On the problem of consistent construction of relaxation spectra from the data of mechanical relaxation of polymers

A new formalism for description of relaxation curves is advanced; this approach is based on the notion of the Stieltjes integral with respect to the relaxation time distribution function of exponential modes. This approach allows a consistent description of all possible types of relaxation spectra, including discrete and continuous spectra, the initial instantaneous decay of a kinetic curve, and its stationary plateau. The key aspect of this approach concerns the existence of one-to-one continuous correspondence between the monotonic relaxation-time function and the time or frequency dependence of the relaxation modulus of elasticity. Owing to this correspondence, the inverse problem of relaxation-time distribution reconstruction becomes mathematically well-posed in its classical meaning and does not require application of a regularization technique. An efficient algorithm of numerical solution is described. The examples of model kinetic data treatment for the modified Rouse chain and the experimental data on the frequency dependence of elastic modulus of linear polymers are presented. Therefore, a method for relaxation-spectrum construction on the basis of the data on mechanical relaxation of polymers is proposed. The correctness of this approach is limited only by the adequacy of the applied mathematical model and by the systematic error and completeness of the experimental data.