Equilibrium rubbery moduli

At present, there are no physical methods to measure the concentration of amines of different types in networks and thus we cannot experimentally prove the computed values. However, the computed results seem reasonable since computer simulations give many features of the real behaviour of the systems under consideration. For example, the calculations gave kinetic curves of different reacting mixtures, sol and gel fractions, and equilibrium rubbery modulus. All results showed very good correlation with experiments 6 9,13,16,ly,31). This situation allows us to correlate the structural features of networks (for example, relative amounts of defects) obtained from computer simulations with macroscopic properties of the polymers. 

The average length (or molecular weight) of network chains in a crosslinked polymer can be experimentally determined from the equilibrium rubbery modulus. This relationship is a direct result of the statistical theory of rubber-like elasticity . In the last decade or so, modem theories of rubber-like elasticity 2127) further refined this relationship but have not altered its basic foundation. In essence, it is... 

The modem theory of mbber-like elasticity theory suggests that there are two types of elastically active network chains which contribute to the overall equilibrium rubbery modulus, G (1) chains attached to the network by chemical crosslinks, G and (2) chains attached by physical crosslinks or entangelements, G . That is,... 

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. 

Fig. 3. Stoichiometric DGEBA/DDS network M, versus prepolymer resin molecular weight. M . (M, calculated from equilibrium rubbery moduli at T = T, -h 45 K). O M, from equihbrium tensile experiments M, from 0.16 hz dynamic mechanical storage modulus measurements (After LeMay >)...

If the rubbery equilibrium shear modulus does not show evidence of crosslink mobility for some other family of thermosets, then the factor (fav-2)/fav would drop out of Equation 6.18, so that the dependence of Tg on network architecture would be expressed more simply, just in tenns of Mc. It could, in that case, be expressed equivalently as in Equations 6.16 and 6.17. For thermosets known to or expected to manifest crosslink mobility, it should, then, generally be possible to combine the functional form of the dependence on fav shown in Equation 6.18 with Equation 6.16, to obtain Equation 6.19 which is an alternative form for the relationship for the Tg of thermosets manifesting crosslink mobility. 

Because of the special swelling and mutual dilution effects encountered in sequential IPN s, special equations were derived for their rubbery modulus and equilibrium swelling. The new equations were used to analyze polystyrene/polystyrene homo-IPN swelling and rubbery modulus data obtained by four different laboratories. In the fully swollen state, there was no evidence for IPN related physical crosslinks, but some data supported the concept of network I domination. In the bulk state, network I clearly dominates network II because of its greater continuity in space. The analysis of the data concerning the possible presence of added physical crosslinks in the bulk state yielded inconclusive results, but this latter is of special interest for modern network theories. 

The volume of the sample is simply V pression for a sample of rubber is comparable to that for a liquid. The equilibrium shear modulus for a liquid is G = 0. For a rubbery solid, the... 

It is well known that the elasticity of polymer networks with constrained chains in the rubbery state is proportional to the number of elastically active chains. The statistical (topological) model of epoxy-aromatic amine networks (see Sect. 2) allows to calculate the number of elastically active chains1 and finally the equilibrium modulus of elasticity Eca,c for a network of given topological structure 9 10). The following Equation 9) was used for the calculations of E, c ... 

The equilibrium small-strain elastic behavior of an "incompressible" rubbery network polymer can be specified by a single number—either the shear modulus or the Young s modulus (which for an incompressible elastomer is equal to 3. This modulus being known, the stress-strain behavior in uniaxial tension, biaxial tension, shear, or compression can be calculated in a simple manner. (If compressibility is taken into account, two moduli are required and the bulk modulus. ) The relation between elastic properties and molecular architecture becomes a simple relation between two numbers the shear modulus and the cross-link density (or the... 

A rubbery polymer is immersed in a liquid and swells, absorbing the liquid until it reaches equilibrium, at which there is a volume fraction 0 of polymer in the mixture of polymer and liquid (see (3.N.2)). Find the ratio GJG oithe shear moduli in swollen and unswollen states in terms of 4>. Hence calculate the shear modulus of the rubber sample of Problem 3.7 when it has absorbed an equal volume of liquid. 

The onset of the transition zone on the frequency scale can be defined as in Fig. 12-9 except that for a cross-linked polymer the left side of the curve for G goes into the equilibrium modulus Ge instead of the plateau G% (except for the very lightly cross-linked systems discussed in Section B4 below). The boundary frequency (j>tr is given by equation 9 of Chapter 12 according to the Rouse-Mooney theory. Various cross-linked rubbery polymers show similar frequency dependences of G as the transition zone is entered, as shown in Fig. 14-1 if the logarithmic scales are arbitrarily shifted to make Gg and o),r coincide for all four polymers shown, the curves nearly coincide. ... 

Here Geo is the equilibrium modulus of the unfilled polymer, is the volume fraction of filler, and , is a maximum volume fraction corresponding to close packing, which may be between 0.74 and 0.80. For < 0.70, this equation is equivalent to the result of a theoretical formulation by van dcr Pocl (which can be evaluated only numerically) relating the shear and bulk moduli of a composite with spherical particles to the shear and bulk moduli and Poisson s ratios of the two component materials. The derivation of van dcr Poel has been corrected and simplified by Smith.For a hard solid in a rubbery polymer, the ratio of the shear moduli is so large that the result is insensitive to its magnitude. An example is shown in Fig. 14-13 for data of Schwarzl, Brcc, and Nederveen for nearly monodisperse sodium chloride particles of several different sizes embedded in a cross-linked polypropylene ether. Extensive comparisons of data with equation 18 have been made by Landcl, -"- - who has also employed an alternative relation ... 

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