Non-equilibrium chemical potential and stress-induced migration of polymers in tubes D. Jou a,c , M. Criado-Sancho b, * , J. Casas-Va Âzquez a a Departament de Fõ Âsica, Universitat Auto Ánoma de Barcelona, 08193 Bellaterra, Catalonia, Spain b Departamento de Ciencias y Te Âcnicas Fisicoquõ Âmicas, Universidad Nacional de Educacion a Distancia, Senda del Rey 9, 28040 Madrid, Spain c Institut d'Estudis Catalans, Carme 47, 08001 Barcelona, Catalonia, Spain Received 26 June 2001; received in revised form 26 September 2001; accepted 2 November 2001 Abstract A non-equilibrium chemical potential depending on the viscous pressure tensor is used to describe shear-induced diffusion in polymer solutions ¯owing along cylindrical tubes. Our results generalize previous ones in three main aspects: a Flory±Huggins expression for the equilibrium contribution to the chemical potential is used instead of the ideal-gas like expression, the full expression for the steady-state compliance of the solution is taken into account instead of only the polymer contribution, and the in¯uence of the solute molecular mass is explicitly considered. As a qualitatively new result of considerable practical interest, it stands the prediction that in some circumstances, a dynamical instability may appear, which accelerates and enhances the separation process. q 2002 Elsevier Science Ltd. All rights reserved. Keywords: Non-equilibrium chemical potential; Polymer migration; Flow in tubes 1. Introduction Stress-induced polymer migration in ¯owing solutions is a relevant phenomenon in rheology, chromatography and engineering [1±5]. A coupling between viscous pressure and diffusion ¯ux produces a migration of the polymer solute towards the center of tubes and consequently a deple- tion of solute near the wall that modi®es the apparent vis- cosity of solution. Furthermore, the sensitivity of this phenomenon to the molecular mass makes it the basis of macromolecular separation methods. One of the topics of theoretical discussion in this ®eld is the role of thermodynamics, in particular, how the ¯ow contributes to the chemical potential. Indeed, many authors [2,3,7,8] have considered that the entropic and energetic changes produced by the stretching of macromolecules under the action of the ¯ow contribute to a thermodynamic force, which drives the migration. For instance, in one of the pioneering approaches in this line [2], a binary ¯uid mixture, initially spatially homogeneous in concentration, was subjected to a stress ®eld which produces a thermody- namic driving force for diffusion of macromolecules toward the zones of lower stress namely, the central region of the tube). This force was taken as the gradient of a potential V as F  27V ; it produces a stress-induced ¯ux given by J s  2 Dc RT 7V 1 where D is the diffusion coef®cient, c the polymer concen- tration and R and T have the usual meaning of the ideal-gas constant and absolute temperature, respectively. On the other hand, concentration inhomogeneities due to this ¯ux produce a diffusion ¯ux described by the classical Fick's law J F  2D7c: The net ¯ux is then J  2D 7c 1 c RT 7V   2 In Ref. [2], the expression adopted for the potential V was related to the free energy of extension of Gaussian chains, given by Ref. [10] DF flow  2 c 2 RT ln det P n cRT   2 Tr P n cRT 2 U     3 in such a way that V  2DF flow =2c: There, P n is the viscous pressure tensor acting on the polymer and U the unit matrix. This model accounts for how the stretching and the orienta- tion of the macromolecules due to the ¯ow modify the free energy of the solution and was able to describe stress- induced migration leading to an accumulation of polymer near the center of the tube where the stretching of the Polymer 43 2002) 1599±1605 0032-3861/02/$ - see front matter q 2002 Elsevier Science Ltd. All rights reserved. PII: S0032-386101)00726-1 www.elsevier.com/locate/polymer * Corresponding author.