Relaxation Time Spectrum, Elasticity, and Viscosity of Rubber. II

In Part I of this work, the occurrence of a relaxation time spectrum in high-polymer materials was described, and a quantitative expression for the relaxation time spectrum for rubber and rubberlike materials was found, based on the creep curve observed for these materials, i.e., the distribution density was given with which the partial elastic moduli are distributed among the relaxation times. In the present paper, conclusions have been derived regarding the elastic and viscous behavior of the materials on the basis of the relaxation time spectrum reported. It has been found that the expected creep curve in such a material, i.e., the curve of the change of length after a definite time at a constantly held stress, must be practically identical with the reciprocal value of the elastic modulus E, which can be determined as a time function from the decrease of stress after rapid deformation. The E modulus observed at time t after rapid deformation is a function of all portions of the relaxation time spectrum in such a way, however, that E(t) is mainly determined by those portions of the spectrum, for which the relaxation time τ is greater than t. The accuracy with which the distribution of the partial elastic moduli can be computed from the available experimental data is not equally large for all regions of the relaxation time spectrum. The possible errors lie both in the region of very large relaxation times τ and very small times. But it appears that the error in the E modulus and in the viscosity, due to the contributions of the inaccurately known portions of the spectrum, is small in all cases and that these inaccuracies represent only a small constant added to the E modulus and the viscosity in the field of practical interest. The dynamic elastic modulus, in the frequency range 10−2 to 104 per second, on the basis of the relaxation time spectrum, is found experimentally to be almost independent of the period. On the other hand, the dynamic viscosity increases proportionally to the period. For a periodTs, those restoring force mechanisms whose relaxation time is somewhat but not much smaller than Ts contribute almost entirely to the magnitude of the dynamic viscosity. The amount of heat developed in a test-sample per cc. and per second by periodic displacement increases proportional to the frequency of the applied deformation. The proportionality factor can be calculated from the deformation-time curve observed at constant load, i.e., from the creep curve. It is evident that the distribution density in the relaxation time spectrum in the region τ<10−4 second increases somewhat more rapidly than the extrapolation of the formula valid for the region 10−2 to 104 second would indicate. It was shown that those restoring force mechanisms for which the relaxation time is greater than 10−2 second are probability mechanisms, whereas in the case of shorter relaxation times, energy mechanisms occur in increasing proportions in addition to, or in place of, the probability mechanisms. The occurrence of probability mechanisms having the relaxation time t* is to be interpreted in such a way that linear sections of molecular weight M*, which must be smaller than the molecular weight Mf of the lattice link or of the total molecule, need a time t* to change their configuration or orientation noticeably in the interior of the mass in which they are embedded. Accordingly, a relationship between t* and M* can be given t* depends on the viscosity η* by which the embedding medium opposes a Brownian movement by the linear section of molecular weight M*. Conclusions can be drawn thatη* increases rapidly with M*, e.g., exponentially. Furthermore, as is shown, the relaxation time of a restoring force mechanism considered not only through its contribution to the viscosity determines the relaxation times of all the other restoring force mechanisms, but also by their contributions to the viscosity is itself determined by the total of all the other relaxation times.