Networks, theoretical moduli

Tests of Theoretical Modulus Values—Model Networks... 

A number of workers have treated non-Gaussian networks theoretically in terms of this finite extensibility problem. The surprising conclusion is that the effect on simple statistical theory is not as severe as might be expected. Even for chains as short as 5 statistical random links at strains of up to 0.25, the equilibrium rubbery modulus is increased by no more than 20-30 percent (typical epoxy elastomers rupture at much lower strains). Indeed, hterature reports of highly crosslinked epoxy M, calculated from equilibrium rubbery moduh are consistently reasonable, apparently confirming this mild finite extmsibiUty effect. 

Data lying to the right of the theoretical line provide an indication of physical crosslinks, since the experimental modulus is larger than the theoretic modulus. A vertical stacking of points containing the same network I (but different network II s), suggest the dominance of network I over network II, since the experimental modulus is relatively constant. 

Figure 7. Ratio of experimentally observed and theoretically calculated modulus, using phantom network theory with f2 and v2, versus branching density z-...

From a theoretical point of view, the equilibrium modulus very probably gives the best characterization of a cured rubber. This is due to the relationship between this macroscopic quantity and the molecular structure of the network. Therefore, the determination of the equilibrium modulus has been the subject of many investigations (e.g. 1-9). For just a few specific rubbers, the determination of the equilibrium modulus is relatively easy. The best example is provided by polydimethylsiloxane vulcanizates, which exhibit practically no prolonged relaxations (8, 9). However, the networks of most synthetic rubbers, including natural rubber, usually show very persistent relaxations which impede a close approach to the equilibrium condition (1-8). 

Figure 10. Dependence of the reduced equilibrium modulus of POP triol - MDI networks prepared in the presence of diluent. POP triol Mu= 708 stress-strain measurements in the presence of diluent (o) and after evaporation of the diluent ( ). Flory theory for the values of the front factor A indicated, theoretical dependence including trapped interchain constraints Numbers at curves Indicate the value of ry.

In the present paper, theoretical arguments and modulus measurements are used to deduce the significant gel structures which lead to inelastic loop formation and to quantify the network defects and reductions in modulus which may be expected, even in the limit of no pre-gel intramolecular reaction. In this limit all the existing theories and computer simulations of polymerisations including intramolecular reactlon(8,10,ll) predict that perfect networks are formed. 

This is a theoretical study on the structure and modulus of a composite polymeric network formed by two intermeshing co-continuous networks of different chemistry, which interact on a molecular level. The rigidity of this elastomer is assumed to increase with the number density of chemical crosslinks and trapped entanglements in the system. The latter quantity is estimated from the relative concentration of the individual components and their ability to entangle in the unmixed state. The equilibrium elasticity modulus is then calculated for both the cases of a simultaneous and sequential interpenetrating polymer network. 

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. 

Fig. 34. Schematic diagram of the difference in elastic behaviour of a network with structuring (1) and the same network without structuring (2). The disappearance of the structuring (e.g. with swelling) is experimentally coupled to a decrease in the modulus atqin(Ax — /t 8)-1. New theoretical approaches must take this into...

The above analysis and Fig. 19-25 provide a theoretical foundation similar to the Thiele-modulus effectiveness factor relationship for fluid-solid systems. However, there are no generalized closed-form expressions of E for the more general case ofa complex reaction network, and its value has to be determined by solving the complete diffusion-reaction equations for known intrinsic mechanism and kinetics, or alternatively estimated experimentally. 

So far, we have considered the elasticity of filler networks in elastomers and its reinforcing action at small strain amplitudes, where no fracture of filler-filler bonds appears. With increasing strain, a successive breakdown of the filler network takes place and the elastic modulus decreases rapidly if a critical strain amplitude is exceeded (Fig. 42). For a theoretical description of this behavior, the ultimate properties and fracture mechanics of CCA-filler clusters in elastomers have to be evaluated. This will be a basic tool for a quantitative understanding of stress softening phenomena and the role of fillers in internal friction of reinforced rubbers. 

We note that, in principle, the main physical discussions related to filler networking in this paper do not change if a sinusoidal tensile or uniaxial compres-sional stress (amplitude 0) is imposed on the rubber material. In some examples the complex dynamic modulus is then denoted with E = E + iE" and the compliance with C = C - iC". All theoretical considerations use the shearing modulus G. ... 

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