Elastically effective cross-link calculation

For semi- and full IPNs made of poly(oxyethylene) and poly(acrylic acid), the effective cross-link densities were determined from the elastic modulus and were compared with values estimated assuming the additivity of crosslink densities of components [113]. Discrepancies between estimated and calculated values are observed. 

FIGURE 17.11 The effect of ri (the number of statistical chain elements in a cord between cross-links) on the relation between stress and strain of a polymer gel in elongation. a0 is the force divided by the original cross-sectional area of a cylindrical test piece, v is twice the cross-link density, L is the length, and L0 the original length of the test piece. (After calculations by L. R. G. Treloar. The Physics of Rubber Elasticity. Clarendon, Oxford, 1975.)... 

Ve is XLD expressed as the number of moles of elastically effective network chains per cubic centimeter of film. Since E" is low at temperatures well above Tg, Ef E. XLD can also be calculated from the extent of swelling of a film by solvent. While cross-linked films do not dissolve in solvent, solvent dissolves in a cross-linked film. As cross-links get closer together the extent of swelling decreases. Equation (5) can be used to predict the storage modulus above Tg from the XLD. 

As a result, the concentration of the elastically effective subchains (i.e., subchains contributing to the elasticity modulus) differs from the concentration of subchains Vch calculated from the amount of cross-linker used in the gel synthesis under the assumption of the formation of an ideal network, where all the cross-linker molecules are incorporated in such a way that they connect elastically effective network subchains. To describe this difference, the elasticity modulus G is represented as a sum of two components Gch and Gg associated with chemical cross-links and entanglements, respectively. This approach was called a two-network model. " °... 

Some early studies in which Vc was estimated stoichiometrically (Bueche, " Mullins,and van der HofP ) qj- jy combining data on systems cross-linked to different extents (Kraus ) led to the conclusion that v calculated from equation 1 is not proportional to Vc, even when corrected for the effect of free ends, and hence that it includes a contribution which is independent of Vc and is attributed to elastically effective strands terminated by entanglements that were present in the polymer before cross-linking and became permanently trapped by the cross-links, as mentioned in Section B4 of Chapter 10. It is assumed that the total concentration of effective strands (reticulation density ) is... 

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