# Shear-thickening

#### Soft materials and liquids

Soft materials (gels, elastomers, pastes, foams, suspensions, liquids ...) are generally from a particular point of view - the material is more liquid, or more solid - chosen by the physicist. In this article, we will focus on the "liquid" aspects of a very specific class of soft materials.

The simplest liquids are Newtonian; this means that they are fully characterized by a scalar quantity called viscosity $\eta$. This viscosity opposes the shear of the fluid, which is defined as a progressive transformation of the elementary fluid elements (bricks) in rhomboedral shapes (parallelograms in three dimensions, say):

Shearing the fluid at a specific rate $\dot \gamma$ necessitates to apply forces (purple arrows above) on the boundaries of the elementary surface. The applied, normalized force $\tau$, is proportional to $\dot\gamma$, providing a definition for $\eta$ :

$\eta = \displaystyle{\frac{\tau}{\dot\gamma}}$

The class of Newtonian fluid is rather small  (water, ethanol, glycerol, honey, silicon oil, decane, ...) and generally work for (chemically) pure solvents of small molecules.

#### Non-Newtonian fluids

A wide array of soft materials (foams, shampoo, hair gel, muds, peanut butter, spreads in general) exhibit a non-constant viscosity when $\dot\gamma$ varies, requiring fluid models that go beyond the Newtonian case. Such non-Newtonian fluids exhibit numerous curious effects. One of them is the Weissenberg effect, associated to the elastic behaviour of such materials in addition of a purely viscous nature.

Non-Newtonian fluids can be sub-divided into two classes. Shear-thinning fluids (most common case) have a decreasing $\eta$ when the shear rate increases. In constrast, shear-thickening fluids get more viscous when we try to shear them faster or harder.  A well-known example of this is cornstarch suspensions when highly concentrated in water. Its viscosity is low at low shear rate, but it is clearly not the case when you try to impose a large force to it !

We can examine with the right apparatus (a rheometer) the response of a 41 % vol. cornstarch suspension as a function of  $\dot\gamma$. The viscosity is clearly not constant (notice the log scale):

Around a specific shear rate (4-5 s$^{-1}$ ), the viscosity dramatically increases: the cornstarch suspension seems to get stuck. What can happen in the fluid to produce such a striking effect ? A part of the answer lies in the shape of the cornstarch particles, which look like small rocks under the microscope.

#### Two models for shear thickening

Until recently, the rheologists postulated that grains suspended in a liquid were never in contact with another, since a (sometimes extremely thin) liquid layer was always present between them [1]. When this layer is very thin, moving the particles relative to one another induces extreme shear rates for the fluid suspending the particles, requiring a lot of energy or a long time to separate them.  Shear-thickening was therefore explained saying that packs of particles very close to each other are formed in the liquid and act as effective solids (called hydroclusters) which hinder the movement of the rest of the suspension. This increases the global suspension viscosity.

This idealized model has recently been challenged. A new model including solid friction and therefore contacts between grains has been developed, when sufficiently compressed one onto another [2]. The authors of this frictional model claim that the thin fluid film can break if the particles are not perfect spheres, which is experimentally always the case.

Numerical simulations performed at the Levich Institute (in New York) [3] have shown that taking into account particle friction between grains provides a satisfactory description of the spectacular and discontinuous viscosity increase in cornstarch suspensions, especially since at the onset of shear-thickening, the number of inter-grain contacts jumps and forms a network spanning the whole sample.

A last experiment validates the frictional model hypothesis versus the hydrocluster model [4]. Shear cycles are performed, shearing in one direction, then reversing (instantaneously) the shear (hence, from $+\dot\gamma$ to $- \dot\gamma$ ). If we consider the hydrocluster model, changing the rotation sense will not change much to the suspension material properties (it will have the same "instantaneous viscosity" without showing any discontinuity at the time of reversal) since the thin films separating the particles will take a long time to grow. Changes in global material properties will only be noticed after a long time.

When such experiments are performed, a sharp change in the material properties is observed at the time of reversal. If we now consider the frictional model, just after the reversal, the previously compressed grains can now release their compressive forces and therefore their friction between each another. The fluid grains will then be redistributed in the fluid, which will ultimately generate new network of particles bearing compressive forces and friction, and reach a viscosity similar to the one measured before shear reversal. The paper from Gadala Maria and Acrivos and more recent numerical simulations [5]) agree on such a subject.

Here, at ENS de Lyon, we focus our research on the complex dynamics of cornstarch in the shear-thickened state. At imposed stresses, the measured response is very irregular and may exhibit interesting statistical properties. The actual shape of the shear-thickening curve (shown above) is still not perfectly understood, even though it looks very similar to the characteristic lines of first-order phase transitions.

#### Bibliography

[1] N.J. Wagner, J.F. Brady,  "Shear thickening in colloidal dispersions", Physics Today, (2009)
[2] M. Wyart, M. Cates, "Discontinuous shear thickening without inertia in dense non-Brownian suspensions", Physical Review Letters (2014)
[3] R. Seto, R. Mari, J. Morris, M. Denn, "Discontinuous Shear Thickening of Frictional Hard-Sphere Suspensions", Physical Review Letters (2013)
[4] F. Gadala Maria, A. Acrivos, "Shear‐Induced Structure in a Concentrated Suspension of Solid Spheres", Journal of Rheology (1980)
[5] F. Peters, G. Ghigliotti, S. Gallier, F. Blanc, E. Lemaire, and L. Lobry, "Rheology of non-Brownian suspensions of rough frictional particles under shear reversal: A numerical study", Journal of Rheolgy (2016)

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