Fundamental characteristics and properties of crosslinked elastomers.

A raw elastomer, as synthesized, is a high molecular weight liquid with little strength and low elasticity. Although highly intertwined, chains can slip past entanglement "junctions" upon stressing giving rise to facile fracture by viscous flow, especially after long times and at high temperatures. Crosslinking (also sometimes termed vulcanization or curing) is the process in which the chains are chemically linked together to form an infinite network, thereby transforming the viscoelastic liquid into a viscoelastic solid (ref. 1).

The mechanical behavior of an elastomer depends strongly on crosslink density. This is shown schematically in figure 1, where various physical properties are shown as a function of crosslink density. Modulus and hardness increase monotonically with crosslink density, and, at the same time, networks become more elastic, or, stated alternatively, less hysteretic. Fracture properties, such as tear and tensile strength, pass through a maximum as crosslinking is increased. To understand this behavior it is instructive to examine the crosslinking process in some detail.

[CHART OMITTED]

Crosslinked structure

Consider a collection of primary chains (ref. 2). "Primary" denotes the chains prior to any crosslinking. Let [y.sub.i] equal the number of sites within the [i.sup.th] chain, which are capable of crosslinking. Now, in a random fashion and only intermolecularly, introduce crosslinks among the chains. Each crosslink connects two chains (sites). Crosslinks are centers of tetrafunctionality, since four chains emanate from each. Further define the crosslink density, v, as the fraction of total sites (all primary chains) which have crosslinked, and the crosslinking index, [gamma], as the number of crosslinked sites per primary molecule, [gamma] = v[[y, bar above].sub.n] (1) where [[y, bar above].sub.n] is the number average value of y. The effect of a few crosslinks is to increase the molecular weight - creating branched molecules and a broader molecular weight distribution. As the number of crosslinks is increased, the gel point is eventually reached, and a three dimensional network is formed. Stockmayer (ref. 3) has theorized that gelation occurs when v reaches a critical value, [v.sub.c], [v.sub.c] = 1/([[y, bar above].sub.w] - 1) [nearly equal to] 1/[[y, bar above].sub.w] (2) where [[y, bar above].sub.w] is the weight average value of y. Combining equations 1 and 2, it is found that the critical value, [[gamma].sub.c] for gelation is given by [[gamma].sub.c] = [[y, bar above].sub.n]/[[y, bar above].sub.w] (3) If the primary chains are monodisperse, then [[y, bar above].sub.n] = [[y, bar above].sub.w] and [[gamma].sub.c] = 1, i.e., gelation commences when there is one crosslinked site per primary molecule (or one crosslink for every two primary molecules). For primary chains with polydispersity, X, X [equivalent] [[y, bar above].sub.w]/[[y, bar above].sub.n] gelation occurs when [[gamma].sub.c] = 1/X. Fewer crosslinks are required to produce gelation, if the molecular weight distribution is broader. For example, if X = 2, one crosslink for every four primary molecules is sufficient for the onset of gelation. Since crosslinking israndom, larger molecules in a polydisperse mixture will acquire proportionately more crosslinks.

After crosslinking, the original primary chains lose their identity. The structure now consists of two types of chains (segments of primary chains) - those terminating with crosslinks at both ends (principal chains), and those which terminate at crosslinks at one end, while the other end is free (dangling chains). For linear primary chains, two dangling chains will occur in the network for each primary molecule. Dangling chains are flaws in the network, since, unlike the principal chains, they will not bear load (at equilibrium) when the network is deformed. The lower the molecular weight of the primary chains, the greater the concentration of flaws. For this reason, networks obtained from primary chains of low molecular weight have lower strength than do those derived from higher molecular weight primary chains.

The total concentration of chains (principal plus dangling), in moles per gram, equals

[MATHEMATICAL EXPRESSION OMITTED] where [[m, bar above].sub.n] = number average molecular weight of primary chains; [[m, bar above].sub.c, n] = number average molecular weight per crosslinked site.

The weight fraction, w, of a network which is composed of dangling chains is given by

[MATHEMATICAL EXPRESSION OMITTED]

At the onset of gelation, although an infinite network is formed, not all molecules are chemically linked to it. The system is composed of gel (network) and sol (not part of network). With further crosslinking, gel increases at the expense of sol. Gel is insoluble in all solvents which do not destroy the primary valence structure, while sol (in principle) can be extracted - although long times may be required to remove very high molecular weight gel. Flory (ref. 4) has shown, for monodisperse primary chains, that the weight fraction, [w.sub.s], of sol is given by -1n [w.sub.s]/(1 - [w.sub.s]) = [gamma] (5) For example, with [gamma] = 1.5, which exceeds the critical value ([[gamma].sub.c] = 1.0) for gelation, equation 5 predicts [w.sub.s] = 0.42. The sol portion dilutes load-bearing network chains and also contributes strongly to mechanical losses (hysteretic heat generation) during dynamic deformation. Thus, most practical rubber vulcanizates are crosslinked well beyond the gel point to reduce the sol content to 5% or less.

Equilibrium properties

Small strain

The starting point for understanding the deformation behavior of rubber is the kinetic theory of rubber elasticity, assuming affine deformation and random walk, phantom chain (Gaussian) statistics. Further, it is assumed that internal energy does not change with extension, such that elasticity is entirely entropic in origin.

If the ends of a linear, phantom chain consisting of n "steps" of length, 1, are fixed a distance, r, apart, a force, f, will be required (ref. 5), f = 2k[Tb.sup.2]r, (6) where k = Boltzman's constant; T = absolute temperature; [b.sup.2] [equivalent] 3/2[nl.sup.2]

Several aspects of equation 6 are noteworthy:

* For a single chain, the extension force varies linearly with r, i.e., the behavior is Hookean.

* The force increases in direct proportion to the thermal energy of the chain, kT; it will be more difficult to deform a chain as temperature is increased.

* And finally, the greater the chain length the lower the force required to fix the ends a distance r apart.

A macroscopic three dimensional network consisting of N moles of Gaussian chains per unit volume is considered next. Upon deformation the crosslink points are considered fixed and the molecular strain is assumed to be the same as the macroscopic strain (affine deformation). For an arbitrary deformation, the elastically stored strain energy, W, per unit volume is given by

[MATHEMATICAL EXPRESSION ODMITTED]

[MATHEMATICAL EXPRESSION ODMITTED] where R = gas constant; [[lambda].sub.i] = principal extension ratios.

Since rubber is nearly incompressible and hence deforms with almost no change in volume, the product of the principal extension ratios equals unity, [[lambda].sub.1][[lambda].sub.2][[lambda].sub.3] = 1. Hence, only two of the three [lambda]'s can be varied independently. Equation 7 now will be applied to uniaxial extension and then to simple shear.

In uniaxial extension, define [[lambda].sub.1] [equivalent] [lambda], where [lambda] is the extension ratio in the direction of the applied force. Taking into account symmetry and constancy of volume, [[lambda].sub.2] = [[lambda].sub.3] = [[lambda].sup.-1/2], whereupon, W = NRT([[lambda].sup.2] - (2/[lambda]) - 3)/2.

The engineering stress, (force per unit underformed crosssection), is the derivative of the stored strain energy with respect to [lambda], [sigma] = dW/d[lambda] = NRT ([lambda] - (1/[[lambda].sup.2])). (8)

Comparing equations 6 (single chain) and 8 (network), it is noted that the resistance to deformation in both cases changes in direct proportion to temperature, but the network, unlike the single chain (in uniaxial tension), is not Hookean. Also, the fewer the crosslinks, the smaller the value of N, and the easier it is to deform the network. The counterpart of this behavior for a single chain is the lower force required to hold a longer chain at fixed r.

It should be remembered that equation 8 is predicted for equilibrium ("zero" strain rate) elastic behavior in which there is no viscous dissipation whatsoever. Also, chains are freely-jointed, have no internal energy change on deforming and there are no interchain interactions - other than the crosslinks. As such, equation 8 gives the lower limit stress required to deform an ideal network. For real networks, which often are deformed far from equilibrium, viscous effects (incomplete physical relaxation) augment the resistance to deformation - especially at rapid test rate or low temperature. Note that lowering temperature increases stiffness when the response is dominated by viscous effects; this is just the opposite of the equilibrium behavior. Lower temperatures (or higher rates) decrease the extent of physical relaxations and this results in increased stiffness. Thus, the resistance to deformation of a network can increase, decrease or stay the same as temperature is increased, depending on the proximity to equilibrium.

Insight into ideal rubbery behavior can be gleaned by comparing equation 8 to the ideal gas law, p = nRT(1/V), (9) where p = pressure; n = moles of gas; V = volume.

Note the close correspondence. The pressure (stress) of an ideal gas (rubber) is due to the RT thermal energy of its constituent gas molecules (network chains).

Next, equation 7 is solved for simple shear deformation, [[lambda].sub.1] [equivalent] [lambda], [[lambda].sub.2] = 1, [[lambda].sub.3] = 1/[lambda], so that

W = NRT([[lambda].sup.2] + (1/[[lambda].sup.2]) - 2)/2. Noting that the shear strain, [gamma], is [gamma] = [lambda] - (1/[lambda]), it is found that W = NRT[[gamma].sup.2][multiplied by]/2 The simple shear stress, [pi], is given by [pi] = dW/d[gamma] = NRT [gamma].

Surprisingly, an ideal rubber is Hookean in shear - with shear modulus equal to NRT - even though it is non-linear in uniaxial tension, as shown previously.

Strength

When an uncrosslinked elastomer is stressed, chains may readily slip past one another and disentangle. At slow rates, fracture occurs by weak, viscous flow without breaking chemical bonds. A few crosslinks give branched molecules, which are more difficult to disentangle and hence strength increases. With sufficient crosslinking the gel point is reached. As noted previously, some chains (sol) may not be attached to the network, but the whole composition will no longer dissolve in a solvent. A gel cannot be fractured without breaking chemical bonds. Thus, strength is higher at the gel point, since chemical bonds must be ruptured to create fracture surface. However, strength does not increase indefinitely with more crosslinking. When an elastomer is deformed by an external force, part of the input energy is stored elastically in the chains and is available (will be released upon crack growth) as a driving force for fracture. The remainder of the energy is dissipated through molecular motions into heat, and in this manner, is made unavailable to break chains. At high crosslink levels, chain motions become restricted, and the "tight" network is incapable of dissipating much energy. This results in relatively easy, brittle fracture at low elongation. Elastomers have an optimum crosslink density range for practical use. Crosslinking levels must be high enough to prevent failure by viscous flow, but low enough to avoid brittle failure.

Lake and Thomas (ref. 6) were the first to derive an expression for the threshold (equilibrium, lower limit) fracture energy, [G.sub.o], of an elastomeric network. In general, when a crack grows within an elastomer, energy not only is expended in breaking those network chains which happen to cross the fracture path, but there also may be energy losses within other network chains, far from the fracture front, which are deformed, but not ruptured during crack growth. The former losses are assumed to be the lower limit which must be expended to fracture an elastomer. Thus, [G.sub.o] is given by the product of the energy required to rupture a chain and the number of chains crossing unit area in the unstrained state. The net result gives [G.sub.o] = [(3/8).sup.1/2] [gamma]1U[rho][[micro].sup.1/2]/m (10) where [gamma] = a factor determined by chain flexibility; 1 = length of a monomer unit; U = energy required to rupture a monomer unit; [rho] = density of rubber; [micro] = number of monomer units in a network chain; m = mass of a monomer unit

The prediction that [G.sub.o] should increase with network chain length raised to the half power has been tested and confirmed experimentally. Typical values of [G.sub.o] are of the order |50 J/[m.sup.2], which is substantially greater than the energy (|2 J/[m.sup.2]) required just to cleave bonds lying across the fracture path. This is due to the fact that in order to break one link (backbone bond) of a network chain sufficient energy must be supplied to extend the entire chain to the breaking point.

Determination of crosslink density

The crosslink density of an elastomer can be determined from swelling or mechanical measurements. An elastomer crosslinked above its gel point will not dissolve in a solvent but rather will imbibe solvent and swell. Swelling will continue until the retractive forces in the extended chains balance the forces tending to swell the network. One expression widely used to relate the amount of swelling to the crosslink density is the Flory -Rehner (ref. 7) equation: N = -(1n (1 - [phi]) + [phi] + X[[phi].sup.2])/2[V.sub.s]([[phi].sup.1/3] -([phi]/2)) (11) where N = moles of crosslinks per unit volume; [V.sub.s] molar volume of the swelling solvent; [phi] = volume fraction of polymer in the swollen gel; X = polymer-solvent interaction parameter

Crosslink density also has been determined from equilibrium stress strain measurements using the Mooney-Rivlin (ref. 8) equation: [sigma] /(2 ([lambda] - [[lambda].sup.-2])) = [C.sub.1] + ([C.sub.2]/[lambda]), (12) where [sigma] = engineering stress, force per unit original crosssectional area; [lambda] = extension ratio; [C.sub.1], [C.sub.2] = constants.

By plotting [sigma]/(2 ([lambda] - [[lambda].sup.-2])) vs. 1/[lambda] and extrapolating to 1/[lambda] = 0, a value of [C.sub.1] can be obtained. By comparison with the theory of rubber elasticity, it has been proposed that [C.sub.1] = NRT. To assure near equilibrium response, stress-strain measurements are carried out at low strain rate, elevated temperature, and sometimes in the swollen state.

References

(1.)G.R. Hamed, in Engineering with Rubber, edited by A.N. Gent, Hanser Publishers, Chapter 2, 1992.

(2.)P.J. Flory, Principles of Polymer Chemistry, Cornell University Press, Chapter IX, 1953.

(3.)W.H. Stockmayer, J. Chem. Phys., 12, 125 (1944).

(4.)P.J. Flory, Ind. En. Chem., 38, 417 (1946).

(5.)W. Kuhn, Zolloid-Z., 76, 258 (1936).

(6.)G.J. Lake and A.G. Thomas, Proc. R. Soc., Ser. A300, 108 (1967).

(7.)P.J. Flory and J. Rehner, J. Chem. Phys., 11, 521 (1943).

(8.)M. Mooney, J. Appl. Phys., 11, 582 (1940).