styrene had a molecular weight of 5 ¥ 105 g/mol. The anisotropy of the Rg

values agreed with those predicted on the basis of a chain affine model.While general conclusions appear to be premature, it appears that the cross-

link sites rearrange themselves during deformation to achieve their lowestfree-energy states; thus the chains deform less than the affine mechanisms pre-dicts. A modified end-pulling mechanism is also possible. A possible molecu-lar mechanism, which results in minimal changes in R^, is illustrated in Figure9.26 (107). The debate over the exact molecular mechanism of deformation issure to continue.

9.11 INTERNAL ENERGY EFFECTS

9.11.1 Thermoelastic Behavior of Rubber

In Section 9.5 some of the basic classical thermodynamic relationships forrubber elasticity were examined. Now the classical and statistical formulationsare combined (108,109).

Rearranging equation (9.8),

(9.85)

Dividing through by f and rearranging,

(9.86)

Rewriting equation (9.79) in terms of force, and substituting equation(9.38), we find that

f

f

f

T

e

L V

= - Êˈ¯1

∂

∂

ln

ln ,

f f Tf

Te

L V

= - ÊË

ˆ¯

∂

∂ ,

9.11 INTERNAL ENERGY EFFECTS 469

Figure 9.26 Model of the end-pulling mechanism, showing how R|| increases, while R^ remains

nearly constant: (A) relaxed and (B ) stretched.

(9.87)

where A0 is the initial cross-sectional area; substituting equation (9.87) intothe right-hand side of equation (9.86) and carrying out the partial derivativeyields

(9.88)

where b is the isobaric coefficient of bulk thermal expansion, (1/V)(∂V/∂T)L,P.Substituting equation (9.88) into equation (9.86),

(9.89)

Returning to equation (9.38), and differentiating the natural logarithm ofthe network end-to-end distance with respect to the natural logarithm of thetemperature obtains

(9.90)

Nothing that the right-hand sides of equations (9.89) and (9.90) are identical,

(9.91)

which expresses the fractional force due to internal energy considerations in terms of the temperature coefficient of the free chains end-to-end distance.

9.11.2 Experimental Values

Values of fe/f are usually derived by applying the equations above toforce–temperature data of the type presented in Figure 9.27 (110). These data,carefully taken after extensive relaxation at elevated temperatures, arereversible within experimental error; that is, the same result is obtainedwhether the temperature is being lowered (usually first) or raised.

Some values of fe/f are shown in Table 9.2. For most simple elastomers, fe/fis a small fraction, near ±0.20 or less. This indicates that some 80% or more of the retractive force is entropic in nature, as illustrated from early data in

f

f

d r

d T T

d r

dT

e = =ln

ln

ln02

021

d r

d T

d G

d T

Tln

ln

ln

ln

02

13

= - -b

f

f

d G

d T

Te = - -13

ln

ln

b

∂

∂

bln

ln

ln

ln,

f

T

d G

d T

T

L V

ÊË

ˆ¯ = +

3

f GAV

V= -Ê

ˈ¯0

02

1a

a

470 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY

9.11 INTERNAL ENERGY EFFECTS 471

Figure 9.27 Force–temperature relationships for natural rubber. Extension ratios, a, are indi-

cated by the numbers associated with the lines (110).

Table 9.2 Thermoelastic behavior of various polymers

Polymer fe/f Reference

Natural rubber 0.12 (a)trans-Polyisoprene 0.17 (b)cis-Polybutadiene 0.10 (c)Polyethylene -0.42 (d)Poly(ethyl acrylate) -0.16 (e)Poly(dimethyl siloxane) 0.15 (e)

References: (a) G. Allen, M. J. Kirkham, J. Padget, and C. Price, Trans. Faraday Soc., 67, 1278

(1971). (b) J. A. Barrie and J. Standen, Polymer, 8, 97 (1967). (c) M. Shen, T. Y. Chem. E. H. Cirlin,

and H. M. Gebhard, in Polymer Networks, Structure, and Mechanical Properties,A. J. Chompff and

S. Newman, eds., Plenum Press, New York, 1978. (d) A. Ciferri, C. A. J. Hoeve, and P. J. Flory, J.

Am. Chem. Soc., 83, 1015 (1961). (e) L. H. Sperling and A. V. Tobolsky, J. Macromol. Chem., 1, 799

(1966).

Figure 9.7. These same values, of course, lead to temperature coefficients ofpolymer chain expansion [equation (9.91)].

9.12 THE FLORY–REHNER EQUATION

9.12.1 Causes of Swelling

The equilibrium swelling theory of Flory and Rehner (76) treats simplepolymer networks in the presence of small molecules. The theory considersforces arising from three sources:

1. The entropy change caused by mixing polymer and solvent. The entropychange from this source is positive and favors swelling.

2. The entropy change caused by reduction in numbers of possible chainconformations on swelling. The entropy change from this source is neg-ative and opposes swelling.

3. The heat of mixing of polymer and solvent, which may be positive, neg-ative, or zero. Usually it is slightly positive, opposing mixing.

The Flory–Rehner equation may be written

(9.92)

where v2 is the volume fraction of polymer in the swollen mass, V1 is the molarvolume of the solvent, and c1 is the Flory–Huggins polymer–solvent dimen-sionless interaction term. Appendix 9.3 describes the application of theFlory–Rehner theory. This theory, of course, is also related to the thermody-namics of solutions (see Section 3.2). As a rubber elasticity phenomenon, it isan extension in three dimensions.

The value of equation (9.92) here lies in its complementary determina-tion of the quantity n [see equation (9.4) for simplicity]. Both equations (9.4) and (9.92) determine the number of elastically active chains per unit volume (containing, implicitly, corrections for front factor changes). Bymeasuring the equilibrium swelling behavior of an elastomer (c1 values areknown for many polymer–solvent pairs), its modulus may be predicted. Viceversa, by measuring its modulus, the swelling behavior in any solvent may bepredicted.

Generally, values from modulus determinations are somewhat higher,because physical cross-links tend to count more in the generally less relaxedmechanical measurements than in the closer-to-equilibrium swelling data.However, agreement is usually within a factor of 2, providing significant inter-play between swelling and modulus calculations (111–113).

- -( ) + +[ ] = -ÈÎÍ

˘˚̇

ln 12

2 2 1 22

1 21 3 2

v v v V n vv

c

472 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY

Simple elastomers may swell a factor of 4 or 5 or so, leading to a quantita-tive determination of n. However, two factors need to be considered beforethe final numerical results are accepted:

1. The front-factor, not explicitly stated in the Flory–Rehner equation, maybe significantly different from unity (114).

2. While step polymerization methods lead to more or less statistical net-works and good agreement with theory, addition polymerization and vul-canization nonuniformities lead to networks that may swell as much as20% less than theoretically predicted (115,116).

9.12.2 Example Calculation of Young’s Modulus from Swelling Data

At equilibrium, a sample of poly(butadiene–stat–styrene) swelled 4.8 times itsvolume in toluene at 25°C. What is Young’s modulus at 25°C?

This material is a typical elastomer, widely used for rubber bands, gaskets,and rubber tires. Table 3.4 gives c1 = 0.39. The molar volume of toluene canbe calculated from its density, 0.8669 g/cm3, Table 3.1. A molecular weight of92 g/mol for toluene yields a molar volume of 106 cm3/mol. The quantity v2 =1/4.8 = 0.208. Substituting into equation (9.92) yields

Young’s modulus is given by equation (9.36),

This is a typical Young’s modulus for such elastomers. Of course, the reversecalculation can be performed, starting with the modulus, and estimating theequilibrium swelling volume.

9.13 GELATION PHENOMENA IN POLYMERS

Gelation in polymers may be brought about in several ways: temperaturechanges, particularly important in protein gelation formation; polymerizationwith cross-links; phase separation in block copolymers; ionomer formation; oreven crystallization. Such materials are usually thermoreversible for physicalcross-links, or thermoset through the advent of chemical cross-links. Of course,there must be at least two cross-link sites per chain to induce gelation.A major

E

E

= ¥ ¥ ( ) ¥ ¥ ◊ ◊( ) ¥ ( )= ¥

-3 1 5 10 8 31 10 298

1 1 10 1 1

4 7

7

. .

. .

mol cm dyn cm mol K K

dyn cm or MPa

3

2

n

n

=- -( ) + + ¥[ ]( ) -{ }

= ¥ -

ln . . . .

. .

.

1 0 208 0 208 0 39 0 208

106 0 208 0 208 2

1 5 10

2

1 3

4

cm mol

mol cm

3

3

9.13 GELATION PHENOMENA IN POLYMERS 473