Abstract
.Two alternative routes are taken to derive, on the basis of the dynamics of a finite number of dumbbells, viscoelasticity in terms of a conformation tensor with fluctuations. The first route is a direct approach using stochastic calculus only, and it serves as a benchmark for the second route, which is guided by thermodynamic principles. In the latter, the Helmholtz free energy and a generalized relaxation tensor play a key role. It is shown that the results of the two routes agree only if a finite-size contribution to the Helmholtz free energy of the conformation tensor is taken into account. Using statistical mechanics, this finite-size contribution is derived explicitly in this paper for a large class of models; this contribution is non-zero whenever the number of dumbbells in the volume of observation is finite. It is noted that the generalized relaxation tensor for the conformation tensor does not need any finite-size correction.Graphical abstract
Highlights
Fluctuations are important when studying small systems
For Newtonian fluids, i.e., fluids with a deformation-independent viscosity and a lack of memory, the dynamics on small scales could be described in terms of the fluctuating Newtonian fluid dynamics developed by Landau and Lifshitz [4]
Another approach towards modeling fluctuating effects in complex fluids has been taken by VazquezQuesada, Ellero, and Espanol [7] and applied to microrheology [8], in which smoothed-particle hydrodynamics is extended by a conformation tensor that describes the conformation of the small number of polymer chains per volume element
Summary
The dynamics of the conformation tensor roots in a finer description, in particular, it can be related to the kinetic theory of dumbbells (e.g., see Chapt. 13 in [16]). The question addressed in this paper is what lessons can be learned from deriving the dynamics for the conformation tensor with fluctuations from an underlying kinetic description for a finite number of dumbbells. It is pointed out that the dumbbell description already contains the relaxation and fluctuation effects that are relevant on the conformation-tensor level. 2, a certain class of kinetic dumbbell models is introduced, based on which a description of fluctuating viscoelasticity in terms of the conformation tensor is derived via a direct route, for a finite number of dumbbells. This route is paralleled, where a thermodynamic approach is taken to arrive at fluctuating viscoelasticity. The dyadic product of two vectors v1 and v2 is written as v1v2
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