Elastic properties of networks

Yamamoto M. The visco-elastic properties of network structure. 1. General formalism. J Phys Soc Jpn 1956 11 413-421. 

Greene, A., and A. Ciferri Elastic properties of networks formed from oriented chain molecules of fibrous natural rubber. Kolloid-Z. 186, 1 (1962). 

The properties of the surplus segment probability p and the effective constraint coordination number z are less well established. It seems possible that p will dep d on polymer species to some extent, since loop projection may be easier for a more locally flexible chain. Weak dependences on concentration and temf rature are likely for the same reason. On the other hand, z characterizes the topology on a fairly large scale and therefore may be virtually a universal constant. Diese however are only some speculations. Values of p and z can be established by various experiments, p from the elastic properties of networks and also from the relaxation of star polymers, z from relative relaxation rates of linear and star molecules in liquids and networks and also from measurements of diffusion rates of stars in linear chain liquids. The adequacy of the... 

In the case of moderately crosslinked and topologically restricted networks, we obtain results which may be used to explain the low (non-affine) values of molecular deformation observed experimentally (Fig. 14). Values of a close to 1 indicate that the elastic properties of networks prepared by the end-linking reaction are very... 

Greene, A. Smith, K. J. Ciferri, A., Elastic Properties of Networks Formed from Oriented Chain Molecules Part-2. Composite Networks. Trans. Faraday Soc. 1965, 61, mi-riZ i. 

The effect of pendent chains on the mechanical properties of model PDMS has been studied. It was foimd that the loss modulus of these networks was a function of the concentration and molecular weight of the dangling chains. The elastic properties of networks with pendent chains were found to be lower than those that were completely cross-linked (352). 

Prediction of the elastic properties of networks using rubber elasticity theory is based upon the knowledge of concentrations of elastically active network junctions (EANJs) and chains (EANCs), respectively and [260, 261]. EANJs are the intersection of at least three chains leading to the gel, whereas EANCs are the chains linking EANJs (see Figure 3.13). 

Scanlan and Case have used another criterion to identify the chains and junctions which should control the elastic properties of networks. In their scheme, a junction is elastically active if three or more of its arms are independently attached to the network. A strand is elastically active if it is attached at both ends to an active junction. However, it can be shown that equation (55) still holds if Vg and Pg are replaced by the numbers of active chains and junctions Pa, respectively. This is stated in equation (56). In the phantom network model, the elastic modulus depends only on the cycle rank and therefore is not influenced by the choice of criterion. Furthermore, there is no difference between the two criteria when applied to perfect networks. All the equations derived above by graph-theoretical arguments hold for odd as well as even values of and may be adapted to networks of variable functionality merely by replacing (j) by its average 0.124,126,127... 

C. Elastic Properties of Networks Sw[Pg.28]

If this is true, this should hold not only for polymer melts but, in the limit of long chains, also for polymer networks. In the simplest case the elastic properties of polymer networks are entirely governed by the entropic... 

Pearson D.S. and Graessley W.W., Elastic properties of well characterized ethylene propylene copol3Tner network. Macromolecules, 13, 1001, 1980. 

Arbabi, S Sahimi, M, Elastic Properties of Three-Dimensional Percolation Networks with Stretching and Bond-Bending Eorces, Physical Review B 38, 7173, 1988. 

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

Figure 4. The elastic properties of some bimodal PDMS networks. Short chains were segregated by pre-reacting them with a limited amount of (tetrafunctional)...

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. 

Flory,P.J. Network structure and the elastic properties of vulcanized rubber. Chem. Rev. 35, 51-75 (1944). 

Even if completely homogeneous and disordered in the relaxed state, a real network differs from the ideal network, defined in Chapter I. Three types of network defects are commonly considered to be present in polymer networks unreacted functionalities, closed loops, and permanent chain entanglements. Within each group there are several possibilities dependent on the arrangement of chains the effect of defects on the elastic properties of the network is thus by no means simple, as has been stressed e.g. by Case (28). Several possible arrangements are shown in Fig. 1, where only nearest neighbour defect structures have been drawn. 

Jackson, J. F., and S. J. Gill Elastic properties of crosslinked poly (vinyl alcohol) gels. Network topology. J. Polymer Sci. Pt A-2, 5, 663 (1967). 

Hermans, J. 1965. Investigation of the elastic properties of the particle network in gelled solutions of hydrocolloids. I. Carboxymethyl cellulose. J. Polym. Sci. A 3 1859-1868. 

Most of the physical properties of networks in the rubbery state can be linked to two groups of quantities that characterize respectively, the equilibrium entropic elasticity and the relaxation kinetics (linked to the segmental mobility). 

It was shown that the stress-induced orientational order is larger in a filled network than in an unfilled one [78]. Two effects explain this observation first, adsorption of network chains on filler particles leads to an increase of the effective crosslink density, and secondly, the microscopic deformation ratio differs from the macroscopic one, since part of the volume is occupied by solid filler particles. An important question for understanding the elastic properties of filled elastomeric systems, is to know to what extent the adsorption layer is affected by an external stress. Tong-time elastic relaxation and/or non-linearity in the elastic behaviour (Mullins effect, Payne effect) may be related to this question [79]. Just above the melting temperature Tm, it has been shown that local chain mobility in the adsorption layer decreases under stress, which may allow some elastic energy to be dissipated, (i.e., to relax). This may provide a mechanism for the reinforcement of filled PDMS networks [78]. 

Fig. 17 Schematic representation of the temperature gradient occurring during the exothermic curing reaction. Owing to the comparatively high thermal conductivity of the continuous metallic filler (e.g. wires consisting of copper), exothermic reaction heat Q is conducted away from the interface. By changing locally the thermal conditions of the curing reaction, the temperature gradient T(N) may influence the resulting network structure D (N) which in turn defines the elastic properties of the epoxy, e.g. its elastic modulus E(N). jw denotes the heat current density...

The low shear rheology measurements also show a rapid increase in the viscoelastic properties (modulus and zero shear viscosity) with increase of bentonite concentration above the gel point (> 30 g dm bentonite). Several models have been proposed to account for the elastic properties of concentrated dispersions, of which that originally proposed by van den Tempel (25) and later developed by Papenhuizen (26) seems to be the most appropriate for the present system. According to this model, if the interaction energy minimum between adjacent particles is sufficiently negative, a three-dimensional network structure may ensue, giving an elastic component. Various models can be used to represent the three dimensional structure, the simplest of which would be either an ideal network where all particles are... 

More detailed calculations of the elastic properties of model networks have confirmed Phillips model. The coordination dependence of the elastic modulus is shown in Fig. 2.12 (He and Thorpe 1985). Both the modulus Cn and the number of zero frequency vibrational modes, /, drop to zero at the critical coordination of 2.4, as predicted by Eq. (2.17). The properties are explained in terms of percolation of rigidity. The coordination of 2.4 represents the lowest network coordination for which locally rigid structmes are fully connected, so that the entire network is rigid, but only just so. The elastic modulus is therefore non-zero and continues to increase as the network becomes more connected. The four-fold amorphous silicon network is far from the critical coordination and is very rigid. 

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