Rheol Acta 35:186-193 (1996) © Steinkopff Verlag 1996 Christian Friedrich Josef Honerkamp Jfirgen Weese New ill-posed problems in rheology Received: 12 October i995 Accepted: 11 December 1995 Dedicated to Prof. Dr. J. Meissner on the occasion of his retirement from the chair of Polymer Physics at the Eidgen0ssische Technische Hochschule (ETH) Zfirich, Switzerland Chr. Friedrich Freiburger Materialforschungszentrum Stefan-Meier-Strage 21 79104 Freiburg, FRG Prof. Dr. J. Honerkamp (~) Fakultgt ftir Physik Albert-Ludwigs-Universit~tt Freiburg Hermann-Herder-Stral3e 3 79104 Freiburg, FRG J. Weese Philips GmbH Forschungslaboratorium Hamburg ROntgenstraBe 2 4 - 26 22315 Hamburg, FRG Abstract Many methods of materi- al characterization by rheologicaI methods lead to ill-posed problems. The nature of ill-posedness is discussed. In this light, some such new problems appearing in molecu- lar weight characterization of homopolymers as well as in the characterization of size distribution of dispersed phases in polymer blends by rheological techniques are analyzed. The working capability of a nonlinear regularization method is demonstrated with examples of blend rheology. The role of Pro- fessor Meissner in motivating the research in the field of ill-posed problems as well as his important contribution to rheology of polymer blends is outlined. Key words Ill-posed problem - nonlinear regularization - viscoelasticity - polymer blend Introduction In 1987, as one of the authors (J. H.) got to know Pro- fessor Meissner, first at a rheology meeting, later on a visit to Ztirich, he still though as a theoretical physicist does: theoreticians should develop mathematical models and compare their consequences with experimental data. The meetings with Prof. Meissner were, however, the starting point for a new way of thinking about what else the task of a theoretician could be. Such a new task was arising because of a problem, which at the first glance ap- peared incidental, namely the critical analysis of the determination of a relaxation time spectrum from ex- perimentally given moduli. However, contemplation about this led to the recognition of an interesting, nearly ubiquitous problem, the ill-posed problem. Methods were already well developed for the treatment of such a problem (Groetsch, 1984; Morozov, 1984). That there is a problem with the usual least-squares method was already discovered by Tanner (1968) some basic papers treating the ilt-posedness were given in the rheolo- gy literature also (Wiff et al., 1975; Wiff, 1978; Friedrich et al., 1983). The way from the mathematical formulation of the proposed methods to the adaptation to rheological problems was however in no way obvious. This way demanded knowledge of stochastic as well as numerical methods and also familiarity with rheological ex- periments. The problem soon proved as so general that one recog- nized it also as an important question in many different areas, in NMR-spectroscopy, NMR-tomography, light- scattering and pharmacokinetics. The problem was even > recognized as a special case of a more general question 2