A fractional K-BKZ constitutive formulation for describing the nonlinear rheology of multiscale complex fluids

Journal of Rheology

Volumn 58, Issue 6, 2014, Pages 1751-1788

  Jaishankar, Aditya ^a   McKinley, Gareth H ^a  

^a Massachusetts Institute of Technology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPLEX FLUIDS; CONSTITUTIVE FORMULATION; NONLINEAR RHEOLOGY;

EID: 84907101353     PISSN: 01486055     EISSN: 15208516     Source Type: Journal    
DOI: 10.1122/1.4892114     Document Type: Article

References (84)

1. Abramowitz, M., and I. A. Stegun, Handbook of Mathematical Functions (U.S. National Bureau of Standards, Washington, 1964).
2. Bagley, R. L., and P. J. Torvik, " A theoretical basis for the application of fractional calculus to viscoelasticity," J. Rheol. 27, 201-210 (1983a). 10.1122/1.549724
3. Bagley, R. L., and P. J. Torvik, " Fractional calculus-A different approach to the analysis of viscoelastically damped structures," AIAA J. 21, 741-748 (1983b). 10.2514/3.8142
4. Bird, R. B., " Useful non-Newtonian models," Annu. Rev. Fluid Mech. 8, 13-34 (1976). 10.1146/annurev.fl.08.010176.000305
5. Bird, R. B., R. C. Armstrong, and O. Hassager, Dynamics of Polymeric Liquids, 2nd ed. (John Wiley & Sons, New York, 1987).
6. Blair, G. W. S., B. C. Veinoglou, and J. E. Caffyn, " Limitations of the Newtonian time scale in relation to non-equilibrium rheological states and a theory of quasi-properties," Proc. R. Soc. A 189, 69-87 (1947). 10.1098/rspa.1947.0029
7. Booij, H. C., and P. Leblans, " Nonlinear viscoelasticity and the Cox-Merz relations for polymeric fluids," J. Polym. Sci. 21, 1703-1711 (1983). 10.1002/pol.1983.180210910
8. Caputo, M., " Linear models of dissipation whose Q is almost frequency independent-II," Geophys. J. Int. 13, 529-539 (1967). 10.1111/j.1365-246X.1967.tb02303.x
9. Cox, W. P., and E. H. Merz, " Correlation of dynamic and steady flow viscosities," J. Polym. Sci. 28, 619-622 (1958). 10.1002/pol.1958.1202811812
10. Craiem, D., and R. L. Magin, " Fractional order models of viscoelasticity as an alternative in the analysis of red blood cell (RBC) membrane mechanics," Phys. Biol. 7, 13001 (2010). 10.1088/1478-3975/7/1/013001
11. Cuvelier, G., and B. Launay, " Concentration regimes in xanthan gum solutions deduced from flow and viscoelastic properties," Carbohydr. Polym. 6, 321-333 (1986). 10.1016/0144-8617(86)90023-8
12. Dealy, J. M., " Weissenberg and Deborah numbers-their definition and use," Rheol. Bull. 79, 14-18 (2010).
13. Dealy, J. M., and K. F. Wissbrun, Melt Rheology and its Role in Plastics Processing: Theory and Applications (Van Nostrand Reinhold, New York, 1990).
14. Dimitriou, C. J., R. H. Ewoldt, and G. H. McKinley, " Describing and prescribing the constitutive response of yield stress fluids using large amplitude oscillatory shear stress (LAOStress)," J. Rheol. 57, 27-70 (2013). 10.1122/1.4754023
15. Djordjević, V. D., J. Jarić, B. Fabry, J. J. Fredberg, and D. Stamenović, " Fractional derivatives embody essential features of cell rheological behavior," Ann. Biomed. Eng. 31, 692-699 (2003). 10.1114/1.1574026
16. Doraiswamy, D., A. N. Mujumdar, I. Tsao, A. N. Beris, S. C. Danforth, and A. B. Metzner, " The Cox-Merz rule extended: A rheological model for concentrated suspensions and other materials with a yield stress," J. Rheol. 35, 647-685 (1991). 10.1122/1.550184
17. Dzuy, N. Q., " Yield stress measurement for concentrated suspensions," J. Rheol. 27, 321-349 (1983). 10.1122/1.549709
18. English, R. J., S. R. Raghavan, R. D. Jenkins, and S. A. Khan, " Associative polymers bearing n-alkyl hydrophobes: Rheological evidence for microgel-like behavior," J. Rheol. 43, 1175-1194 (1999). 10.1122/1.551026
19. Ewoldt, R. H., and G. H. McKinley, " Creep ringing in rheometry or how to deal with oft-discarded data in step stress tests! " Rheol. Bull. 76, 4-6 (2007).
20. Fabry, B., G. Maksym, J. Butler, M. Glogauer, D. Navajas, and J. Fredberg, " Scaling the microrheology of living cells," Phys. Rev. Lett. 87, 148102 (2001). 10.1103/PhysRevLett.87.148102
21. Friedrich, C., " Relaxation functions of rheological constitutive equations with fractional derivatives: Thermodynamical constraints," in Rheological Modelling: Thermodynamical and Statistical Approaches (Springer-Verlag, Berlin, 1991), pp. 321-330.
22. Friedrich, C., " Rheological material functions for associating comb-shaped or H-shaped polymers. A fractional calculus approach," Philos. Mag. Lett. 66, 287-292 (1992). 10.1080/09500839208219047
23. Friedrich, C., and H. Braun, " Generalized cole-cole behavior and its rheologial relevance," Rheol. Acta 31, 309-322 (1992). 10.1007/BF00418328
24. Friedrich, C., H. Schiessel, and A. Blumen, " Constitutive behavior modeling and fractional derivatives," in Advances in the Flow and Rheology of Non-Newtonian Fluids, edited by D. A. Siginer, D. De Kee, and R. P. Chhabra (Elsevier, New York, 1999), pp. 429-466.
25. Gallegos, C., J. M. Franco, and P. Partal, " Rheology of food dispersions," Rheol. Rev. 2004, 19-65.
26. Glockle, W. G., and T. F. Nonnenmacher, " Fractional integral operators and Fox functions in the theory of viscoelasticity," Macromolecules 24, 6426-6434 (1991). 10.1021/ma00024a009
27. Goh, K. K. T., L. Matia-Merino, C. E. Hall, P. J. Moughan, and H. Singh, " Complex rheological properties of a water-soluble extract from the fronds of the black tree fern, Cyathea medullaris," Biomacromolecules 8, 3414-3421 (2007). 10.1021/bm7005328
28. Guskey, S. M., and H. H. Winter, " Transient shear behavior of a thermotropic liquid crystalline polymer in the nematic state," J. Rheol. 35, 1191-1207 (1991). 10.1122/1.550171
29. Heymans, N., and I. Podlubny, " Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives," Rheol. Acta 45, 765-771 (2005). 10.1007/s00397-005-0043-5
30. Jaishankar, A., and G. H. McKinley, " Power-law rheology in the bulk and at the interface: Quasi-properties and fractional constitutive equations," Proc. R. Soc., A 469, 20120284 (2013). 10.1098/rspa.2012.0284
31. Ketz, R. J., R. K. Prud'homme, and W. W. Graessley, " Rheology of concentrated microgel solutions," Rheol. Acta 27, 531-539 (1988). 10.1007/BF01329353
32. Khan, S. A., C. A. Schnepper, and R. C. Armstrong, " Foam rheology: III. Measurement of shear flow properties," J. Rheol. 32, 69-92 (1988). 10.1122/1.549964
33. Koeller, R. C., " Applications of fractional calculus to the theory of viscoelasticity," J. Appl. Mech. 51, 299-307 (1984). 10.1115/1.3167616
34. Krieger, I., " Correspondence: Comments on a manuscript Doraiswamy et al.," J. Rheol. 36, 215-217 (1992). 10.1122/1.550339
35. Krishnamoorti, R., and E. P. Giannelis, " Rheology of end-tethered polymer layered silicate nanocomposites," Macromolecules 30, 4097-4102 (1997). 10.1021/ma960550a
36. Lapasin, R., and S. Pricl, Rheology of Industrial Polysaccharides: Theory and Applications (Blackie Academic and Professional, Glasgow, 1995).
37. Larson, R. G., " Constitutive relationships for polymeric materials with power-law distributions of relaxation times," Rheol. Acta 24, 327-334 (1985). 10.1007/BF01333961
38. Larson, R. G., Constitutive Equations for Polymer Melts and Solutions (Butterworths, Boston, 1988).
39. Laun, H. M., " Prediction of elastic strains of polymer melts in shear and elongation," J. Rheol. 30, 459-501 (1986). 10.1122/1.549855
40. Leblans, P. J. R., J. Sampers, and H. C. Booij, " The mirror relations and nonlinear viscoelasticity of polymer melts," Rheol. Acta 24, 152-158 (1985). 10.1007/BF01333243
41. Lion, A., " On the thermodynamics of fractional damping elements," Continuum Mech. Thermodyn. 9, 83-96 (1997). 10.1007/s001610050057
42. Ma, L., and G. V. Barbosa-Canovas, " Simulating viscoelastic properties of selected food gums and gum mixtures using a fractional derivative model," J. Texture Stud. 27, 307-325 (1996). 10.1111/j.1745-4603.1996.tb00077.x
43. Macosko, C. W., Rheology Principles, Measurements and Applications (Wiley-VCH, New York, 1994).
44. Mainardi, F., Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models (Imperial College Press, London, 2010).
45. Markovitz, H., " Boltzmann and the beginnings of linear viscoelasticity," Trans. Soc. Rheol. 21, 381-398 (1977). 10.1122/1.549444
46. Mason, T., " Estimating the viscoelastic moduli of complex fluids using the generalized Stokes-Einstein equation," Rheol. Acta 39, 371-378 (2000). 10.1007/s003970000094
47. Mason, T. G., and D. A. Weitz, " Linear viscoelasticity of colloidal hard sphere suspensions near the glass transition." Phys. Rev. Lett. 75, 2770-2773 (1995). 10.1103/PhysRevLett.75.2770
48. Metzler, R., E. Barkai, and J. Klafter, " Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach," Phys. Rev. Lett. 82, 3563-3567 (1999). 10.1103/PhysRevLett.82.3563
49. Metzler, R., and J. Klafter, " The random walk's guide to anomalous diffusion: A fractional dynamics approach," Phys. Rep. 339, 1-77 (2000). 10.1016/S0370-1573(00)00070-3
50. Metzler, R., and J. Klafter, " From stretched exponential to inverse power-law: Fractional dynamics, Cole-Cole relaxation processes, and beyond," J. Non-Cryst. Solids 305, 81-87 (2002). 10.1016/S0022-3093(02)01124-9
51. Metzler, R., W. Schick, H.-G. Kilian, and T. F. Nonnenmacher, " Relaxation in filled polymers: A fractional calculus approach," J. Chem. Phys. 103, 7180-7186 (1995). 10.1063/1.470346
52. Miller, K. S., and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (John Wiley & Sons, Inc., New York, 1993).
53. Muthukumar, M., " Dynamics of polymeric fractals," J. Chem. Phys. 83, 3161-3168 (1985). 10.1063/1.449222
54. Ng, T. S., and G. H. Mckinley, " Power law gels at finite strains: The nonlinear rheology of gluten gels," J. Rheol. 52, 417-449 (2008). 10.1122/1.2828018
55. Nonnenmacher, T. F., " Fractional relaxation equations for viscoelasticity and related phenomena," in Lecture Notes in Physics Rheological Modelling: Thermodynamical and Statistical Approaches (Springer-Verlag, Berlin, 1991), pp. 309-320.
56. Oertel, R., and W. M. Kulicke, " Viscoelastic properties of liquid crystals of aqueous biopolymer solutions," Rheol. Acta 30, 140-150 (1991). 10.1007/BF01134603
57. Oldham, K. B., and J. Spanier, The Fractional Calculus-Theory and Applications of Differentiation and Integration to Arbitrary Order (Academic, New York, 1974).
58. Podlubny, I., Fractional Differential Equations (Academic, San-Diego, 1999).
59. Rees, D. A., and E. J. Welsh, " Secondary and tertiary structure of polysaccharides in solutions and gels," Angew. Chem. 16, 214-224 (1977). 10.1002/anie.197702141
60. Renardy, M., " Qualitative correlation between viscometric and linear viscoelastic functions," J. non-Newtonian Fluid Mech. 68, 133-135 (1997). 10.1016/S0377-0257(96)01466-8
61. Rich, J. P., G. H. McKinley, and P. S. Doyle, " Size dependence of microprobe dynamics during gelation of a discotic colloidal clay," J. Rheol. 55, 273-299 (2011). 10.1122/1.3532979
62. Rochefort, W. E., and S. Middleman, " Rheology of xanthan gum: Salt, temperature, and strain effects in oscillatory and steady shear experiments," J. Rheol. 31, 337-369 (1987). 10.1122/1.549953
63. Rolón-Garrido, V. H., and M. H. Wagner, " The damping function in rheology," Rheol. Acta 48, 245-284 (2009). 10.1007/s00397-008-0308-x
64. Ross-Murphy, S. B., " Structure-property relationships in food biopolymer gels and solutions," J. Rheol. 39, 1451-1463 (1995). 10.1122/1.550610
65. Ross-Murphy, S. B., V. J. Morris, and E. R. Morris, " Molecular viscoelasticity of xanthan polysaccharide," Faraday Symp. Chem. Soc. 18, 115-129 (1983). 10.1039/fs9831800115
66. Sanchez, C., D. Renard, P. Robert, C. Schmitt, and J. Lefebvre, " Structure and rheological properties of acacia gum dispersions," Food Hydrocolloids 16, 257-267 (2002). 10.1016/S0268-005X(01)00096-0
67. Schiessel, H., R. Metzler, A. Blumen, and T. Nonnenmacher, " Generalized viscoelastic models: Their fractional equations with solutions," J. Phys. A Math. Gen. 28, 6567-6584 (1995). 10.1088/0305-4470/28/23/012
68. Sharma, R., and B. J. Cherayil, " Polymer melt dynamics: Microscopic roots of fractional viscoelasticity," Phys. Rev. E 81, 021804 (2010). 10.1103/PhysRevE.81.021804
69. Sharma, V., and G. H. McKinley, " An intriguing empirical rule for computing the first normal stress difference from steady shear viscosity data for concentrated polymer solutions and melts," Rheol. Acta 51, 487-495 (2012). 10.1007/s00397-011-0612-8
70. Sokolov, I. M., J. Klafter, and A. Blumen, " Fractional kinetics," Phys. Today 55 (11), 48-54 (2002). 10.1063/1.1535007
71. Sollich, P., " Rheological constitutive equation for a model of soft glassy materials," Phys. Rev. E 58, 738-759 (1998). 10.1103/PhysRevE.58.738
72. Soskey, P. R., and H. H. Winter, " Large step shear strain experiments with parallel-disk rotational rheometers," J. Rheol. 28, 625-645 (1984). 10.1122/1.549770
73. Surguladze, T., " On certain applications of fractional calculus to viscoelasticity," J. Math. Sci. 112, 4517-4557 (2002). 10.1023/A:1020574305792
74. Tako, M., " Molecular origin for rheological characteristics of xanthan gum," in ACS Symposium Series (1992), Vol. 489, pp. 268-281.
75. Tanner, R. I., and J. M. Simmons, " Combined simple and sinusoidal shearing in elastic liquids," Chem. Eng. Sci. 22, 1803-1815 (1967). 10.1016/0009-2509(67)80211-2
76. Torvik, P. J., and R. L. Bagley, " On the appearance of the fractional derivative in the behavior of real materials," J. Appl. Mech. 51, 294-298 (1984). 10.1115/1.3167615
77. Valério, D., J. J. Trujillo, M. Rivero, J. A. T. Machado, and D. Baleanu, " Fractional calculus: A survey of useful formulas," Eur. Phys. J. Spec. Top. 222, 1827-1846 (2013). 10.1140/epjst/e2013-01967-y
78. Venkataraman, S. K., and H. H. Winter, " Finite shear strain behavior of a crosslinking polydimethylsiloxane near its gel point," Rheol. Acta 29, 423-432 (1990). 10.1007/BF01376793
79. Wagner, M., T. Raible, and J. Meissner, " Tensile stress overshoot in uniaxial extension of a LDPE melt," Rheol. Acta 18, 427-428 (1979). 10.1007/BF01515835
80. Wagner, M. H., and S. E. Stephenson, " The irreversibility assumption of network disentanglement in flowing polymer melts and its effects on elastic recoil predictions," J. Rheol. 23, 489-504 (1979). 10.1122/1.549529
81. Whitcomb, P. J., and C. Macosko, " Rheology of xanthan gum," J. Rheol. 22, 493-505 (1978). 10.1122/1.549485
82. Winter, H. H., and M. Mours, " Rheology of polymers near liquid-solid transitions," in Advances in Polymer Science (Springer-Verlag, Berlin, 1997), Vol. 134, pp. 165-234.
83. Yalpani, M., L. D. Hall, M. A. Tung, and D. E. Brooks, " Unusual rheology of a branched, water-soluble chitosan derivative," Nature 302, 812-814 (1983). 10.1038/302812a0
84. Yang, P., Y. C. Lam, and K.-Q. Zhu, " Constitutive equation with fractional derivatives for the generalized UCM model," J. Non-Newtonian Fluid Mech. 165, 88-97 (2010). 10.1016/j.jnnfm.2009.10.002