Rubber elasticity cross-linked polymer network

Arruda, E. M. and Boyce, M. C. (1993) A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials, /. Mech. Phys. Solids, 41, 389-412. Flory, P. J. and Rehner, J., Jr. (1943) Statistical mechanics of cross-linked polymer networks, J. Chem. Phys., 11, 512-520. 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

Another vivid example of the exceptional role of network topology is the unexpectedly high deformation abUity of hypercrosslinked polystyrenes under loading, which is usuaUy characteristic of conventional slightly cross-linked networks or linear polymers in the rubber elasticity state. Hypercrosslinked polymers, however, differ from the latter in that they retain their mobUity even at very low temperatures. In fact, hypercrosslinked materials do not exhibit typical features of polymeric glasses, nor are they typical elastomers. Their physical state thus cannot be described in terms of generaUy accepted notions. More likely, the hypercrosslinked networks demonstrate distinctly different, unique deformation and relaxation properties. 

Rubbers and gels are three-dimensional networks composed of mutually cross-linked polymers. They behave like solids, but they still have high internal degrees of freedom that are free from constraints of external force the random coils connecting the cross-links are free in thermal Brownian motion. The characteristic elasticity of polymeric materials appears from the conformational entropy of these random coils. In this chapter, we study the structures and mechanical properties of rubbers on the basis of the statistical-mechanical models of polymer networks. 

Our present theories of rubber elasticity are mostly of the kinetic theory type. In some of these theories the elasticity of a single chain is multiplied by the number of effective chains in the network to provide the total elasticity. In these theories no information concerning the statistical properties of the network structure can be inferred since either no structural aspects are evaluated or a simple structural nature is assumed. These structuril properties should be of great interest in furthering our understanding of polymers in bulk. Thus the strain dependence of X-ray scattering measurements from networks, which have heavy atoms at the cross-links and/or the chain ends, should provide some correlation between the structural and elastic properties of polymer networks. 

Characterization of network stnictimes 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). 

The polymer is cross-linked. In this case the dotted line in Figure 8.2 is followed, and improved rubber elasticity is observed, with the creep portion suppressed. The dotted line follows the equation E = 3nRT, where n is the number of active chain segments in the network and RT represents the gas constant times the temperature see equation (9.36). An example of a cross-linked polymer above its glass transition temperature obeying this relationship is the ordinary rubber band. Cross-linked elastomers and rubber elasticity relationships are the primary subjects of Chapter 9. 

While the rubber elasticity theory to be described below presumes a randomly cross-linked polymer, it must be noted that each method of network formation described above has distinctive nonuniformities, which can lead to significant deviation of experiment from theory. For example, chain polymerization leads first to microgel formation (9,10), where several chains bonded together remain dissolved in the monomer. On continued polymerization, the microgels grow in number and size, eventually forming a macroscopic gel. However, excluded volume effects, sUght differences in reactivity between the monomer and cross-linker, and so on lead to systematic variations in crosslink densities at the 100- to 500-A level. 

Let us consider a polymeric network that contains solvent, usually called a polymeric gel. There are several types of gels. A previously cross-linked polymer subsequently swollen in a solvent follows the Flory-Rehner equation (Section 9.12). If the network was formed in the solvent so that the chains are relaxed, the Flory-Rehner equation will not be followed, but rubber elasticity theory can still be used to count the active network segments. 

TTie elastic energy contribution Uei describes the rubber elasticity of the cross-linked polymer chains, and is proportional to the cross-link density which is the number density of elastic strands in the undeformed polymer network. We use the Flory model [35] to specify U ... 

It is somewhat difficult conceptually to explain the recoverable high elasticity of these materials in terms of flexible polymer chains cross-linked into an open network structure as commonly envisaged for conventionally vulcanised rubbers. It is probably better to consider the deformation behaviour on a macro, rather than molecular, scale. One such model would envisage a three-dimensional mesh of polypropylene with elastomeric domains embedded within. On application of a stress both the open network of the hard phase and the elastomeric domains will be capable of deformation. On release of the stress, the cross-linked rubbery domains will try to recover their original shape and hence result in recovery from deformation of the blended object. 

The elasticity of a polymer is its ability to return to its original shape after being stretched. Natural rubber has low elasticity and is easily softened by hearing. Flowever, the vulcanization of rubber increases its elasticity. In vulcanization, rubber is heated with sulfur. The sulfur atoms form cross-links between the poly-isoprene chains and produce a three-dimensional network of atoms (Fig. 19.17). Because the chains are covalently linked together, vulcanized rubber does not soften as much as natural rubber when the temperature is raised. Vulcanized rubber is also much more resistant to deformation when stretched, because the cross-... 

Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

Rubber elasticity, which is a unique characteristic of polymers, is due to the presence of long chains existing in a temperature range between the Tg and the Tm. The requirements for rubbery elasticity are (1) a network polymer with low cross-link density, (2) flexible segments which can rotate freely in the polymer chain, and (3) no volume or internal energy change during reversible deformation. 

In 1944, Flory (3) noted that the moduli of cross-linked butyl rubbers generally differ somewhat from values calculated from the crosslink density according to the kinetic theory of rubber elasticity. In many cases, the modulus also depends on the primary (uncross-linked) molecular weight distribution of the polymer. He attributed both observations to three kinds of network defects chain ends, loops, and chain entanglements. The latter are latent in the system prior to cross-linking and become permanent features of the network when cross-links are added. 

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