Dependence of mechanical relaxation spectra of linear amorphous polymers on the distribution of molecular weights

Abstract

An expression is obtained in the form of an integral equation of the Fredholm type which correlates the mechanical relaxation distribution of a linear amorphous polymer with its molecular weight distribution. A method of approximate solution of this integral equation is given on the basis of the assumptions that: (1) the relaxation spectrum of a monodisperse sample in the rubbery region is approximated by a box‐shaped function; (2) the steady‐flow viscosity of a monodisperse sample increases with the 3.4 power of molecular weight above a certain critical molecular weight M̄_c; and (3) the relaxation modulus (or the real part of the complex dynamic elasticity), Ē, in the initial zone of the rubbery region is independent of molecular weight and may be calculated in terms of Alfrey's approximation. The method permits predicting the molecular weight distribution from experimental relaxation data obtained over the entire region of rubbery timescale, provided the values of M_c, Ē, and M̄_z (z‐average molecular weight) of the given sample are known. Stress‐relaxation data on unfractionated samples of polystyrene and polyvinyl acetate are analyzed in terms of this method, and it is found that the predicted molecular weight distributions agree quite satisfactorily with the observed data for both polymers, except in regions of low molecular weight where the theory gives rise to a fictitious maximum and minimum, neither of which is observed in the experimental data. This anomaly is attributed to the effect of the transition region on the initial zone of the rubbery region which is neglected in the present theory. The experimental data on polyvinyl acetate used for the tests are taken from our previous study, and those on polystyrene are newly obtained for this work. Since no complete stressrelaxation data covering the entire region from glassy state to rubbery flow are yet available on polystyrene, those obtained in this work are presented in rather great detail. The relaxation spectrum in the transition region is compared with that of Grandine and Ferry and of Becker from dynamic mechanical measurements in an audiofrequency range, with a fair agreement with the former of the two. The shift factor a_T determined over the temperature range from about 80 to 150°C. follows a similar curve to that found previously for polyvinyl acetate when reduced to an appropriate reference temperature, and yields a maximum activation energy for relaxation processes of 210 kcal. mole⁻¹ at a temperature of 92 ± 1°C. This temperature is in the middle of the reported second order transition temperatures on polystyrene of high molecular weight.