The application of non-Gaussian chain statistics to ultralow density polyethylenes and other thermoplastic elastomers

A conventional rubber elasticity equation based on the inverse Langevin fun ction, combined with a yield stress (Y-0) has been used for the purpose of studying published tensile stress-strain curves for thermoplastic elastomer s. In order to simplify the calculation a Fade approx approximation has bee n employed [Cohen A. Rheol. Acta. 1991;30:270] for the inverse Langevin fun ction which leads to the following equation, relating f, the nominal or eng ineering stress, to the extension ratio lambda: f = Y-0/lambda + (Cr/3)[lambda(3 - lambda(2)/n)/(1 - lambda(2)/n) - (1/lamb da(2))(3 - 1/lambda n)/(1 - 1/lambda n)]. When suitable values of the parameters are selected the equation may be use d to model nominal stress-strain curves for the new ultralow density polyet hylenes, ethylene vinyl acetate copolymers and SBS block copolymers. The pa rameters Cr and n selected in this way represent identifiable physical enti ties; Cr the initial modulus and n(1/2) the limit of extensibility. However Cr does not increase with temperature as with a conventional rubber, but d eclines as the temperature is raised. With the polyethylenes this may be re lated to the gradual melting of the crystals which are believed to act as c ross links [Bensason S, Stepanov EV, Chum S, Hiltner A, Baer E. Macromols 1 997;30:2436]. However, with an SBS block copolymer the reason for the fall in Ct and the rise in n are not clear. Generally, for instance when the tem perature is reduced and the materials become stiff, Cr will increase and n decrease. However when it is plotted against crystallinity with the ultralo w density polyethylenes, n does not follow CI but shows a minimum at a crys tallinity of 30% after which it appears to increase. With polyethylenes n i s more sensitive to molecular weight than Cr and gives a linear Flory plot for n(1/2) against 1/T at 0 degrees C. At 25 degrees C the values of n obta ined are very high and when the molecular weight falls to 32 000 and the st ress-strain curve is found to follow a Gaussian equation. This supports the mathematical requirement that the equation reduces to a Gaussian form when n is very large. The same result can be predicted from a series approximat ion suggested by Treloar. (C) 1999 Elsevier Science Ltd. All rights reserve d.