^{1,2}and A. J. Giacomin

^{1,2,3,a)}

### Abstract

Recent work has focused on deepening our understanding of the molecular origins of the higher harmonics that arise in the shear stress response of polymeric liquids in large-amplitude oscillatory shear flow. For instance, these higher harmonics have been explained by just considering the orientation distribution of rigid dumbbells suspended in a Newtonian solvent. These dumbbells, when in dilute suspension, form the simplest relevant molecular model of polymer viscoelasticity, and this model specifically neglects interactions between the polymer molecules [R. B. Bird et al., “Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response,” J. Chem. Phys. 140, 074904 (2014)]. In this paper, we explore these interactions by examining the Curtiss-Bird model, a kinetic molecular theory designed specifically to account for the restricted motions that arise when polymer chains are concentrated, thus interacting and specifically, entangled. We begin our comparison using a heretofore ignored explicit analytical solution [X.-J. Fan and R. B. Bird, “A kinetic theory for polymer melts. VI. Calculation of additional material functions,” J. Non-Newtonian Fluid Mech. 15, 341 (1984)]. For concentrated systems, the chain motion transverse to the chain axis is more restricted than along the axis. This anisotropy is described by the link tension coefficient, ϵ, for which several special cases arise: ϵ = 0 corresponds to reptation, ϵ > 1/8 to rod-climbing, 1/5 ≤ ϵ ≤ 3/4 to reasonable predictions for shear-thinning in steady simple shear flow, and ϵ = 1 to the dilute solution without hydrodynamic interaction. In this paper, we examine the shapes of the shear stress versus shear rate loops for the special cases , and we compare these with those of rigid dumbbell and reptation model predictions.

P. H. Gilbert is grateful for an International Tuition Award from the School of Graduate Studies of Queen’s University at Kingston. A. J. Giacomin is indebted to the Faculty of Applied Science and Engineering of Queen’s University at Kingston, for its support through a Research Initiation Grant (RIG). This research was undertaken, in part, thanks to funding from the Canada Research Chairs program of the Government of Canada for the Natural Sciences and Engineering Research Council of Canada (NSERC) Tier 1 Canada Research Chair in Rheology. We are also grateful to the Canadian Foundation of Innovation for support through the John R. Evans Leaders Fund.

I. INTRODUCTION II. METHOD III. COMPARISONS A. Reptation theories (

*ϵ*= 0) B. Chains and dumbbells with hydrodynamic interaction () C. Chains with hydrodynamic interaction: Best fit () D. Dilute solutions of chains, dumbbells, and polygons (

*ϵ*= 1) IV. CONCLUSION

^{2}ω

^{2}and not 8/7λ

^{2}ω

^{2}, and in the final line should be; in Eq. (2.23), the multiplicative factor should be 2/7λ

^{2}ω

^{2}, not 8/7λ

^{2}ω

^{2}, and sλω should be 2λω; Eq. (2.29) is then reduced to 1/4 of the published result; in the second unnumbered equation following Eq. (A.3), sinhshould be sinh andcosh should be cosh.

_{2}” and “10De

_{2}− 50De

_{4}” should be “20De” and “10De–50De

_{3}. ''

^{2}” and “10De

^{2}− 50De

^{4}” should be “20De” and “10De − 50De

^{3}”; after Eq. (119), “” should be “”; In Eq. (147), “n − 1” should be “n = 1”; In Eqs. (76) and (77),Ψ′ andΨ″ should be and ; throughout,, and should be, and ; in Eqs. (181) and (182), “1,21” should be “1,2. ''

^{2}= α

^{2}{1 + cos2ωτcosωτ + sin2ωtsinωτ}(1 − cosωτ); Eq. (6.41a) should bep

_{11}− p

_{22}= α

^{2}{A + Bcos2ωt + Csin2ωt}; Eq. (6.41b) should bep

_{21}= α{Dcosωt + Asinωt}; in line 4 of p. 113,αAcosωt should beαDcosωt; in the sentence preceding Eq. (6.43), and also in Eq. (6.43), “the out-of-phase part ofp

_{21}” should be “the part ofp

_{21}that is in-phase with s. ''

_{2}” should be “throughϕ

_{4}. ''

^{iωs}should be e

^{−iωs}in Eq. (A2); after Eq. (A10),α should be; and in Eq. (A11), cosx should be coshx; in Eq. (A7), should be .

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### Abstract

Recent work has focused on deepening our understanding of the molecular origins of the higher harmonics that arise in the shear stress response of polymeric liquids in large-amplitude oscillatory shear flow. For instance, these higher harmonics have been explained by just considering the orientation distribution of rigid dumbbells suspended in a Newtonian solvent. These dumbbells, when in dilute suspension, form the simplest relevant molecular model of polymer viscoelasticity, and this model specifically neglects interactions between the polymer molecules [R. B. Bird et al., “Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response,” J. Chem. Phys. 140, 074904 (2014)]. In this paper, we explore these interactions by examining the Curtiss-Bird model, a kinetic molecular theory designed specifically to account for the restricted motions that arise when polymer chains are concentrated, thus interacting and specifically, entangled. We begin our comparison using a heretofore ignored explicit analytical solution [X.-J. Fan and R. B. Bird, “A kinetic theory for polymer melts. VI. Calculation of additional material functions,” J. Non-Newtonian Fluid Mech. 15, 341 (1984)]. For concentrated systems, the chain motion transverse to the chain axis is more restricted than along the axis. This anisotropy is described by the link tension coefficient, ϵ, for which several special cases arise: ϵ = 0 corresponds to reptation, ϵ > 1/8 to rod-climbing, 1/5 ≤ ϵ ≤ 3/4 to reasonable predictions for shear-thinning in steady simple shear flow, and ϵ = 1 to the dilute solution without hydrodynamic interaction. In this paper, we examine the shapes of the shear stress versus shear rate loops for the special cases , and we compare these with those of rigid dumbbell and reptation model predictions.

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