Consititutive Modeling of Time-Dependent Flows in Frictional Suspensions

EPJ Web of Conferences 249, 01002 (2021) https://doi.org/10.1051/epjconf/202124901002 Powders and Grains 2021

Consititutive modeling of time-dependent flows in frictional suspensions

1 Michael Cates 1DAMTP, Centre for Mathematical Sciences, University of Cambridge, CB3 0WA, United Kingdom

Abstract. This paper summarizes recent joint work towards a constitutive modelling framework for dense granular suspensions. The aim is to create a time-dependent, tensorial theory that can implement the physics described in steady state by the Wyart-Cates model. This model of shear thickening suspensions supposes that lubrication films break above a characteristic normal force so that frictional contact forces come into play: the resulting non-sliding constraints can be enough to rigidify a system that would flow freely at lower stresses [1]. Implementing this idea for time-dependent flows requires the introduction of new concepts including a configuration-dependent ‘jamming coordinate’, alongside a decomposition of the velocity gradient tensor into compressive and extensional components which then enter the evolution equation for particle contacts in distinct ways. The resulting approach [2, 3] is qualitatively successful in addressing (i) the collapse of stress during flow reversal in shear flow, and (ii) the ability of transverse oscillatory flows to unjam the system. However there is much work required to refine this approach towards quantitative accuracy, by incorporating more of the physics of contact evolution under flow as determined by close interrogation of particle-based simulations.

1 Introduction 2 Wyart-Cates Theory Recent years have seen a shift in our understanding of the In the absence of friction, or if the interparticle fric- rheology of dense suspension of hard, non-Brownian par- tion is linear (i.e., Coulombic), dense hard-core suspen- ticles in a Newtonian solvent. Naive application of Stoke- sions without Brownian motion or inertia are ‘quasi- sian physics would imply that such particles, if spherical Newtonian’. In simple shear flow all stresses are then lin- and governed by a no-slip boundary condition, never come ear in the strain-rate γ˙ ≡ dγ/dt. This includes normal into direct contact because the force needed to create a fi- stresses, hence the ‘quasi’: such stresses vanish only in nite interparticle normal velocity diverges as the fluid film the pure Stokes limit to give a truly Newtonian fluid. The between them becomes thin [4]. However, this is an ap- quasi-Newtonian outcome is readily established by dimen- proximation too far: real particles have aspherities and a sional analysis given that the physics of the problem does finite (nanometre scale) slip length, and therefore suspen- not admit an intrinsic time scale (no Brownian motion or sions do contain direct interparticle contacts. In a Stoke- inertia) or stress scale (hard-core forces are scale-free). sian system, sudden flow reversal changes the sign but not Now suppose that there is a soft-core repulsion which the magnitude of the shear stress. In reality, the magnitude is overcome at large normal interparticle forces. One can drops discontinuously before slowly recovering to steady then consider two quasi-Newtonian branches, a friction- ∗ state, because the direct contact forces that are present less one occupied at low stress σ ≤ b1σ , say, and a fric- ∗ carry compressive but not tensile forces (unlike a lubri- tional one at high stress, σ ≥ b2σ where b1 < b2 and both cation film). This was demonstrated long ago in the ex- are of order unity. Here σ∗ is stress scale set by the soft- periments of Gadala-Maria and Acrivos [5] on flow rever- core forces, the particle volume fraction φ, and the mean sal, and recently confirmed in forensic detail by Lin et. particle radius. Crucially, as the number density increases, al. [6]. Although it is possible to recover some aspects the viscosity η2(φ) of the frictional branch diverges at a J J of the Stokesian picture by replacing the word ‘contact’ in jamming point φ2 that is lower than the jamming point φ1 the above discussion with a more general concept involv- of the frictionless branch whose viscosity is η1(φ). Hence J ing intact lubrication films and explicit aspherities [7], we for φ approaching φ2 from below, η2 η1. Accordingly, stick with the simpler picture of direct particle contacts. the underlying steady-state flow curve has an unavoidably Once these contacts are present, one has essentially a sigmoidal shape as shown in figure 1 [1]. The region of wet granular system and friction enters the stage with a negative slope is unstable so that at some stress at or below ∗ central role. To people familiar with granular systems this b1σ there will be a discontinuous jump in the observed may appear obvious, yet the idea that friction matters has stress σ(γ˙) from low to high values. This is discontinu- only recently achieved consensus in the dense suspension ous shear thickening (DST). Backing off the density so the community through a combination of numerical (e.g. [8]), flow curve is no longer sigmoidal, one still expects a very experimental (e.g. [6, 9]) and theoretical (e.g. [1]) studies large, but now smooth increase in viscosity on traversing that are by now already too numerous for a full survey. the crossover, giving continuous shear thickening (CST). A video is available at https://doi.org/10.48448/1ztk-h896 © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). EPJ Web of Conferences 249, 01002 (2021) https://doi.org/10.1051/epjconf/202124901002 Powders and Grains 2021

bution, which we describe by a microstructure (fabric) tensor hnni created from the vectors n between neighbours. These are taken to be interparticle vectors that lie within some coarse-graining shell, thin compared to the particle radius a and thick compared to the range of any direct interparticle force F(h) with h the interparticle separation. Specifically, the compressive strain rate Ec advects, into the coarse-graining shell, previously non-contacting particle pairs whose separation-vector distribution can be modelled as isotropic. It imports preferentially contacts Figure 1. The core insight of the Wyart-Cates theory [1]. For along the compressive axis (or axes). Conversely the densities just below the frictional jamming point φJ , the slope 1 extensional strain rate E advects existing particle pairs, of the upper (frictional) branch of the flow curve is vastly larger e whose separation vectors are anisotropically distributed, than that of the lower (frictionless) branch. Thus any reasonable stress-dependent friction law will create a sigmoidal curve. out of the coarse graining shell. It exports preferentially those oriented along the extensional axis (or axes). Combining these ideas with a standard advective model for the n vectors, Gillissen and Wilson were thereby A simple quantification of these ideas is to choose led to the following equation for the fabric tensor [13, 14]: J −2 η = ηsν(φ − φ) with ηs the solvent viscosity and ν some constant, and then set T ∂thnni = L · hnni + hnni · L − 2L : hnnnni ∗ φ φJ( f ) = φJ(1 − f ) + φJ f ; f (Π) = exp(−Π /Π) (1) − β E : hnnnni + (2E + Tr(E )δ) (2) 1 2 e 15 c c Here f (Π) is the fraction of contacts at which friction is with δ the unit tensor. (The precise prefactor of the φ term mobilized enough to enforce a rolling rather than a sliding is inessential and can be absorbed by a rescaling [2].) motion. This is written here as a function of the particle To complete their model, Gillissen and Wilson adopted pressure Π = −TrΣ/3, with Σ the particle stress tensor and ∗ ∗ a standard Hinch-Leal closure [15] to express hnnnni in Π of order σ , although a similar dependence on the shear terms of hnni – a choice inspired by the fact that the fab- stress σ = Σxy would be sufficient for pure shear flows. ric should be almost isotropic close to jamming [13, 14]. The Wyart-Cates theory captures much of the phe- Finally they proposed an instantaneous dependence of the nomenology of shear thickening in dense suspensions [10] particle stress on microstructure and strain rate as and it has also been extended to allow for Brownian motion [11], interparticle adhesion [12] and other effects. Σ = ηs αE + χEc : hnnnni (3) However a fundamental limitation is that it is a steady- state theory only. In this equation the α term encodes lubrication forces and is therefore linear in E. The χ term represents direct interparticle forces comprising any soft repulsion F(h), hard- 3 Gillissen-Wilson Theory core terms, and friction; this neglects tangential forces Let us set aside the physics of shear thickening, and con- which are known to be subdominant when calculating the sider a quasi-Newtonian dense suspension belonging to stress [16]. (Note that this choice does not preclude a cru- (say) the frictional branch – effectively setting σ∗ = 0 in cial role for tangential forces, such as friction, in control- figure 1. How can one construct a constitutive model of ling the jamming point.) Because Ec and Ee interchange the time-dependent rheology of such a material? If this on flow reversal, this contact term can capture the sudden question is answered, we can hope to include the concept loss in stress when compressive contacts suddenly open of stress-dependent friction afterwards. And, of course, up. In contrast the lubrication term changes sign at fixed there are many dense suspensions that are indeed quasi- magnitude as required for purely Stokesian physics. Newtonian rather than shear thickening, for which such a model can be directly useful [9]. 4 Shear Thickening Suspensions The problem is that a direct particle contact, unlike a lubrication film, is a highly nonlinear object: it supports A marriage of the Wyart-Cates and the Gillissen-Wilson compressive stress but not tensile ones. This issue is of theory was proposed in [2]. To achieve this one should al- course familiar to those working on granular matter. A low the stress parameter χ in (3) to depend on f , the frac- key insight, however, is that this nonlinearity is so ex- tion of contacts that are frictional. In this context, we can treme it can be treated like a switch [13]. At the level of retain the choice of f (φ) in (1) but can no longer pretend macroscopic constitutive equations, this idea is encoded that the viscosity depends only on φ − φJ( f ) as assumed by a decomposition of the macroscopic rate of strain ten- there. Such a form makes sense for steady states but in un- T sor E = (L + L )/2, where Li j = ∂ jvi (with v the fluid ve- steady flow the structure undergoes changes at fixed vol- locity) is the velocity gradient tensor, into its compressive ume fraction and the viscosity clearly depends on these and extensile components, E = Ec + Ee. These have dis- changes. We thus need a scalar function of hnni (which tinct effects on the evolution of the particle contact distri- can also involve the flow) to describe the distance from

2 EPJ Web of Conferences 249, 01002 (2021) https://doi.org/10.1051/epjconf/202124901002 Powders and Grains 2021

jamming in the way φ − φJ( f ) does for steady states. We 103 call this the ‘jamming coordinate’. Proximity to jamming is linked to the number of 102 coarse-grained contacts that are close to becoming frictional. Such contacts are the subset that lie within the 1 range of the soft-core interactions; most of these are ori- 10 ented along the compression axis or axes. In [2] we proposed on this basis the following jamming coordinate: 100 0 1 2 3 4 hnni : E ξ ≡ − √ c (4) Ec : Ec 103 and showed that, in particle simulations, ξ evolves simi- 2 larly to a coordination number Z that counts direct (h < ) 10 contacts only. (The latter might be an equally good choice for the jamming coordinate, but it cannot help us close the 1 description at coarse-grained level until approximated by 10 something like ξ.) Replacing (1) with 0 J J J ∗ 10 ξ ( f ) = (1 − f )ξ1 + f ξ2 , f (Π) = exp(−Π /Π) (5) 0 1 2 3 4

J the extremal jamming coordinates ξ1,2 can be found from J φ1,2 by solution of the fabric evolution equations [2]. Figure 2. Upper panel: reduced viscosity as a function of strain To complete the picture, we replace the viscosity law after ﬂow reversal of simple shear for various strain rates, from with a similar divergence, on the approach to jamming, of the constitutive model of [2]. Lower panel: corresponding data the non-hydrodynamic stress coeﬃcient in (3): from particle-based simulations. See [2] for further details.

J−2 χ = χ0 1 − ξ/ξ (6)

Here the exponent −2 is retained for simplicity, with the would be [17, 18]. Interestingly a transverse oscillation empirical support of particle-based simulations [2]. is much more eﬀective than superposing an oscillation in the same direction as the main ﬂow. The unjamming ef- 5 Results fect is strongly dependent on friction, and indeed barely detectable when non-frictional materials (σ∗ → ∞) are

For reversing shear flows (Li j = ±γδ˙ iyδ jx) we define a simulated; but there is no requirement that the material be reduced viscosity ηr = Σxy/(˙γηs). Sample results from shear-thickening [18]. The transverse vibration, by open- the model are shown in figure 2 alongside corresponding ing contacts, is broadly similar in its effects to a reduction results for particle-based simulations at various reduced in the volume fraction φ. ∗ shear ratesγ ˙r ≡ γη˙ s/Π . These results use α = 120 and The model of [2], as outlined above, is used in [3] χ0 = 2.4 which are fitted using steady-state flow curve to address such steady shear flows with superposed trans- data. Those fits are done with β = 50 which is in turn fit- verse oscillations. That paper also provides a more mi- ted to the simulated reversal data in figure 2 itself, to match croscopic derivation of the governing equations and intro- the observed strain scale for recovery of the stress after re- duces slight technical modifications, without changing the ∗ J versal. Other parameters are φ, Π , and φ1,2, with the latter conceptual basis of the model. For instance α, which can J J fixing χ via matching of the steady-state solutions (see be treated as constant in the neighbourhood of φ2, is re- 1,2 J 2 above). These are already present in the Wyart-Cates the- placed with α0(1 − φ/φ1) . This captures the growth of J ory and since our χ0 replaces its ν parameter, the exten- lubrication forces near φ1 which, with transverse shear sion from steady shear to arbitrary time-dependent flow applied, becomes a more accessible region of parameter requires only two new parameters, α and β. Given this, the space (since jamming at φ2 is now avoided) even in sys- agreement between the constitutive model and the simu- tems that are frictional from the outset (σ∗ → 0). lation data is good. However, various discrepancies for This setting offers a stringent test of the new modelling other quantities, particularly normal stress differences, are framework because there is no further parameter fitting found , as described in [2]. available and one is testing a model developed to describe A topical aspect of time-dependent flow in shear- flows of fixed orientation but variable rate (reversal of sim- thickening suspensions concerns the imposition of a trans- ple shear) by applying to flow where the extensional and verse oscillatory flow on top of a steady shear. It is compression axes are fully time dependent. The results known that the oscillatory component can mitigate or pre- are somewhat mixed. Figure 3 shows the reduced viscos- vent shear thickening, essentially because the compres- ity ηr as a function ofγ ˙⊥ = ωγ/γ˙ (where ω and γ refer to sive contacts leading to large frictional forces are opened the transverse flow andγ ˙ to the steady shear) alongside its by the transverse oscillation, effectively moving the sys- contact and hydrodynamic contributions for both the con- tem further from the jamming point than it otherwise stitutive model and the particle-based simulations.

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Acknowledgements: This paper summarizes joint work 2 with Jurriaan Gillissen, Chris Ness, Joe Peterson and He- 10 len Wilson. The author thanks them all for a most enjoy- 1 able collaboration. Work funded in part by the European 10 Research Council under the Horizon 2020 Programme, 100 ERC grant number 740296. The author holds a Royal So- ciety Professorship. 10-1 10-1 100 101 102 References

[1] M. Wyart, M. E. Cates, Phys. Rev. Lett. 112, 098302 Figure 3. Reduced viscosity as a function of nondimensional (2014) transverse strain rate from the constitutive model of [3]. Full [2] J. J. J. Gillissen, C. Ness, J. D. Peterson, H. J. Wilson, curve: total value; dashed curve, contact contribution; dotted 123 curve, hydrodynamic contribution. Circles, squares and inverted M. E. Cates, Phys. Rev. Lett. , 214504 (2019) triangles: the same quantities measured in particle-based simula- [3] J. J. J. Gillissen, C. Ness, J. D. Peterson, H. J. Wilson, tions. See [3] for further details. M. E. Cates, J. Rheol. 64, 353-365 (2020) [4] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, 1967 6 Discussion [5] F. Gadala Maria, A. Acrivos, J. Rheol 24, 799-841 (1980) The results shown in figure 2 for flow reversal are most [6] N. Y. C. Lin, B. M. Guy, M. Hermes, C. Ness, J. Sun, encouraging, but those for steady normal stress differences W. C. K. Poon, I. Cohen, Phys. Rev. Lett. 115, 228304 (see [2]), like the data from [3] in figure 3 for transverse (2015) oscillations, show qualitative agreement at best when con- fronted with particle-based simulation data. (Such data, as [7] S. Jamali, J. F. Brady, Phys. Rev. Lett. 123, 138002 emphasized in [19], offers stringent tests for constitutive (2019) modelling approaches to dense suspensions in general.) [8] R. Seto, R. Mari, J. F. Morris, M. M. Denn, Phys. Rev. Discrepancies reported in [2] include poor prediction of Lett. 111, 218301 (2013) normal stress differences which were attributed to a sys- [9] F. Boyer, E. Guazzelli, O. Pouliquen, Phys. Rev. Lett. tematic over-prediction of microstructural anisotropy for 107, 188301 (2011) any given flow conditions. As reported in [3] the model [10] M. Hermes, B. M. Guy, W. C. K. Poon, G. Poy, M. accounts well for the decrease in contacts on increasing E. Cates, W. C. K. Poon, J. Rheol. 60, 457-468 (2016) oscillation frequency ω, and for both the build-up of trans- [11] B. M. Guy, M. Hermes, W. C. K. Poon, Phys, Rev. verse shear stress under transient conditions and its phase Lett 115, 088304 (2015) relation to the transverse shear rate. On the other hand, [12] B. M. Guy, J. A. Richards, D. J. M. Hodgson, E. figure 3 shows a qualitatively incorrect prediction that the Blanco, W. C. K. Poon, Phys. Rev. Lett. 121, 128001 contact contribution to the viscosity collapses to negligible (2018) levels at highγ ˙⊥. In practice it does fall substantially but [13] J. J. J. Gillissen, H. J. Wilson, Phys. Rev. E 98, continues to dominate the lubrication contribution. 033119 (2018) An encouraging element of the work summarized [14] J. J. J. Gillissen, H. J. Wilson, Phys. Rev. Fluids 4, above is the way that close interrogation of microstruc- 013301 (2019) tural data from simulations can inform and constrain the [15] E. J. Hinch, L. G. Leal, J. Fluid Mech. 76, 187-208 continuum modelling. By looking more closely at the (1976) birth and death statistics of contacts, among other quan- [16] R. Seto, G. G. Giusteri, J. Fluid Mech. 857, 200-215 tities, one can hope to improve upon the modelling as- (2018) sumptions laid out above without either altering their basic [17] N. Y. C. Lin, C. Ness, M. E. Cates, J. Sun, I. Cohen, structure or introducing a proliferation of uncontrolled pa- Proc. Nat. Acad. Sci. USA 113, 10774-10778 (2016) rameters. Among obvious targets for improvement are the [18] C. Ness, R. Mari, M. E. Cates, Sci. Adv. 4, eear3296 absence of any frictional term in the microstructural evolu- (2018) tion equation (2); the simplified treatment of contact birth and death processes reflected in that equation; the ansatz [19] R. N. Chako, R. Mari, S. M. Fielding, M. E. Cates, J. of a near-isotropic (Hinch-Leal [15]) closure; and the spe- Fluid Mech. 847, 700-734 (2018) cific definition of the jamming coordinate ξ (4) in terms of the fabric tensor and the compressional flow.