The concept of a Deborah number is widely used in the study of viscoelastic materials to represent the ratio of a material relaxation time to the time scale of observation and to demarcate transitions between predominantly viscous or elastic material responses. However, this construct does not help quantify the importance of long transients and nonmonotonic stress jumps that are often observed in more complex time-varying systems. Many of these nonintuitive effects are lumped collectively under the term thixotropy; however, no proper nouns are associated with the key phenomena observed in such materials. Thixotropy arises from the ability of a complex structured fluid to remember its prior deformation history, so it is natural to name the dimensionless group representing such behavior with respect to the ability to remember. In Greek mythology, Mnemosyne was the mother of the nine Muses and the goddess of memory. We, thus, propose the definition of a Mnemosyne number as the dimensionless product of the thixotropic time scale and the imposed rate of deformation. The Mnemosyne number is, thus, a measure of the flow strength compared to the thixotropic time scale. Since long transient responses are endemic to thixotropic materials, one also needs to consider the duration of flow. The relevant dimensionless measure of this duration can be represented in terms of a mutation number, which compares the time scale of experiment/observation to the thixotropic time scale. Collating the mutation number and the Mnemosyne number, we can construct a general two-dimensional map that helps understand thixotropic behavior. We quantify these ideas using several of the simplest canonical thixotropic models available in the literature.
REFERENCES
1.
Reiner
, M.
, “The Deborah number
,” Phys. Today
17
(1
), 62
(1964
). 2.
Dealy
, J. M.
, “Weissenberg and Deborah numbers—Their definition and use
,” Rheol. Bull.
79
(2
), 14
–18
(2010
).3.
White
, J. L.
, “Dynamics of viscoelastic fluids, melt fracture, and the rheology of fiber spinning
,” J. Appl. Polym. Sci.
8
(5
), 2339
–2357
(1964
). 4.
Porteous
, K. C.
, and M. M.
Denn
, “Linear stability of plane Poiseuille flow of viscoelastic liquids
,” Trans. Soc. Rheol.
16
(2
), 295
–308
(1972
). 5.
Wagner
, N. J.
, and J. F.
Brady
, “Shear thickening in colloidal dispersions
,” Phys. Today
62
, 27
–32
(2009
). 6.
Barnes
, H. A.
, “Thixotropy—A review
,” J. Non-Newtonian Fluid Mech.
70
(1
), 1
–33
(1997
). 7.
Mewis
, J.
, “Thixotropy—A general review
,” J. Non-Newtonian Fluid Mech.
6
(1
), 1
–20
(1979
). 8.
Mewis
, J.
, and N. J.
Wagner
, “Thixotropy
,” Adv. Colloid Interface Sci.
147–148
, 214
–227
(2009
). 9.
Larson
, R. G.
, “Constitutive equations for thixotropic fluids
,” J. Rheol.
59
(3
), 595
–611
(2015
). 10.
Larson
, R. G.
, and Y.
Wei
, “A review of thixotropy and its rheological modeling
,” J. Rheol.
63
(3
), 477
–501
(2019
). 11.
Cheng
, D. C.
, “Thixotropy
,” Int. J. Cosmet. Sci.
9
(4
), 151
–191
(1987
). 12.
de Souza Mendes
, P. R.
, and R. L.
Thompson
, “Time-dependent yield stress materials
,” Curr. Opin. Colloid Interface Sci.
43
, 15
–25
(2019
). 13.
Agarwal
, M.
, S.
Sharma
, V.
Shankar
, and Y. M.
Joshi
, “Distinguishing thixotropy from viscoelasticity
,” J. Rheol.
65
(4
), 663
–680
(2021
). 14.
Bauer
, W. H.
, and E. A.
Collins
, “Chapter 8—Thixotropy and dilatancy
,” in Rheology
, edited by F. R.
Eirich
(Academic
, New York
, 1967
), pp. 423
–459
.15.
Péterfi
, T.
, “Die abhebung der Befruchtungsmembran bei seeigeleiern
,” Wilhelm Roux'Arch. Entwicklungsmech. Org.
112
(1
), 660
–695
(1927
). 16.
Mewis
, J.
, and N. J.
Wagner
, Colloidal Suspension Rheology
, Cambridge Series in Chemical Engineering (Cambridge University
, New York
, 2012
).17.
Wagner
, N. J.
, and J.
Mewis
, Theory and Applications of Colloidal Suspension Rheology
(Cambridge University
, Cambridge
, 2021
).18.
Schalek
, E.
, and A.
Szegvari
, “Die langsame koagulation konzentrierter eisenoxydsole zu reversiblen gallerten
,” Kolloid-Zeitschrift
33
(6
), 326
–334
(1923
). 19.
Freundlich
, H.
, “Ueber thixotropie
,” Kolloid-Zeitschrift
46
(4
), 289
–299
(1928
).20.
Hauser
, E.
, “Über die thixotropie von dispersionen geringer konzentration
,” Kolloid-Zeitschrift
48
(1
), 57
–62
(1929
). 21.
Coleman
, B. D.
, and W.
Noll
, “Foundations of linear viscoelasticity
,” Rev. Mod. Phys.
33
(2
), 239
–249
(1961
). 22.
Green
, H.
, and R.
Weltmann
, “Analysis of thixotropy of pigment-vehicle suspensions—Basic principles of the hysteresis loop
,” Ind. Eng. Chem., Anal. Ed.
15
(3
), 201
–206
(1943
). 23.
Bird
, R. B.
, and B. D.
Marsh
, “Viscoelastic hysteresis. Part I. Model predictions
,” Trans. Soc. Rheol.
12
(4
), 479
–488
(1968
). 24.
Fredrickson
, A. G.
, Principles and Applications of Rheology
(Prentice-Hall
, Englewood Cliffs
, 1964
).25.
Divoux
, T.
, V.
Grenard
, and S.
Manneville
, “Rheological hysteresis in soft glassy materials
,” Phys. Rev. Lett.
110
(1
), 018304
(2013
). 26.
Wei
, Y.
, M. J.
Solomon
, and R. G.
Larson
, “Letter to the editor: Modeling the nonmonotonic time-dependence of viscosity bifurcation in thixotropic yield-stress fluids
,” J. Rheol.
63
(4
), 673
–675
(2019
). 27.
Varchanis
, S.
, G.
Makrigiorgos
, P.
Moschopoulos
, Y.
Dimakopoulos
, and J.
Tsamopoulos
, “Modeling the rheology of thixotropic elasto-visco-plastic materials
,” J. Rheol.
63
(4
), 609
–639
(2019
). 28.
Ewoldt
, R. H.
, and G. H.
McKinley
, “Mapping thixo-elasto-visco-plastic behavior
,” Rheol. Acta
56
(3
), 195
–210
(2017
). 29.
Eliade
, M.
, “Mythologies of memory and forgetting
,” Hist. Relig.
2
(2
), 329
–344
(1963
). 30.
Mours
, M.
, and H. H.
Winter
, “Time-resolved rheometry
,” Rheol. Acta
33
(5
), 385
–397
(1994
). 31.
Geri
, M.
, B.
Keshavarz
, T.
Divoux
, C.
Clasen
, D. J.
Curtis
, and G. H.
McKinley
, “Time-resolved mechanical spectroscopy of soft materials via optimally windowed chirps
,” Phys. Rev. X
8
(4
), 041042
(2018
). 32.
Pryce-Jones
, J.
, “Experiments on thixotropic and other anomalous fluids with a new rotation viscometer
,” J. Sci. Instrum.
18
(3
), 39
–48
(1941
). 33.
Choi
, J.
, M.
Armstrong
, and S. A.
Rogers
, “The role of elasticity in thixotropy: Transient elastic stress during stepwise reduction in shear rate
,” Phys. Fluids
33
(3
), 033112
(2021
). 34.
Radhakrishnan
, R.
, T.
Divoux
, S.
Manneville
, and S. M.
Fielding
, “Understanding rheological hysteresis in soft glassy materials
,” Soft Matter
13
(9
), 1834
–1852
(2017
). 35.
Jamali
, S.
, R. C.
Armstrong
, and G. H.
McKinley
, “Time-rate-transformation framework for targeted assembly of short-range attractive colloidal suspensions
,” Mater. Today Adv.
5
, 100026
(2020
). 36.
Jamali
, S.
, R. C.
Armstrong
, and G. H.
McKinley
, “Multiscale nature of thixotropy and rheological hysteresis in attractive colloidal suspensions under shear
,” Phys. Rev. Lett.
123
(24
), 248003
(2019
). 37.
Ong
, E. E. S.
, S.
O’Byrne
, and J. L.
Liow
, “Yield stress measurement of a thixotropic colloid
,” Rheol. Acta
58
(6
), 383
–401
(2019
). 38.
Rathinaraj
, J. D. J.
, J.
Hendricks
, G. H.
McKinley
, and C.
Clasen
, “Orthochirp: A fast spectro-mechanical probe for monitoring transient microstructural evolution of complex fluids during shear
,” J. Non-Newtonian Fluid Mech.
301
, 104744
(2022
). 39.
Helal
, A.
, T.
Divoux
, and G. H.
McKinley
, “Simultaneous rheoelectric measurements of strongly conductive complex fluids
,” Phys. Rev. Appl.
6
(6
), 064004
(2016
). 40.
Goodeve
, C. F.
, “A general theory of thixotropy and viscosity
,” Trans. Faraday Soc.
35
, 342
–358
(1939
). 41.
Coussot
, P.
, Q. D.
Nguyen
, H. T.
Huynh
, and D.
Bonn
, “Viscosity bifurcation in thixotropic, yielding fluids
,” J. Rheol.
46
(3
), 573
–589
(2002
). 42.
Bird
, R. B.
et al, Dynamics of Polymeric Liquids, Volume 2: Kinetic Theory
(Wiley
, New York
, 1987
).43.
Dimitriou
, C. J.
, and G. H.
McKinley
, “A comprehensive constitutive law for waxy crude oil: A thixotropic yield stress fluid
,” Soft Matter
10
(35
), 6619
–6644
(2014
). 44.
Armstrong
, M. J.
, A. N.
Beris
, S. A.
Rogers
, and N. J.
Wagner
, “Dynamic shear rheology of a thixotropic suspension: Comparison of an improved structure-based model with large amplitude oscillatory shear experiments
,” J. Rheol.
60
(3
), 433
–450
(2016
). 45.
Geri
, M.
, R.
Venkatesan
, K.
Sambath
, and G. H.
McKinley
, “Thermokinematic memory and the thixotropic elasto-viscoplasticity of waxy crude oils
,” J. Rheol.
61
(3
), 427
–454
(2017
). 46.
de Souza Mendes
, P. R.
, “Thixotropic elasto-viscoplastic model for structured fluids
,” Soft Matter
7
(6
), 2471
–2483
(2011
). 47.
Le-Cao
, K.
, N.
Phan-Thien
, N.
Mai-Duy
, S. K.
Ooi
, A. C.
Lee
, and B. C.
Khoo
, “A microstructure model for viscoelastic–thixotropic fluids
,” Phys. Fluids
32
(12
), 123106
(2020
). 48.
Fredrickson
, A. G.
, “A model for the thixotropy of suspensions
,” AIChE J.
16
(3
), 436
–441
(1970
). 49.
Blackwell
, B. C.
, and R. H.
Ewoldt
, “A simple thixotropic–viscoelastic constitutive model produces unique signatures in large-amplitude oscillatory shear (LAOS)
,” J. Non-Newtonian Fluid Mech.
208–209
, 27
–41
(2014
). 50.
White
, D. E.
, G. D.
Moggridge
, and D.
Ian Wilson
, “Solid–liquid transitions in the rheology of a structured yeast extract paste, Marmite™
,” J. Food Eng.
88
(3
), 353
–363
(2008
). 51.
Mwasame
, P. M.
, A. N.
Beris
, R. B.
Diemer
, and N. J.
Wagner
, “A constitutive equation for thixotropic suspensions with yield stress by coarse-graining a population balance model
,” AIChE J.
63
(2
), 517
–531
(2017
). 52.
Dinh
, S. M.
, and R. C.
Armstrong
, “Non-isothermal channel flow of non-Newtonian fluids with viscous heating
,” AIChE J.
28
(2
), 294
–301
(1982
). 53.
Sen
, S.
, and R. H.
Ewoldt
, “Thixotropic spectra and Ashby-style charts for thixotropy,” arXiv:2201.10004
(2022). 54.
Stadler
, F. J.
et al, “Multiple interval thixotropic test (miTT)—An advanced tool for the rheological characterization of emulsions and other colloidal systems
,” Rheol. Acta
61
, 229
–242
(2022
). 55.
See supplementary material at https://www.scitation.org/doi/suppl/10.1122/8.0000432 for additional results for similar sweeping ramp down/up flow protocols as studied in the manuscript but instead using a Coussot-Bonn viscosity-bifurcating constitutive model.
© 2022 The Society of Rheology.
2022
The Society of Rheology
You do not currently have access to this content.
