K, Young’s modulus E, and shear modulus G, respectively (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/V0 )

(1)

where V = V 0 − V; K is the reciprocal of the compressibility and can be determined 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 L0 is stretched uniaxially 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 relative deformation, linear dilation, or extension) and the elongation or extension ratio λ are related by = L/L0 = λ − 1

(2)

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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 per unit length is called Poisson’s ratio (47): νp = (w/w0 )/(L/L0 )

(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 illustrated 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

5

f/A, N/mm2

4

3

2

1 0

1

2

3

5

4

6

7

8

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

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

Rubber

Metal

Glass

500 500 10c

Large, plastic 2 5 × 104

3 3 5 × 103

Breaking extension, % Elastic limit, % Tensile strength, MN/m2b a Ref.

14. convert MN/m2 (MPa) to psi, multiply by 145. c >103 if based on area at break. b To

Force gauge

Recorder

N2

Constant T

Cathetometer

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 atmosphere 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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Vol. 2 y

y

f A

A U

H

H

x

f

(a)

x (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 tensiontesting machine. Typical values of moduli and Poisson’s ratios for metals, ceramics, and polymers 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 interrelated (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 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 Polymersa Speciﬁc properties Material

221

Metals Cast iron Steel (mild) Aluminum Copper Lead Mercury Inorganics Quartz Vitreous silica Glass Granite Whiskers Alumina Carborundum Graphite Polymers Polystyrene Poly(methyl Methacrylate) Nylon-6,6 Polyethylene (low density) Ebonite Rubber Liquids Water Organic liquids a Ref.

νp

Young’s modulus E, Shear modulus G, Bulk modulus K, Density ρ, E/ρ, 106 G/ρ, 106 K/ρ, 106 109 N/m2b 109 N/m2b 109 N/m2b g/cm3 m2 /s2 m2 /s2 m2 /s2

0.27 0.28 0.33 0.35 0.43 0.5

90 220 70 120 15 0

35 86 26 44.5 5.3 0

66 166 70 134 36 25

7.5 7.8 2.7 8.9 11.0 13.55

12.0 28.0 26.0 13.5 13.6 0.0

4.7 11.0 9.6 4.5 4.8 0.0

8.8 21.0 26.0 15.0 33.0 1.85

0.07 0.14 0.23 0.30

100 70 60 30

47 30.5 24.5 11.5

39 32.5 37 25

2.65 2.20 2.5 2.7

38.0 32.0 24.0 11.1

17.8 14.0 9.8 4.3

14.7 14.7 14.9 9.2

2000 1000 1000

1000 500 500

667 333 333

3.96 3.15 2.25

510 315 440

253 160 220

225 106 150

0.33 0.33 0.33 0.45 0.39 0.49

3.2 4.15 2.35 1.0 2.7 0.002

1.2 1.55 0.85 0.35 0.97 0.0007

3.0 4.1 3.3 3.33 4.1 0.033

1.05 1.17 1.08 0.91 1.15 0.91

3.05 3.55 2.21 1.1 2.35 0.002

1.15 1.33 0.79 0.385 0.86 0.00075

2.85 3.5 2.3 3.7 3.6 0.04

0.5 0.5

0.0 0.0

0.0 0.0

2.0 1.33

1.0 0.9

0.0 0.0

0.0 0.0

2.0 1.5

39. Courtesy of Elsevier. = Pa. To convert Pa to psi, multiply by 0.000145.

b N/m2

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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 active 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 deformation 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 intermolecular 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

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(b)

(a)

(c)

223

(d)

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 Em 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 interatomic distances, and therefore the forces required are considerably lower. The number of available conﬁgurations for a network chain is reduced in the deformation 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 number of available conﬁgurations is drastically reduced in this region, and chains reach the limits of their extensibility. For polymers with regular structures, crystallization 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 direction 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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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 number 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 ri . 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 r2 deﬁned by N 1 r2 N i=1 i

r2 =

(7)

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

n

Ij

(8)

j =1

Ij r

Fig. 7. Ideal chain formed with n links of length l. r = End-to-end distance, I j = link vector.

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225

and r2 i is the scalar product of ri with itself: ri2 =

n

I j2 + 2

j =1

Ik·I j

(9)

k< j

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 ri2 = nl2

(10)

The mean-square end-to-end distance of a freely orienting chain is deduced from equations (7) and (10) (73–76) as follows: r2 = nl2

(11)

Another interesting quantity is the probability that a chain has a given endto-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( − b2r 2 )dxdydz

(12)

where r 2 = x 2 + y2 + z2 , and b2 = 3/2r2 . 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( − b2r 2 )4πr 2 dr

(13) 1

which is illustrated in Figure 8 (3). The maximum occurs at r = (2nl2 /3) 2 . The mean-square end-to-end distance is the second moment of the radial distribution function ∞ ∞ r2 = r 2 W(r)dr/ W(r)dr (14) 0

0

which yields the result of equation (11) r2 = 3/2b2 = nl2

(15)

The Gaussian distribution (eq. (12)) was obtained in the aforementioned treatment with the assumption that chains are far from their full extension. Moreover, 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

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0.04

W(r), nm−1

0.03

0.02

0.01

0

0

10

30

20

40

50

r, nm

Fig. 8. Radial distribution function W(r) of the chain displacement vectors for chain molecules consisting of 104 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: r −1 W(r)dr = const.×exp − L (r/nl)dr/l 4πr 2 dr

(16)

0

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 aforementioned 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 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)

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227

0.4

W(r)

0.3

0.2

A

B 0.1

0.0

0

1

2

3

4

5

6

Fig. 9. Distribution function W(r) for a six-link random chain, with length l = 1 in arbitrary units: A, Gaussian limit; B, inverse Langevin function (7). Courtesy of Oxford University Press.

In the particular case of a tetrahedrally bonded chain, θ = 109.5◦ and r2 = 2nl2 , 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: r2 = r2e = nele2

(20)

R= Re = nele

(21)

and

Thus, only one model chain is equivalent to the real one, obeying the same Gaussian distribution function and composed of ne segments of length le given by ne = R2 /r2

(22)

le = r2 /R

(23)

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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): r2

= 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 excludedvolume interactions, have been measured in various solvents by light scattering, xray 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 dA= −dW = f dr

(25)

dA= dU − TdS

(26)

with

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 = 2kTb2r

(30)

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ELASTICITY, RUBBER-LIKE f = 2kTr/r2

229 (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/r2 . The use of a nonGaussian 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 contribution 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 accompanying 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

∂H f= ∂L

∂S −T ∂ L T, p

(38) T, p

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

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0.5 60 50

0.4

Stress, N/mm2

40 0.3

30

20

0.2

15 10

0.1

5 3 1 0

20

60

40

80

Temperature, °C

Fig. 10. Stress–temperature curves for sulfur-vulcanized natural rubber (99,103). Courtesy 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 unstrained 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:

∂U f= ∂L

∂S −T ∂L T,V

(39) T,V

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The variation in the Helmholtz free energy has the following expression: dA= −SdT − pdV + f dL

(40)

The second derivative obtained by differentiating ( ∂∂ LA )V,T with respect to T at constant V and L is identical with that obtained by differentiating ( ∂∂ TA )V,L with respect to L at constant V and T: ∂f ∂S = − ∂ T V,L ∂ L V,T

(41)

Thus, equation (39) can be written as f=

∂U ∂L

+T T,V

∂f ∂T

(42) V,L

The energetic and entropic components of the elastic force, f e and f s , respectively, are obtained from thermoelastic experiments using the following equations:

∂U fe = ∂L

∂f = f −T ∂ T T,V

∂S fs = −T ∂L

∂f =T ∂ T T,V

(43) V,L

(44) V,L

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 stereoregular 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 rearranging its conﬁgurations by Brownian motion. Classical thermodynamics explains 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 mechanics 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 models based on afﬁne and phantom networks were studied. The cross-linking points

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2.4 2.2 2.0 1.8

Stress, N/mm2

1.6 1.4

f 1.2

fs

1.0 0.8 0.6 0.4 0.2

fe

0 −0.2

0

50

100

150

200

250

300

350

400

Elongation, %

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/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 undergo 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 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 probability W(ri ) or relative number of conﬁgurations. Representation of W(r) by a Gaussian function according to equation (13) fails as r approaches the maximum extension rmax . 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 comparisons 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/A0 = (1/3)vkTn1/2 L − 1 (λn − 1/2 ) − λ − 3/2 L − 1 (λ − 1/2 n − 1/2 )

(45)

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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 random 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¨ theory) of the polymeric chain (120). Another random link (of the Kuhn–Grun approach to a non-Gaussian theory utilizes information provided by rotational isomeric 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 simulate 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 account 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(ri ) 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 treatment 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 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 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)

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L0

L

V0

V

Isotropic state

Deformed state

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 λx as λ = λx = L/L0

(50)

Since the volume changes from V 0 to V, λx λ y λz = V/V0

(51)

Because there is symmetry about the x axis, λy = λz . Combining equations (50) and (51) leads to λ y = λz = (V/λV0 )1/2

(52)

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

∂A f= ∂L

∂A = (1/L0 ) ∂λ T,V

(53) T,V

Using equation (46), f = (vkT/L0 ) λ − V/ V0 λ2

(54)

It is also possible to deduce the expression for the force as a function of the extension α measured relative to the length Lv i of the unstretched sample when its volume is ﬁxed at the same volume V as occurs in the stretched state: α = L/Liv = λL0 /Liv

(55)

L0 = Liv (V/V0 ) − 1/3

(56)

and

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Equation (54) is then transformed to f = vkT/Liv (V/V0 )2/3 (α − 1/α 2 ).

(57)

Energetic Contributions. Equation (57) can be used for the molecular interpretation of the ratio f e /f . Thus, equation (43) can be rewritten as fe = f − T

∂f ∂T

= −fT V,L

∂ln( f/T) ∂T

(58) V,L

or fe ∂ln( f/T) = −T f ∂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 fe / f = (2/3)T(dln V0 /dT) = T dln r20 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 r20 , 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) conformation of the polyethylene chain is the all-trans form, since gauche states at rotational angles of ±120◦ cause steric repulsions between CH2 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 ideality 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 behavior 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 unusually large Si O Si bond angles reduce steric repulsion in general, and the trans conformation places CH3 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 extension. 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).

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L0

Li

L

V0

V

V

Isotropic swollen state 2

Deformed swollen state 3

Isotropic dry state 1

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/Liv is the extension ratio for the isotropic swollen state relative to the deformed swollen state. The ratio V 0 /V is the volume fraction v2 of polymer in the swollen system, and A0 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 Aiv = A0 (V/V0 )2/3

(61)

Equation (56), of course, still holds, and L0 A0 = V 0 . In an experiment, the force f is measured as a function of elongation. Under these conditions, − 1/3

f/A0 = (νkT/L0 A0 )v2

(α − 1/α 2 )

(62)

1/3

Hence, the quantity [ f ∗]≡v2 /A0 (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 Aiv is the cross-sectional area of the undeformed dry specimen, the force per unit undeformed area is f/Aiv = (νkT/V)(V/V0 )2/3 (α − 1/α 2 )

L0

V0 Reference state for cross-linking process

(63)

Li

L

V

V

Dry, undeformed state

Dry, deformed state

Fig. 14. Simple extension of an unswollen network cross-linked in the diluted state (V 0 > V).

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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/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’s ratio for elastomers is close to 0.5. Speciﬁcally, Eaff = 3vkT/V0 = 3Gaff

(66)

Gaff = vkT/V0

(67)

and hence,

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 crosslink points (in g/mol)], then Gaff = vRT/V0

(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, Gaff = ρ RT/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 M c = 11,300 g/mol is Gaff = 0.212 × 106 N/m2 . However, imperfections such as chain ends exist in typical networks, as illustrated in Figure 6. The following correction can be made to account for this circumstance (16). Before the cross-linking reaction, it is assumed that vm chains of length M n are present in the melt, with vm = ρ/M n . Each chain has two ends, thus there are 2ρ/M n chain ends. After the cross-linking process, there are v0 chains of length M c , v0 = ρ/M c , along with the 2ρ/M n chain ends. Therefore the number of chains that are elastically effective is v = ρ/Mc − 2ρ/M n = (ρ/Mc )(1 − 2Mc /M n ) This correction is small for Mc M n , which is frequently the case.

(70)

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Phantom Networks. In this idealized model, chains may move freely through one another (135). Junctions ﬂuctuate around their mean positions because of Brownian motion, and these ﬂuctuations are independent of deformation. The mean square ﬂuctuations r2 of the end-to-end distance r are related to the mean square of the end-to-end separation of the free unperturbed chains r2 (102,136,137) by r2 /r2 0 = 2/φ

(71)

The ﬂuctuation range (r2 = r20 /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 Ael (ph) = (1/2)ξ kT(I1 − 3) = (1/2)(ν − µ)kT(I1 − 3)

(72)

In simple extension, the equivalent of equation (62) is now − 1/3

f/A0 = (ν − µ)kTV0− 1 v2

(α − 1/α 2 )

(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)ρ RT/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 measurements in uniaxial extension can be compared with the prediction of an afﬁne Gaussian network (eq. (64)), as illustrated in Figure 15 (113). The Gaussian relationship in the afﬁne limit is valid only at small deformations. The best ﬁt is obtained 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 the parameters of the non-Gaussian stress–strain relationship (eq. (45)) (84,117,138). These disagreements between experiments and the simple predictions of statistical mechanics have led some workers to develop a phenomenological theory of rubber-like elasticity.

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Tensile force per unit unstrained area, N/mm2

7.0

6.0

5.0

4.0

3.0

2.0 Theoretical 1.0

0.0 1

2

3

5 4 Extension ratio

6

7

8

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, 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 λx , λy , λz . A rotation of the material through 180◦ , ie, a change of sign of two of the λi (i = x, y, z), does not alter W (144). The three simplest even-powered functions satisfying these conditions are the strain invariants I1 , I2 , and I3 deﬁned as I1 = λ2x + λ2y + λ2z I2 = λ2x λ2y + λ2y λ2z + λ2z λ2x

(78)

I3 = λ2x λ2y λ2z The most general form of the strain–energy function, which vanishes at zero strain, for an isotropic material is the power series

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∞

Vol. 2

Ci jk(I1 − 3)i (I2 − 3) j (I3 − 1)k

(79)

i, j,k = 0

In equation (79), the experimental results obtained at sufﬁciently small strains are 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): 1/3

f υ2 /A0 = 2(C1 + C2 /α)(α − 1/α 2 )

(81)

A convenient and standard way to treat stress–strain data is to plot the reduced force [ f ∗]≡ f ∗ υ2 /(α − 1/α 2 ) 1/3

(82)

versus 1/α, where f ∗ is the nominal stress f /A0 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 horizontal lines (eqs. (62) and (73)) from equations [f ∗] = vRT/V 0 and [f ∗] = (v − µ)RT/V 0 , respectively. It has been reported that C2 decreases as υ 2 decreases, whereas C1 is approximately constant, as illustrated in Figure 16 (112). In view 0.225 2 1.00

[ f ∗], N/mm2

0.200

0.74 0.175 0.55

0.40

0.150

0.29 0.125 0.4

0.20 0.8

0.6

1.0

1/

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

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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: − 1+m/2

W/V0 = C1 (I1 − 3)+C2 (I2 I3

− 3)

(83) (4/3) − m

α − 1 . The paThe reduced force can be deduced (145) as [ f ∗] = 2C1 +2C2 v2 rameter C1 of the swollen network is equal to C1 of the dry network, whereas (4/3) − m . Experimentally, m was found to be 0 or C2 (swollen) is equal to C2 (dry) v2 1 (145). The constant C increases with the cross-linking density of the rubber 1 2 (114,147), and 2C1 + 2C2 is approximately the shear modulus at small deformation (148). Although there have been many attempts (149–155), a molecular explanation 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 determination 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 molecular theory has been formulated (102,158–161). In this model, the restrictions of junction ﬂuctuations due to neighboring chains are represented by domains of constraints. 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 ∗] = 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 to entanglement constraints to that for the phantom network. A speciﬁc expression for f c /f ph in uniaxial extension is fc / fph = (µ/ξ ) α K α12 − α − 2 K α22 /(α − 1/α 2 ) − 1/3

(85)

− 1/3

,α2 = α − 1/2 v2 , and µ/ξ is the ratio of the number of effective where α1 = αv2 junctions to the cycle rank. For a perfect network, µ/ξ =2/(φ − 2). The function K(x2 ) is given by . . . K(x 2 ) = B[ B (B + 1) − 1 + g( g B + g B )(gB + 1) − 1 ]

(86)

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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 modulus decreases in the direction of stretching, but increases in the normal direction 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).

Mn 4000 0.1 0 0.20

0.12

Mn 18500 30 0.01

Mn 9500 3.4 0

[ f *], N/mm2

21.5 0

0.08

23 0

27.5 0

Mn 25600 24.5 0 31.5 0.01

Mn 32900 16.2 0 0.04

0

0.2

0.6

0.4

0.8

1

1/

Fig. 17. Determination of the parameters κ and ζ of the Flory–Erman theory for perfect trifunctional poly(dimethylsiloxane) networks (155). To convert N/mm2 to psi, multiply by 145. Courtesy of Springer-Verlag.

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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 deformation 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 proportional 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 3/2 (µ/V0 ) κ = I r20

(87)

where I is a constant of proportionality, (r20 )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 r20 can be taken proportional to M c (95), the molecular weight between cross-links, and µ/V 0 to Mc− 1 for a perfect 1/2 tetrafunctional network; therefore κ Mc (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 conﬁgurational entropy of the network chains by dilation. Therefore, an equilibrium degree of swelling is established, which increases as the cross-linking density decreases (168–170). The free energy change A for this process is usually assumed to 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 accounts for the constraints and is given by Ac = µkT/2

{[1 + g(λi )]B(λi ) − ln[(B(λi ) + 1)(g(λi )B(λi ) + 1)]}.

(90)

i=x,y,z

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

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The chemical potential of the solvent in the swollen network is ∂Am ∂Ael ∂λ 0 µ 1 − µ1 = N +N ∂n1 T, p ∂λ T, p ∂n1 T, p

(91)

where n1 is the number of moles of solvent, and the isotropic extension ratio λ is − 1/3

λ = λx = λ y = λz = [(n1 V1 + V0 )/V0 ]1/3 = v2

(92)

where V 1 is the molar volume of the solvent and V 0 the volume of the dry network. At swelling equilibrium, µ1 = µ01 . Hence, the standard expression for Am (3) leads to ∂Ael ∂λ 2 (RT) − 1 (93) ln(1 − υ2m) + υ2m + χ υ2m = − ∂λ T, p ∂n1 T, p where v2m is the volume fraction of polymer at swelling equilibrium. The interaction parameter χ 1 may be determined, for example, by vapor pressure or by osmometry measurements (145,175–178). It depends on the concentration of polymer 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 − 2/3 1/3 (94) H = − (ξ/V0 )V1 v2m 1 + (µ/ξ )K v2m 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 : 1/3 − 2/3 H Mcr = (2/φ − 1)ρV1 υ2m 1 + (ϕ/2 − 1) − 1 K V1 υ2m

(95)

where the subscript r is employed here for real networks. For a network deforming afﬁnely, κ = ∞, K(λ2 ) = 1 − λ − 2 , and 1/3 2/3 H Mca = − ρV1 υ2m 1 − 2υ2m /φ

(96)

For a phantom network, κ = 0, K(λ2 ) = 0, and 1/3 Mcp = (2/φ − 1)ρV1 υ2m 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 experiments 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).

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The modern theory of real networks now permits a more accurate determination 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 measurements, 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 constraints that restrict junction ﬂuctuations. Another viewpoint considers entanglements 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 G0N 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 universal law has been proposed for the dependence of G0N 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 = Gc + Ge Te

(98)

where Gc is the contribution of the chemical cross-links. Taking into account the restrictions 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 deformation; h can also be expressed as a function of the Flory parameters κ and ζ (155,212). The additional contribution Ge T e is said to arise from permanently trapped (interchain) entanglements in the network. The modulus Ge is thought to have a value close to G0N, and T e , the “trapped entanglement factor,” is the probability 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 developed 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

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typical, highly speciﬁc reaction used for this purpose is the end-linking of functionally terminated polymer chains. Speciﬁc examples would be hydroxyl-terminated or vinyl-terminated chains of poly(dimethylsiloxane), [Si(CH3 )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 characteristics, 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 chromatography 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 polymerizations 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 reaction, 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 networks 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 models 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. Contacts between a pair of entangled chains are transitory and of short duration owing to the diffusion of segments and associated time-dependent changes of conﬁgurations. Such trapped entanglements as previously described are possibly of minor importance in equilibrium stress measurements.

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J. P. QUESLEL Manufacture Michelin, CERL – GPA J. E. MARK University of Cincinnati