Single System Dual Split Equations

In 1981, I rewrote the original equations used in my PhD to include a dissipative term and they became what they look like now in the Grain-Field-Statistics when the solution applies to unentangled systems, i.e when only one Dual-Phase is involved (below M_c). I was mostly interested then in characterizing the molecular relaxations of glasses induced by the return to equilibrium after quenching from the rubbery state. I got involved in the development of a fine instrument, a dielectric spectrometer, which produced a current which could be converted into a relaxation time via a formula due to Bucci. The electrical signal was deconvoluted into elementary Debye relaxation modes which demonstrated the existence of a strong interactive character between them, not just below T_g, in the glassy state, but above T_g as well. The nature of the strong interaction between the relaxation modes could be analyzed and interpreted by my newly written equations, and I could differentiate the glassy and rubbery behavior by the distinct interactive character between the Debye relaxation modes below and above T_g. I wrote a book about these findings (1993). I presented the dielectric relaxation data which, not only validated the spectroscopic evidence of a boson type of local cohesion below T_g, but suggested that the relaxation modes above T_g were modulated by a very low frequency compensation, the signature of global cohesion, which I identified with "entanglements". In 1997, I published a paper in the Journal Macromolecular Sci. Physics Review "Do we need a new theory in polymers physics?" in which I examined the experimental evidence (mostly spectroscopic evidence) for a nanometric density fluctuation to describe the local cohesion of glasses. In summary, at that stage of my research, I had integrated my new modelization of the bonds interactions with spectroscopic and thermal analysis results. Yet there was no force involved, no understanding of visco-elastic deformation resulting in flow. Additionally, I was facing another difficulty in understanding the effect of molecular weight on the flow properties. It took me another 20 years to integrate the effect of molecular weight into the Dual-Phase model, which became the Cross-Dual-Phase. This finally gave me the mathematical tools to tackle flow in the liquid state.

The Dual-Split Equations for a single system are given in the slide (jpeg file) downloadable at the bottom of this page.