Fokker–Planck equation for the one-molecule distribution function in polymer mixtures and its solution

• Published in: The Journal of chemical physics 1997-06, Vol.106 (23), p.9899-9921
• Main Authors: Curtiss, C. F., Byron Bird, R.
• Format: Article
• Language: English
• ISSN: 0021-9606; 1089-7690
• DOI: 10.1063/1.473878

It is shown how a Fokker–Planck equation in the phase space of a single polymer molecule in a multicomponent mixture can be obtained from the Liouville equation in the phase space of a mixture of polymeric liquids. This result is a generalization of the Schieber–Öttinger equation for a dilute solution of a single polymer species in a solvent, or the Öttinger–Petrillo equation for nonisothermal systems. The Fokker–Planck equation is solved as a series in powers of a small parameter ε, thereby displaying quantitatively the deviation of the velocity distribution from the Maxwellian. It is then shown how moments of the singlet distribution function needed for the evaluation of the transport coefficients can be obtained. In addition, expressions for the first three moments of the Brownian force are developed. It is further shown how the present discussion is related to the Curtiss–Bird theory for multicomponent diffusion. Throughout the development the polymer molecules are modeled as arbitrary bead-spring structures, with all inter-bead forces (representing both intra- and intermolecular forces) derivable from a potential and directed along the bead–bead vectors. These models can describe flexible chain macromolecules, ring-shaped polymers, starlike polymers, and branched polymers.