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Progress and challenges in suspension rheology

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Abstract

Developments in the last century, and especially in the last 50 years, have advanced understanding of suspension rheology greatly. Here, a limited review of suspension work over this period is presented, emphasizing advances over the last three decades in understanding of the particle pressure and strong shear thickening, which were motivated by crucial experimental observations, computational advances, and a critical review, all from the 1980s. This review serves as a preview to some outstanding challenges in suspension mechanics. This article considers primarily dispersions of spherical particles, which serve not only as a model material for understanding the rheology of more complex fluids of practical relevance, but also as a basic system for the study of nonequilibrium statistical physics.

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References

  • Alder BJ, Wainwright TE (1957) Phase transition for a hard sphere system. J Chem Phys 27:1208–1209

    CAS  Google Scholar 

  • Bagnold RA (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc R Soc Lond A 225:49–63

    Google Scholar 

  • Ball RC, Melrose JR (1995) Lubrication breakdown in hydrodynamic simulations of concentrated colloids. Adv Colloid Interface Sci 59:19–30

    CAS  Google Scholar 

  • Banchio AJ, Brady JF (2003) Accelerated Stokesian Dynamics: Brownian motion. J Chem Phys 118:10323–10332

    CAS  Google Scholar 

  • Banetta L, Zaccone A (2019) Radial distribution function of Lennard-Jones fluids in shear flows from intermediate asymptotics. Phys Rev E 99:052606

    CAS  Google Scholar 

  • Barnes HA (1989) Shear-thickening (“dilatancy”) in suspensions of nonaggregating solid particles dispersed in Newtonian liquids. J Rheol 33:329–366

    CAS  Google Scholar 

  • Batchelor GK (1970) The stress system in a suspension of force-free particles. J Fluid Mech 41:545–570

    Google Scholar 

  • Batchelor GK (1977) The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J Fluid Mech 83:97–117

    Google Scholar 

  • Batchelor GK, Green JT (1972a) The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J Fluid Mech 56:375–400

    Google Scholar 

  • Batchelor GK, Green JT (1972b) The determination of the bulk stress in a suspension of spherical particles to order c2. J Fluid Mech 56:401–427

    Google Scholar 

  • Bender J, Wagner NJ (1996) Reversible shear thickening in monodisperse and bidisperse colloidal dispersions. J Rheol 40:899–916

    CAS  Google Scholar 

  • Bingham E, Robertson J (1929) Eine methode zur gleichzeitigen messing von plastizität und elastizität. Kolloid-Zeitschrift 47:1–5

    CAS  Google Scholar 

  • Boersma WH, Laven J, Stein HN (1990) Shear thickening (dilatancy) in concentrated dispersions. AIChE J 36:321–332

    CAS  Google Scholar 

  • Boersma WH, Baets PJM, Laven J, Stein HN (1991) Time-dependent behavior and wall slip in concentrated shear thickening dispersions. J Rheol 35:1093–1120

    CAS  Google Scholar 

  • Bossis G, Brady JF (1984) Dynamic simulation of sheared suspensions. I. General method. J Chem Phys 80:5141–5154

    CAS  Google Scholar 

  • Boyer F, Pouliquen Q, Guazzelli E (2011) Dense suspension in rotating-rod flows: normal stress and particle migration. J Fluid Mech 686:5–25

    Google Scholar 

  • Brader JM, Cates ME, Fuchs M (2008) First-principles constitutive equation for suspension rheology. Phys Rev Lett 101:138301

    CAS  Google Scholar 

  • Brady JF (1993) Brownian motion, hydrodynamics, and the osmotic pressure. J Chem Phys 98:3335–3341

    CAS  Google Scholar 

  • Brady JF, Bossis G (1985) The rheology of concentrated suspensions of spheres in simple shear flow by numerical simulation. J Fluid Mech 155:105–129

    Google Scholar 

  • Brady JF, Bossis G (1988) Stokesian Dynamics. Ann Rev Fluid Mech 20:111–157

    Google Scholar 

  • Brady JF, Morris JF (1997) Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J Fluid Mech 348:103–139

    CAS  Google Scholar 

  • Brady JF, Vicic M (1995) Normal stresses in colloidal dispersions. J Rheol 39(3):545–566

    CAS  Google Scholar 

  • Chu HC, Zia RN (2019) Toward a nonequilibrium Stokes-Einstein relation via active microrheology of hydrodynamically interacting colloidal dispersions. J Colloid Interface Sci 539:388–399

    CAS  Google Scholar 

  • Cloitre M, Bonnecaze RT (2017) A review on wall slip in high solid dispersions. Rheol Acta 56:283–305

    CAS  Google Scholar 

  • Cwalina CD, Wagner NJ (2014) Material properties of the shear-thickened state in concentrated near hard-sphere colloidal dispersions. J Rheol 58:949–967

    CAS  Google Scholar 

  • Da Cunha FR, Hinch EJ (1996) Shear-induced dispersion in a dilute suspension of rough spheres. J Fluid Mech 309:211–223

    Google Scholar 

  • Dasan J, Ramamohan T, Singh A, Nott PR (2002) Stress fluctuations in sheared Stokesian suspensions. Phys Rev E 66:021409

    CAS  Google Scholar 

  • Deboeuf A, Gauthier G, Martin J, Yurkovetsky Y, Morris JF (2009) Particle pressure in a sheared suspension: a bridge from osmosis to granular dilatancy. Phys Rev Lett 102:108301

    Google Scholar 

  • Drazer G, Koplik J, Khusid B, Acrivos A (2002) Deterministic and stochastic behaviour of non-Brownian spheres in sheared suspensions. J Fluid Mech 460:307–335

    CAS  Google Scholar 

  • Einstein A (1906) Eine neue bestimmun der molekuldimensionen. Ann Phys 19:289–306

    CAS  Google Scholar 

  • Frank M, Anderson D, Weeks ER, Morris JF (2003) Particle migration in pressure-driven flow of a Brownian suspension. J Fluid Mech 493:363–378

    Google Scholar 

  • Freundlich H, Röder HL (1938) Dilatancy and its relation to thixotropy. Trans Faraday Soc 34:308–316

    CAS  Google Scholar 

  • Gadala-Maria F (1979) The rheology of concentrated suspensions. PhD thesis, Stanford University

  • Gadala-Maria F, Acrivos A (1980) Shear-induced structure in a concentrated suspension of solid spheres. J Rheol 24:799–814

    CAS  Google Scholar 

  • Gamonpilas C, Morris JF, Denn MM (2016) Shear and normal stress measurements in non-Brownian monodisperse and bidisperse suspensions. J Rheol 60:289–296

    CAS  Google Scholar 

  • Garland S, Gauthier G, Martin J, Morris JF (2013) Normal stress measurements in sheared non-Brownian suspensions. J Rheol 57:71–88

    CAS  Google Scholar 

  • Gurnon AK, Wagner NJ (2015) Microstructure and rheology relationships for shear thickening colloidal dispersions. J Fluid Mech 769:242–276

    CAS  Google Scholar 

  • Guy BM, Hermes M, Poon WCK (2015) Towards a unified description of the rheology of hard-particle suspensions. Phys Rev Lett 115:088304

    CAS  Google Scholar 

  • Guy BM, Richards JA, Hodgson DJM, Blanco E, Poon WCK (2018) Constraint-based approach to granular dispersion rheology. Phys Rev Lett 121(12):128001

    CAS  Google Scholar 

  • Guy BM, Ness C, Hermes M, Sawiak LJ, Sun J, Poon WC (2020) Testing the Wyart-Cates model for non-Brownian shear thickening using bidisperse suspensions. Soft Matter 16:229–237

    CAS  Google Scholar 

  • Hanley H, Rainwater J, Hess S (1987) Shear-induced angular dependence of the liquid pair correlation function. Phys Rev A 36:1795

    CAS  Google Scholar 

  • Happel J, Brenner H (2012) Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media. Springer, Berlin

    Google Scholar 

  • Heussinger C (2013) Shear thickening in granular suspensions: inter-particle friction and dynamically correlated clusters. Phys Rev E 88:050201

    Google Scholar 

  • Hoffman RL (1972) Discontinuous and dilatant viscosity behavior in concentrated suspensions. I. Observation of a flow instability. Trans Soc Rheol 16:155–173

    CAS  Google Scholar 

  • Hsiao LC, Jamali S, Glynos E, Green PF, Larson RG, Solomon MJ (2017) Rheological state diagrams for rough colloids in shear flow. Phys Rev Lett 119:158001

    Google Scholar 

  • Hsu C-P, Ramakrishna SN, Zanini M, Spencer ND, Isa L (2018) Roughness-dependent tribology effects on discontinuous shear thickening. Proc Nat Acad Sci 115:01066

    Google Scholar 

  • Jeffrey DJ (1992) The calculation of the low Reynolds number resistance functions for two unequal spheres. Phys Fluids A 4:16–29

    CAS  Google Scholar 

  • Jeffrey DJ, Onishi Y (1984) Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J Fluid Mech 139:261–290

    Google Scholar 

  • Jeffrey DJ, Morris JF, Brady JF (1993) The pressure moments for two rigid spheres in low-Reynolds-number flow. Phys Fluids A 5:2317–2325

    CAS  Google Scholar 

  • Jenkins JT, McTigue DF (1990) In: eds D. D. Joseph, Schaeffer, D.G. (eds.) Transport processes in concentrated suspensions: the role of particle fluctuations. Springer, New York

  • Kaldasch J, Senge B (2009) Shear thickening in polymer stabilized colloidal suspensions. Colloid Polym Sci 287:1481–1485

    CAS  Google Scholar 

  • Khan M, More RV, Banaei AA, Brandt L, Ardekani AM (2023) Rheology of concentrated fiber suspensions with a load-dependent friction coefficient. Phys Rev Fluids 8:044301

    Google Scholar 

  • Krieger IM, Dougherty TJ (1959) A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans Soc Rheol 3:137–152

    CAS  Google Scholar 

  • Ladd AJ (1988) Hydrodynamic interactions in a suspension of spherical particles. J Chem Phys 88:5051–5063

    CAS  Google Scholar 

  • Le AVN, Izzet A, Ovarlez G, Colin A (2023) Solvents govern rheology and jamming of polymeric bead suspensions. J Colloid Interface Sci 629:438–450

    Google Scholar 

  • Lee Y-F, Luo Y, Brown SC, Wagner NJ (2020) Experimental test of a frictional contact model for shear thickening in concentrated colloidal suspensions. J Rheol 64:267–282

    CAS  Google Scholar 

  • Leighton DT, Acrivos A (1987) The shear-induced migration of particles in concentrated suspensions. J Fluid Mech 181:415–439

    CAS  Google Scholar 

  • Lemaire E, Blanc F, Claudet C, Gallier S, Lobry L, Peters F (2023) Rheology of non-Brownian suspensions: a rough contact story. Rheol Acta 62:253–268

  • Lin NYC, Guy BM, Hermes M, Ness C, Sun J, Poon WCK, Cohen I (2015) Hydrodynamic and contact contributions to continuous shear thickening in colloidal suspensions. Phys Rev Lett 115:228304

    Google Scholar 

  • Lin NY, Ness C, Cates ME, Sun J, Cohen I (2016) Tunable shear thickening in suspensions. Proc Nat Acad Sci 113:10774–10778

    CAS  Google Scholar 

  • Lootens D, van Damme H, Hémar Y, Hébraud P (2005) Dilatant flow of concentrated suspensions of rough particles. Phys Rev Lett 95:268302

    Google Scholar 

  • Lyon M, Leal L (1998) An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 2. Bidisperse systems. J Fluid Mech 363:57–77

    CAS  Google Scholar 

  • Malbranche N, Chakraborty B, Morris JF (2023) Shear thickening in dense bidisperse suspensions. J Rheol 67:91–104

    CAS  Google Scholar 

  • Malkin AY, Kulichikhin VG (2015) Structure and rheology of highly concentrated emulsions: a modern look. Russ Chem Reviews 84:803–825

    CAS  Google Scholar 

  • Malkin AY, Masalova I, Slatter P, Wilson K (2004) Effect of droplet size on the rheological properties of highly-concentrated w/o emulsions. Rheol Acta 43:584–591

    CAS  Google Scholar 

  • Maranzano BJ, Wagner NJ (2001) The effects of particle size on reversible shear thickening of concentrated colloidal dispersions. J Chem Phys 114:10514–10527

    CAS  Google Scholar 

  • Mari R, Seto R, Morris JF, Denn MM (2014) Shear thickening, frictionless and frictional rheologies in non-Brownian suspensions. J Rheol 58:1693–1724

    CAS  Google Scholar 

  • Mari R, Seto R, Morris JF, Denn MM (2015a) Discontinuous shear thickening in Brownian suspensions by dynamic simulation. Proc Natl Acad Sci USA 112:15326–15330

    CAS  Google Scholar 

  • Mari R, Seto R, Morris JF, Denn MM (2015b) Nonmonotonic flow curves of shear thickening suspensions. Phys Rev E 91:052302

    Google Scholar 

  • Maron SH, Pierce PE (1956) Application of Ree-Eyring generalized flow theory to suspensions of spherical particles. J Colloid Sci 11:80–95

    CAS  Google Scholar 

  • Miller RM, Morris JF (2006) Normal stress-driven migration and axial development in pressure-driven flow of concentrated suspensions. J Non-Newtonian Fluid Mech 135:149–165

    CAS  Google Scholar 

  • Miller RM, Singh JP, Morris JF (2009) Suspension flow modeling for general geometries. Chem Eng Sci 64:4597–4610

    CAS  Google Scholar 

  • Monsorno D, Varsakelis C, Papalexandris M (2017) Poiseuille flow of dense non-colloidal suspensions: the role of intergranular and nonlocal stresses in particle migration. J Non-Newtonian Fluid Mech 247:229–238

    CAS  Google Scholar 

  • Morris JF (2018) The lubricated-to-frictional shear thickening scenario in dense suspensions. Phys Rev Fluids 3:110508

    Google Scholar 

  • Morris JF (2020a) Shear thickening of concentrated suspensions: recent developments and relation to other phenomena. Ann Rev Fluid Mech 52:121–144

    Google Scholar 

  • Morris JF (2020b) Toward a fluid mechanics of suspensions. Phys Rev Fluids 5:110519

    Google Scholar 

  • Morris JF, Boulay F (1999) Curvilinear flows of noncolloidal suspensions: the role of normal stresses. J Rheol 43:1213–1237

    CAS  Google Scholar 

  • Morris JF, Brady JF (1998) Pressure-driven flow of a suspension: buoyancy effects. Int J Multiphase Flow 24:105–130

    CAS  Google Scholar 

  • Morris JF, Katyal B (2002) Microstructure from simulated Brownian suspension flows at large shear rate. Phys Fluids 14:1920–1937

    CAS  Google Scholar 

  • Nazockdast E, Morris JF (2012) Microstructural theory and the rheology of concentrated colloidal suspensions. J Fluid Mech 713:420–452

    Google Scholar 

  • Ness C (2021) Simulating dense non-Brownian suspension rheology using LAMMPS. arXiv preprint arXiv:2108.04606

  • Nott PR, Brady JF (1994) Pressure-driven flow of suspensions: simulation and theory. J Fluid Mech 275:157–199

    CAS  Google Scholar 

  • Papanikolaou S, O’Hern CS, Shattuck MD (2013) Isostaticity at frictional jamming. Phys Rev Lett 110:198002

    Google Scholar 

  • Parsi F, Gadala-Maria F (1987) Fore-and-aft asymmetry in a concentrated suspension of solid spheres. J Rheol 31:725–732

    CAS  Google Scholar 

  • Pednekar S, Chun J, Morris JF (2018) Bidisperse and polydisperse suspension rheology at large solid fraction. J Rheol 62:513–526

    CAS  Google Scholar 

  • Phillips RJ, Armstrong RC, Brown RA, Graham AL, Abbott JR (1992) A constitutive equation for concentrated suspensions that accounts for shear-induced particle migration. Phys Fluids A 4:30–40

    CAS  Google Scholar 

  • Prasad D, Kytömaa HK (1995) Particle stress and viscous compaction during shear of dense suspensions. Int J Multiphase Flow 21:775–785

    CAS  Google Scholar 

  • Rampall I, Smart JR, Leighton DT (1997) The influence of surface roughness on the particle-pair distribution function of dilute suspensions of non-colloidal spheres in simple shear flow. J Fluid Mech 339:1–24

    CAS  Google Scholar 

  • Roussel N, Lemaître A, Flatt RJ, Coussot P (2010) Steady state flow of cement suspensions: a micromechanical state of the art. Cem Concr Res 40:77–84

    CAS  Google Scholar 

  • Russel WB, Saville DA, Schowalter WR (1995) Colloidal Dispersions. Cambridge University Press, New York

    Google Scholar 

  • Scacchi A, Krüger M, Brader JM (2016) Driven colloidal fluids: construction of dynamical density functional theories from exactly solvable limits. J Phys Cond Matt 28:244023

    Google Scholar 

  • Semwogerere D, Weeks ER (2008) Shear-induced particle migration in binary colloidal suspensions. Phys Fluids 20:043306

    Google Scholar 

  • Seth JR, Mohan L, Locatelli-Champagne C, Cloitre M, Bonnecaze RT (2011) A micromechanical model to predict the flow of soft particle glasses. Nat Mater 10:838–843

    CAS  Google Scholar 

  • Seto R, Mari R, Morris JF, Denn MM (2013) Discontinuous shear thickening of frictional hard-sphere suspensions. Phys Rev Lett 111:218301

    Google Scholar 

  • Singh A, Mari R, Denn MM, Morris JF (2018) A constitutive model for simple shear of dense frictional suspensions. J Rheol 62:457–468

    CAS  Google Scholar 

  • SOR Nomenclature (2013) Official symbols and nomenclature of the Society of Rheology. J Rheol 57:1047–1055

    Google Scholar 

  • Thomas DG (1965) Transport characteristics of suspension: VIII. A note on the viscosity of Newtonian suspensions of uniform spherical particles. J Colloid Sci 20:267–277

    CAS  Google Scholar 

  • Vollebregt H, van der Sman R, Boom R (2010) Suspension flow modelling in particle migration and microfiltration. Soft Matter 6:6052–6064

    CAS  Google Scholar 

  • Wallis GB (1969) One-dimensional Two-phase Flow. McGraw-Hill, New York

    Google Scholar 

  • Wang M, Jamali S, Brady JF (2020) A hydrodynamic model for discontinuous shear-thickening in dense suspensions. J Rheol 64:379–394

    CAS  Google Scholar 

  • Wilson HJ (2005) An analytic form for the pair distribution function and rheology of a dilute suspension of rough spheres in plane strain flow. J Fluid Mech 534:97–114

    Google Scholar 

  • Wyart M, Cates ME (2014) Discontinuous shear thickening without inertia in dense non-Brownian suspensions. Phys Rev Lett 112:098302

    CAS  Google Scholar 

  • Yeo K, Maxey MR (2011) Numerical simulations of concentrated suspensions of monodisperse particles in a Poiseuille flow. J Fluid Mech 682:491–518

    Google Scholar 

  • Yurkovetsky Y, Morris JF (2008) Particle pressure in sheared Brownian suspensions. J Rheol 52:141–164

    CAS  Google Scholar 

  • Zarraga IE, Hill DA, Leighton DT (2000) The characterization of the total stress of concentrated suspensions of noncolloidal spheres in Newtonian fluids. J Rheol 44:185–220

    CAS  Google Scholar 

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Acknowledgements

The research described here was in large measure the accomplishment of students and collaborators with whom I have been fortunate to work. For my start in suspensions and the push in the direction of their statistical physics, I owe John Brady (Caltech) a debt of gratitude, and for our many years of deeply stimulating work together, I owe a similar debt to Élisabeth Guazzelli (CNRS). The lecture upon which this work is based was dedicated in memoriam to James Swan and Yevgeny Yurkovetsky, two rheologists from whom I learned a great deal and who passed away far too soon.

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The preparation of this paper was supported by NSF CBET-2228680.

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Morris, J.F. Progress and challenges in suspension rheology. Rheol Acta 62, 617–629 (2023). https://doi.org/10.1007/s00397-023-01421-z

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