Modulus of networks

Figure 3. Effect of lignin content on the modulus of networks crosslinked with HDI. Formulation type A ( ) type B ( ) and type C (0).

Therefore, it is well established that topological entanglements dominate and control the modulus of polymer networks with long network strands. The Edwards tube model explains the non-zero intercept in plots of network modulus against number density of strands (see Figs 7.11 and 7.12). The modulus of networks with very long strands between crosslinks approaches the plateau modulus of the linear polymer melt. The modulus of the entangled polymer network can be approximated as a simple sum. 

The elastic modulus of networks with pendant chains, measured at low frequencies, shows very good agreement with values calculated from theory of elasticity when contribution of molecular entanglements is taken into account. 

Figures 4 and 5 show curves of dynamic elastic modulus (G ) as a function of frequency for networks synthesized with monofunctional prepolymers of relatively low and high molecular weight respectively and a tetra functional cross linker [11]. The elastic modulus of a network prepared without pendant chains is also plotted. Results show a marked reduction in elastic modulus of networks with increasing amounts of pendant chains. This is due to the reduction in the concentration of...

The compressional modulus of networks being in equilibrium with pure diluent can be determined by means of a method based on a decrease in the equilibrium swelling using different swelling agents The compressional modulus is obtained from the variation of the polymer volume fraction with the diluent activity a, through the relation... 

To understand the global mechanical and statistical properties of polymeric systems as well as studying the conformational relaxation of melts and amorphous systems, it is important to go beyond the atomistic level. One of the central questions of the physics of polymer melts and networks throughout the last 20 years or so dealt with the role of chain topology for melt dynamics and the elastic modulus of polymer networks. The fact that the different polymer strands cannot cut through each other in the... 

Analysis of data pertaining to the modulus of PEO gels obtained by the polyaddition reaction [90] shows that even in this simplified case the network structure substantially deviates from the ideal one. For all samples studied, the molecular weight between crosslinks (M p) exceeds the molecular weight of the precursor (MJ. With decreasing precursor concentration the M xp/Mn ratio increases. Thus, at Mn = 5650 a decrease in precursor concentration from 50 to 20% increases the ratio from 2.3 to 12 most probably due to intramolecular cycle formation. 

Every quantity in Eq. (3.1) is known or measurable except Mc. Therefore, if experiments furnish the modulus of a rubberlike network, Mc of the polymer can be derived by means of the above equation. 

The ratios of mean-squared dimensions appearing in Equation (13) are microscopic quantities. To express the elastic free energy of a network in terms of the macroscopic (laboratory) state of deformation, an assumption has to be made to relate microscopic chain dimensions to macroscopic deformation. Their relation to macroscopic deformations imposed on the network has been a main area of research in the area of rubber-like elasticity. Several models have been proposed for this purpose, which are discussed in the following sections. Before that, however, we describe the macroscopic deformation, stress, and the modulus of a network. 

Equation (40) shows that the small deformation shear modulus of an affine network increases indefinitely over the phantom network modulus as junction functionality approaches 2. 

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