Ideal rubber elastic modulus

This is a theoretical study on the entanglement architecture and mechanical properties of an ideal two-component interpenetrating polymer network (IPN) composed of flexible chains (Fig. la). In this system molecular interaction between different polymer species is accomplished by the simultaneous or sequential polymerization of the polymeric precursors [1 ]. Chains which are thermodynamically incompatible are permanently interlocked in a composite network due to the presence of chemical crosslinks. The network structure is thus reinforced by chain entanglements trapped between permanent junctions [2,3]. It is evident that, entanglements between identical chains lie further apart in an IPN than in a one-component network (Fig. lb) and entanglements associating heterogeneous polymers are formed in between homopolymer junctions. In the present study the density of the various interchain associations in the composite network is evaluated as a function of the properties of the pure network components. This information is used to estimate the equilibrium rubber elasticity modulus of the IPN. 

Eqnation 9.51 differs from the result of ideal rubber elasticity presented below by a factor of 4/5. The reasons for this factor are complexities in the model [19] that are beyond the scope of this text, but one would expect a factor of less than unity becanse of chain ends that do not contribute to the elasticity of the entangled polymer network. Given that the plateau modulus is independent of molecular weights, M, for M > Me, the value of in Equation 9.51 is a constant for a given melt but will vary from polymer to polymer depending on the structural properties of the polymer such... 

Note the analogy to Equation 9.70 where the phenomenological coefficient 2Cj can be equated to RTN, the elastic modulus of the ideal rubber elasticity model. If we also keep the next term in the series in Equation 9.85, for i = 0 and j = 1, we get... 

The elastic stress curve in figure perfectly follows elastic strain [2]. This constant is the elastic modulus of the material. In this idealized example, this would be equal to Young s modulus. Here at this point of maximum stretch, the viscous stress is not a maximum, it is zero. This state is called Newton s law of viscosity, which states that, viscous stress is proportional to strain rate. Rubber has some properties of a liquid. At the point when the elastic band is fully stretched and is about to return, its velocity or strain rate is zero, and therefore its viscous stress is also zero. 

However, in doing so one tests two theories the network formation theory and the rubber elasticity theory and there are at present deeper uncertainties in the latter than in the former. Many attempts to analyze the validity of the rubber elasticity theories were in the past based on the assumption of ideality of networks prepared usually by endllnklng. The ideal state can be approached but never reached experimentally and small deviations may have a considerable effect on the concentration of elastically active chains (EANC) and thus on the equilibrium modulus. The main issue of the rubber elasticity studies is to find which theory fits the experimental data best. This problem goes far beyond the network... 

Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

Stress relaxation is the time-dependent change in stress after an instantaneous and constant deformation and constant temperature. As the shape of the specimen does not change during stress relaxation, this is a pure relaxation phenomenon in the sense defined at the beginning of this section. It is common use to call the time dependent ratio of tensile stress to strain the relaxation modulus, E, and to present the results of the experiments in the form of E as a function of time. This quantity should be distinguished, however, from the tensile modulus E as determined in elastic deformations, because stress relaxation does not occur upon deformation of an ideal rubber. 

This is the well-known Einstein-Smallwood equation, where c is the volume fraction of the filler and Gn, is the elastic modulus of the rubber matrix. The equation is obtained based on three idealized assumptions, such as (i) freely dispersed particles, i.e. low volume fraction, (ii) a spherical shape (leading to the constant 2.5) and (iii) entirely non-elastic filler particles, i.e. their elastic modulus has to be infinitely large. The reinforcement term contains two factors one is a simple number related only to the geometry of the particles, the other is linear in the volume fraction of the filler particles. 

Polymers come in many forms including plastics, rubber, and fibers. Plastics are stiffer than rubber, yet have reduced low-temperature properties. Generally, a plastic differs from a rubbery material due to the location of its glass transition temperature Tg). A plastic has a Tg above room temperature, while a rubber will have a Tg below room temperature. Tg is most clearly defined by evaluating the classic relationship of elastic modulus to temperature for polymers as presented in Fig. 1.1. At low temperatures, the material can best be described as a glassy solid. It has a high modulus and behavior in this state is characterized ideally as a purely elastic sohd. In this temperature regime, materials most closely obey Hooke s law ... 

The modulus of an ideal rubbery network depends only on the length of the chain segment between associations, and should be affected by the molecular weight of the uncrosslinked molecule only insofar as the chain ends do not contribute to the network. Consequently, the shift factor b should reflect this free chain end effect. This has been tested using the simplest expression for the modulus in the theory of rubber elasticity. 

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