No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
oa
Fourier decomposition of polymer orientation in large-amplitude oscillatory shear flow
Abstract
In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear
steady shear flow (where the Deborah number
is zero and the Weissenberg number
is above unity), (ii) nonlinear viscoelasticity (where both
and
exceed unity), and (iii) linear viscoelasticity (where
exceeds unity and where
approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.
© 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License.
Received 04 January 2015
Accepted 24 February 2015
Published online 19 March 2015
Acknowledgments:
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 of the Tier 1 Canada Research Chair in Rheology.
Article outline:
I. INTRODUCTION
A. The flow field
B. Orientation distribution function
II. METHOD
III. RESULTS
A. Zeroth harmonic
B. First harmonic
C. Second harmonic
D. Third harmonic
E. Fourth harmonic
IV. CONCLUSION
/content/aca/journal/sdy/2/2/10.1063/1.4914411
4.
4. J. D. Ferry, Viscoelastic Properties of Polymers, 1st ed. ( John Wiley & Sons, Inc., New York, 1961).
5.
5. J. M. Dealy, Rheometers for Molten Plastics: A Practical Guide to Testing and Property Measurement ( Van Nostrand, New York, 1982).
6.
6. J. M. Dealy and J. Wang, Melt Rheology and its Applications in the Plastics Industry, 2nd ed. ( Springer, Dordrecht, 2013).
8.
8. K. Weissenberg, in Proceedings of First International Congress on Rheology (Holland, 1948/9), pp. 1–29.
9.
9. K. Hyun, M. Wilhelm, C. O. Klein, K. S. Cho, J. G. Nam, K. H. Ahn, S. J. Lee, R. H. Ewoldt, and G. H. McKinley, “ A review of nonlinear oscillatory shear tests: Analysis and application of large amplitude oscillatory shear (LAOS),” Prog. Polym. Sci. 36, 1697–1753 (2011).
http://dx.doi.org/10.1016/j.progpolymsci.2011.02.002
11.
11. A. J. Giacomin, “ A sliding plate melt rheometer incorporating a shear stress transducer,” Ph.D. thesis (Department of Chemical Engineering, McGill University, Montréal, 1987).
12.
12. A. J. Giacomin and J. M. Dealy, “ A new rheometer for molten plastics,” S.P.E. Tech. Papers, XXXII, in Proceedings of 44th Annual Technical Conference, Society of Plastics Engineers, Boston, MA (1986), pp. 711–714.
13.
13. J. E. Roberts, “ The early development of the rheogoniometer,” in Karl Weissenberg: 80th Birthday Celebration Essays, edited by J. Harris ( East African Literature Bureau, Kampala, 1973), pp. 153–163.
15.
15. A. J. Giacomin, R. B. Bird, L. M. Johnson, and A. W. Mix, “ Large-amplitude oscillatory shear flow from the corotational Maxwell model,” J. Non-Newtonian Fluid Mech. 166(19–20), 1081–1099 (2011). Errata: after Eq. (20), Ref. 10 should be Ref. 13; in Eq. (66), “ ” and “ ” should be “ ” and “ ” and so Figs. 15–17 of Ref. (40) below replace Figs. 5–7; on the ordinates of Figs. 5–7, should be 2; after Eq. (119), “ ” should be “ ;” in Eq. (147), “ ” should be “ ;” in Eqs. (76) and (77), and should be and ; throughout, , , and should be , , and ; in Eq. (127), “ ” should be “ ;” after Eq. (136), “Eq. (134)” should be “Eqs. (133) and (135)” and “Eq. (135)” should be “Eqs. (134) and (135)”; after Eq. (143), should be ; in Eqs. (181) and (182), “1,21” should be “1,2”; in Eqs. (184) and (185), “ ” should be “ ;” Eq. (65) should be ; see also Ref. 16 below.
16.
16. A. J. Giacomin, R. B. Bird, L. M. Johnson, and A. W. Mix, ‘ Corrigenda: “Large-amplitude oscillatory shear flow from the corotational Maxwell model,” [J. Non-Newtonian Fluid Mech. 166, 1081–1099 (2011)],’
http://dx.doi.org/10.1016/j.jnnfm.2011.04.002
17.
17. R. B. Bird and R. C. Armstrong, “ Time-dependent flows of dilute solutions of rodlike macromolecules,” J. Chem. Phys. 56, 3680 (1972). Addendum: In Eq. (8), the term should be inserted just before the “+ additional terms.”
http://dx.doi.org/10.1063/1.1677746
18.
18. R. B. Bird, O. Hassager, R. C. Armstrong, and C. F. Curtiss, Dynamics of Polymeric Liquids, 1st ed. ( Wiley, New York, 1977), Vol. 2. Erratum: In Problem 11.C.1 d., “and ” should be “through ;” On p. N-2, under entry “Length of a rigid dumbbell connector” should be “Bead center to center length of a rigid dumbbell.”
21.
21. A. J. Giacomin and J. M. Dealy, “ Using large-amplitude oscillatory shear,” in Rheological Measurement, 2nd ed., edited by A. A. Collyer and D. W. Clegg ( Kluwer Academic Publishers, Dordrecht, Netherlands, 1998), Chap. 11, pp. 327–356.
22.
22. A. J. Giacomin and J. M. Dealy, “ Large-amplitude oscillatory shear,” Techniques in Rheological Measurement, edited by A. A. Collyer ( Chapman and Hall, London, New York, 1993), Chap. 4, pp. 99–121; Kluwer Academic Publishers, Dordrecht (1993), pp. 99–121; Erratum: Corrections to Figs 11.5–11.7 are in Ref. 21 above.
25.
25. O. O. Park, “ Dynamics of rigid and flexible polymer chains. I. Transport through confined geometries,” Ph.D. thesis ( Chemical Engineering, Stanford University, Stanford, CA, 1985).
26.
26. R. B. Bird, H. R. Warner, Jr., and D. C. Evans, “ Kinetic theory and rheology of dumbbell suspensions with Brownian motion,” Adv. Polym. Sci. (Fortschr. Hochpolymeren-Forschung.) 8, 1–90 (1971).
http://dx.doi.org/10.1007/3-540-05483-9_9
27.
27. R. B. Bird, C. F. Curtiss, R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, 2nd ed. ( John Wiley & Sons, Inc., New York, 1987), Vol. 2. Erratum: On p. 409 of the first printing, the in the denominator should be ; In Table 16.4-1, under entry “length of rod” should be “bead center to center length of a rigid dumbbell.” In the Figure 14.1-2 caption, “Multibead rods of length ” should be “Multibead rods of length .”
28.
28. J. G. Kirkwood and R. J. Plock, “ Non-Newtonian viscoelastic properties of rod-like macromolecules in solution,” J. Chem. Phys. 24, 665–669 (1956).
http://dx.doi.org/10.1063/1.1742594
29.
29. J. G. Kirkwood and R. J. Plock, “ Non-Newtonian viscoelastic properties of rod-like macromolecules in solution,” in Macromolecules, edited by P. L. Auer ( Gordon and Breach, New York, 1967), pp. 113–121. Errata: On the left side of Eq. (1) on p. 113, should be . See also Eq. (1) of Ref. 28. In Eq. (2a), should be , and in Eq. (2b), should be . See Eqs. (117a) and (117b) of Ref. 32.
30.
30. R. J. Plock, “ I. non-Newtonian viscoelastic properties of rod-like macromolecules in solution. II. The Debye-Hückel, Fermi-Thomas theory of plasmas and liquid metals,” Ph.D. thesis ( Yale University, New Haven, CT, 1957). Errata: In Eq. (2.4a), should be , and in Eq. (2.4b), should be . See Eqs. (117a) and (117b) of Ref. 32.
31.
31. E. Paul, “ Non-Newtonian viscoelastic properties of rodlike molecules in solution: comment on a paper by Kirkwood and Plock,” J. Chem. Phys. 51, 1271–1290 (1969).
http://dx.doi.org/10.1063/1.1672148
32.
32. E. W. Paul, “ Some non-equilibrium problems for dilute solutions of macromolecules. I. The plane polygonal polymer,” Ph.D. thesis ( Department of Chemistry, University of Oregon, Eugene, OR, 1970).
33.
33. N. A. K. Bharadwaj, “ Low dimensional intrinsic material functions uniquely identify rheological constitutive models and infer material microstructure,” Masters thesis ( Mechanical Engineering, University of Illinois at Urbana-Champaign, IL, 2012).
34.
34. E. W. Paul and R. M. Mazo, “ Hydrodynamic properties of a plane-polygonal polymer, according to Kirkwood-Riseman theory,” J. Chem. Phys. 51, 1102–1107 (1969).
http://dx.doi.org/10.1063/1.1672109
35.
35. C. Y. Mou and R. M. Mazo, “ Normal stress in a solution of a plane-polygonal polymer under oscillating shearing flow,” J. Chem. Phys. 67, 5972–5973 (1977).
http://dx.doi.org/10.1063/1.434774
37.
37. R. B. Bird, A. J. Giacomin, A. M. Schmalzer, and C. Aumnate, “ Dilute rigid dumbbell suspensions in large-amplitude oscillatory shear flow: Shear stress response,” J. Chem. Phys. 140, 074904 (2014); Corrigenda: In Eq. (91), should be . In caption to Fig. 3, “ ” should be “ ” and “ ” should be “ .”
http://dx.doi.org/10.1063/1.4862899
38.
38. A. M. Schmalzer, R. B. Bird, and A. J. Giacomin, “ Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions,” PRG Report No. 002, QU-CHEE-PRG-TR–2014-2, Polymers Research Group, Chemical Engineering Department, Queen's University, Kingston, CANADA, 2014.
39.
39. A. M. Schmalzer, R. B. Bird, and A. J. Giacomin, “ Normal stress differences in large-amplitude oscillatory shear flow for dilute rigid dumbbell suspensions,” J. Non-Newtonian Fluid Mech. (2014); Errata: Above Eqs. (14) and (25), “significant figures” should be “16 significant figures.”
http://dx.doi.org/10.1016/j.jnnfm.2014.09.001
41.
41. A. M. Schmalzer and A. J. Giacomin, “ Orientation in large-amplitude oscillatory shear,” PRG Report No. 005, QU-CHEE-PRG-TR–2014-5, Polymers Research Group, Chemical Engineering Department, Queen's University, Kingston, CANADA, 2014.
42.
42. A. M. Schmalzer, “ Large-amplitude oscillatory shear flow of rigid dumbbell suspensions,” Ph.D. thesis ( University of Wisconsin, Mechanical Engineering Department, Madison, WI, 2014).
43.
43. Y. Bozorgi, “ Multiscale simulation of the collective behavior of rodlike self-propelled particles in viscoelastic fluids,” Ph.D. thesis ( Chemical and Biological Engineering, Rensselaer Polytechnic Institute, Troy, NY, 2014).
45.
45. C. Saengow, A. J. Giacomin, and C. Kolitawong, “ Exact analytical solution for large-amplitude oscillatory shear flow,” PRG Report No. 008, QU-CHEE-PRG-TR–2014-8, Polymers Research Group, Chemical Engineering Dept., Queen's University, Kingston, 2014.
http://aip.metastore.ingenta.com/content/aca/journal/sdy/2/2/10.1063/1.4914411
Article metrics loading...
/content/aca/journal/sdy/2/2/10.1063/1.4914411
2015-03-19
2016-10-27
Abstract
In our previous work, we explored the dynamics of a dilute suspension of rigid dumbbells as a model for polymeric liquids in large-amplitude oscillatory shear flow, a flow experiment that has gained a significant following in recent years. We chose rigid dumbbells since these are the simplest molecular model to give higher harmonics in the components of the stress response. We derived the expression for the dumbbell orientation distribution, and then we used this function to calculate the shear stress response, and normal stress difference responses in large-amplitude oscillatory shear flow. In this paper, we deepen our understanding of the polymer motion underlying large-amplitude oscillatory shear flow by decomposing the orientation distribution function into its first five Fourier components (the zeroth, first, second, third, and fourth harmonics). We use three-dimensional images to explore each harmonic of the polymer motion. Our analysis includes the three most important cases: (i) nonlinear
steady shear flow (where the Deborah number
is zero and the Weissenberg number
is above unity), (ii) nonlinear viscoelasticity (where both
and
exceed unity), and (iii) linear viscoelasticity (where
exceeds unity and where
approaches zero). We learn that the polymer orientation distribution is spherical in the linear viscoelastic regime, and otherwise tilted and peanut-shaped. We find that the peanut-shaping is mainly caused by the zeroth harmonic, and the tilting, by the second. The first, third, and fourth harmonics of the orientation distribution make only slight contributions to the overall polymer motion.
Full text loading...
/deliver/fulltext/aca/journal/sdy/2/2/1.4914411.html;jsessionid=ITBBHv7tkTbz1DBSvfyPCjHG.x-aip-live-02?itemId=/content/aca/journal/sdy/2/2/10.1063/1.4914411&mimeType=html&fmt=ahah&containerItemId=content/aca/journal/sdy
Most read this month
Article
content/aca/journal/sdy
Journal
5
3
true
true
Commenting has been disabled for this content