Fluctuating viscoelasticity based on a finite number of dumbbells
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THE EUROPEAN PHYSICAL JOURNAL E
Regular Article
Fluctuating viscoelasticity based on a finite number of dumbbells Markus H¨ utter1,a , Peter D. Olmsted2,b , and Daniel J. Read3,c 1
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Eindhoven University of Technology, Department of Mechanical Engineering, Polymer Technology, PO Box 513, NL-5600 MB Eindhoven, The Netherlands Georgetown University, Department of Physics, Institute for Soft Matter Synthesis and Metrology, 37th and O Streets, N.W., Washington, D.C. 20057, USA School of Mathematics, University of Leeds, Leeds LS2 9JT, UK Received 12 June 2020 / Received in final form 16 October 2020 / Accepted 2 November 2020 Published online: 20 November 2020 c The Author(s) 2020. This article is published with open access at Springerlink.com 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.
1 Introduction Fluctuations are particularly important when studying small systems. This also holds for fluids, including complex fluids, e.g., macromolecular and polymeric liquids. Small scales are involved, e.g., in microrheology [1] and micro- and nanofluidic devices [2, 3]. 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]. However, this is not sufficient for complex fluids, and thus extensions are needed. For example, the stress tensor has been related to the rate-of-strain tensor by a memory kernel, and correspondingly colored noise has been introduced on the stress tensor [5, 6]. Another approach towards modeling fluctuating effects in complex fluids has been taken by V´azquezQuesada, Ellero, and Espa˜ nol [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. The concept of fluctuating dynamics for the conformation tensor has been extended recently [9], to make it applicable not only to the Maxwell model [10, 11], as in [7, 8], but to a wider class of models, e.g. the FENE-P a b c
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