Experimental study of hydrogel sphere packings close to jamming

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1 Master Science de la matière Avril-Juillet 2013 Ecole Normale Supérieure de Lyon Université Claude Bernard Lyon I M2 Physique "Concepts & applications" Experimental study of hydrogel sphere packings close to jamming Abstract: We studied elastic properties of confined packings of hydrogel spheres close to jamming. We showed that friction between beads and container walls plays a major role in our measurements of the elastic moduli of the packing. We measured the static friction coefficient and showed an ageing effect at the bead-wall contact. We built a model including the elasticity of the beads and the friction to explain the behaviour observed experimentally. Finally, we performed measurements of the shear modulus using a Couette geometry, where the effect of friction is minimized. Key words: Soft granular materials - Jamming - Elasticity - Friction Internship under the direction of: Professor Martin van Hecke mvhecke@physics.leidenuniv.nl / Phone +31 (0) Leiden Institute of Physics - Leiden University Niels Bohrweg CA Leiden - NETHERLANDS

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3 Acknowledgments I want to thank Martin van Hecke for inviting me in his research team, for his guidance and helpful ideas during this internship. I also want to thank Geert Wortel for his help, patience and listening skills, Simon Dagois-Bohy for our discussions, Jeroen Mesman for helping me out with the technical parts of the setup, and the rest of the Granular Matter group in general for their support. A special thank deserves to Danielle Dujin for her help with paperwork and housing. Contents 1 Jamming 2 2 Elastic properties of a confined packing Experimental situation About hydrogel particles Setup Linear elasticity of a confined cylinder Previous results Clues about anisotropy and friction 8 4 Quantitative study of friction Shear experiment Compression experiment Couette geometry experiment Setup and motivations Experimental results Appendix : Single bead measurements 21

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5 Introduction In the 1990 s, there was a regain of interest in matter with granularity after the work of Bagnold in the 40 s. Some examples of such matter are ground coffee, sand, colloidal suspensions like toothpaste, oil-in-water emulsions such as mayonnaise, and packings of foam bubbles such as shaving foam. They are non-thermal (the thermal energy scale is much smaller than the potential energy required for particle rearrangement) and dissipative systems. In 1996 Jaeger, Nagel and Behringer published a key review article describing complex and often non-linear behaviour of granular materials in different situations [1]. They showed that these materials can exhibit a wide variety of behaviours, varying from solid, to liquid- like, to gas-like. Granular media also show a lot of different bulk properties, such as convection, wave patterns, segregation, crystallization or jamming. In 1999, De Gennes published an overview of knowledge in this field [2]. Granular materials can exhibit a jamming transition when compressed. While increasing the pressure, one can reach a critical point where the material switches from a gas-like to a solid-like behaviour. Then, one can ask the following question: what are the physical parameters describing this transition? For example, if we think about elastic properties of the packing, how do elastic moduli scale close to jamming? Over the past 15 years the knowledge on the jamming transition has greatly increased. In 1998, Liu and Nagel proposed a tentative phase diagram describing disordered materials [3], with temperature, load or density as control parameters. In this project, we have studied the more particular case of jamming in packings of soft, athermal spherical particles. In 2010, Van Hecke and Liu & Nagel published review articles on this more specific field [4, 5]. A lot of numerical simulations and theoretical studies have been done [4, 6, 7, 8, 9, 10], that predict the elastic behaviour of the packing close to jamming as a function of φ. In comparison, there have been relatively few experimental studies, especially on the elastic response close to jamming. In 1985, Princen & Kiss [11] performed measurements on the shear modulus G of polydisperse oil in water emulsions, at different packing fraction φ. They found that G φ 1/3 (φ φ c ), and determined φ c In 2005, Majmudar & Behringer [12] studied the jamming transition of ensembles of spheres using photoelastic discs. In this way, they managed to image force networks and show how important friction is. In 2009, Clusel et al. [13] made a fluorescently labelled oil-in-water emulsion. Using 3D imaging, they extracted the statistics of the local ordering of grains, but did not study the elasticity of the packing. Another approach commonly used in this field is to use a two-dimensional monolayer of foam bubbles [14, 15]. This produces a system with minimal friction and no gravity in the horizontal plane. However, such systems are never perfectly frictionless. Van Hecke [4] focused on the effect of friction at the jamming transition. In the light of these studies, we decided to experimentally study the elastic properties of packings of hydrogel particles immersed in liquid, when they are close to jamming. The main aim of our project is to see if we can experimentally find the scalings predicted for G and K. We are also interested in frictional properties of the beads, that directly influence the behaviour of a strained packing. Our study will be a follow up on results obtained by Dieleman [16]. In Section 1, we will introduce jamming theory, which predicts the elastic behaviour of packings close to jamming. Then, we will describe our experimental setup and the previous results obtained by Dieleman in Section 2. In Section 3, we will prove the existence of anisotropy and friction in our system, and then in Section 4 study quantitatively friction. Finally, we will show another experimental setup using the Couette geometry and the first results we obtain in Section 5. 1

6 1 Jamming In this section, we will discuss the jamming scenario for packings of soft, frictionless spheres, and the elastic properties that these packings have near the jamming transition. There are different parameters used in jamming theory: the average contact number z between particles, the packing fraction φ and the pressure P. While compressing the system, one increases its volume fraction (less free space), its pressure (spheres are deformed) and its contact number (beads are touching more neighbours). Figure 1-(left) shows the evolution of a 2D-packing while increasing φ. Initially, particles are not in contact and z = 0. At the critical packing fraction φ c, the contact number abruptly jumps because contacts appear suddenly between particles. The system is now in a jammed state, and φ c corresponds to the jamming point. Above φ c, particles are deformed and z, φ and P increase. The transition between the jammed and the unjammed state is a critical one; by decreasing φ, the packing suddenly loses its rigidity. The critical value for z is called the isostatic point (z isostatic ) and corresponds to the minimum number of contacts needed to have a stable packing. Bolton and Weaire in 1990 [17], and then Durian in 1995 [18] showed numerically that the contact number approaches z = 4 when unjamming a 2D system (z = 2d in d-dimensions). He also showed that z 4 φ (with φ = φ φ c ) (see Figure 1-(right)). In the case of frictionless spheres, z isostatic can be obtained theoretically by imposing there are no floppy modes (dynamic modes in which particles move but at a constant elastic energy) [4]. The critical packing fraction φ c is more difficult to derive, but it is believed to be linked with the Random Close Packing Fraction φ RCP 0.64 in 3D. z 2d 0 Φ c Φ Figure 1: (Left) Evolution of a 2D-packing with respect to the packing fraction. (Right) Evolution of the contact number z with respect to the packing fraction. (adapted from Hecke [4]). Then, we would like to predict the behaviour of the elastic moduli (K and G) of such packings just above jamming. By simulating packings in 2D & 3D, one can study K and G without any side effect like friction, gravity, non-trivial sphere-sphere interaction potentials. Moreover, one can easily study finite size effects. First, one need to define the sphere-sphere interaction potential in the case of soft, frictionless spheres. It can be written as V δ α, where δ is the particle overlap, and α depends on the particles properties. In the case of harmonic particles α = 2, and α = 5/2 for Hertzian particles. Then, the contact force f and the stiffness of this contact k are f dv dδ δα 1 and k d2 V dδ 2 δα 2 (1) Second, we need to express laws with variables that we are able to measure in our experiment. For example, the excess contact number z = z z isostatic is not accessible but the pressure P is. One 2

7 can assume the pressure scales like f, where brackets denote the average over the system. Then, we note that the typical particle overlap δ scales as φ. One obtains the following relations: P f δ 3 2 ( φ) 3 2 ( z) 3 (2) Finally, we need an assumption about how the global forcing links local deformation. In the Effective Medium Theory (EMT), we assume that the packing locally deforms affinely following the global strain field. Consequently, the elastic moduli simply scale like the stiffness of the contacts: K and G k. Using EMT and the equations (1) & (2), one obtain K G P (α 2) (α 1). However, Makse et al. showed this scaling works well for bulk modulus K, but fails for the shear modulus G [9]. Indeed, weakly jammed systems exhibit highly non-affine deformations, which violate the affine assumption needed for EMT to be valid. Moreover, below the critical packing fraction, particles are not in contact, so the elastic response should be equal to zero. Taking these corrections into account, it has been shown that K k and G k z [7, 8, 9, 10], which leads to: K P (α 2) (α 1) and G P 2α 3 2(α 1) (3) 2 Elastic properties of a confined packing In this section, we study the elasticity of a packing of spheres confined in a cylindrical beaker. We describe the beads and the setup we used, and the results obtained previously. We use a rheometer that can exert a shear stress and a normal strain on the packing. Therefore, we measure the following variables: the height L of the packing, the normal force F z, the torque applied T and the angle of deflection θ. 2.1 Experimental situation About hydrogel particles Our main goal is to test the predicted scalings with experiments. So we are looking for macroscopic, soft and frictionless spheres. We decide to use "Growing Spheres" made of polymer material PolyAcrylamide (PMMA), sold by Educational Innovations INC [19] and produced by JRM Chemicals (sold under trademark names such as Soil Moist, Aquagel and Aquabeads). Hydrogels have hydrophilic properties that allow them to absorb hundreds of times their own weight of water. We observe the following behaviour [16]: with an initial diameter of 2.5 mm, beads swell up a factor of in volume, and become soft and transparent, the swelling ratio depends a lot on the salt concentration of the solution used (see Figure 2-(left)). Density Matching Systems with macroscopic particles will naturally be jammed due to gravity. We would like to control the pressure applied to our system as precisely and as close as possible to the jamming point. By putting particles in a fluid column, we can try to nullify the effects of gravity by density matching the fluid to the particles. Our swollen hydrogel particles consist of % of the liquid they are immersed in because they are highly porous. By using Natrium Thiosulfate salt, we manage to reach the point of density matching: particles are between floating and sinking, as shown in Figure 2-(right). It is difficult to control perfectly this density matching because of the slow evaporation of water: we estimate the residual density mismatch about g.cm 3. This density matching point is obtained with a concentration of 385 g/l of Na 2 S 2 O 3, resulting in a density of g.cm 3. Then, the particle diameter is about 8 mm. 3

8 Experimental study of hydrogel sphere packings close to jamming Figure 2: (Left) Hydrogel particles: unswollen particle / particle swollen in a solution containing salt / particle swollen in demineralized water. (Right) Particles density matched immersed in the salty solution. Time Dependence It has been shown that swollen hydrogels exhibit a complex time-dependent relaxation behaviour when compressed [20]. This is mainly due to poro-elasticity, which corresponds to the migration of solvent out of strained areas. The main consequence of poro-elasticity is the decay of the size of the beads and a change of their stiffness when they are compressed: the contact force (bead-bead and bead-object) decreases when beads are compressed. This process cannot be avoided and can have a significant effect on elasticity of a packing of hydrogel spheres. However, we make sure its influence on our measurements is minimized (see Section 2.1.2). Interaction Potential To determine the proper scaling for our packing we need to know α in equation (3). We therefore need to know elastic response of a single swollen bead. Heeres [21] measured the instantaneous response of PMMA spheres. The Stress (σ) versus Strain (γ) curve can be well fitted for strains below 10 % by σ γ β. He found a value close to β = 1.5 over a wide range of sphere sizes, which means our swollen hydrogel beads follow the Hertz s contact law i.e. α = 52. Consequently, the scalings predicted by theory (equation (3)) are: 1 K P and 2 G P3 (4) Setup Normal Force Torque R Top plate L Beads in solution Bottom plate Figure 3: Schematic and picture of the setup. Beads are confined within the beaker walls, and rough upper and bottom disks. In order to shear and compress packings of beads, the container must have rough boundary conditions at the top and the bottom. We use a glass beaker of radius R = 4.3 cm and height L = 17 cm 4

9 and two PVC disks (of thickness 2 cm) with conical holes at the top and the bottom. In this beaker, we put our density matched solution with N spheres, and we can compress the whole packing with the upper disk attached to the shaft of our rheometer (Anton Paar MCR 501), as shown in Figure 3. This setup therefore enables us to exert a shear stress by imposing a finite torque, and a normal strain by moving the probe up or down. Consequently, one can measure two sets of conjugate variables: the gap z between probe and platform (which controls the height of the packing z = L) & the normal force F z, and the torque T exerted by the shaft & the deflection angle θ. N or mal For ce (N ) Time (min) Torque (mn.m) Time (s) z (mm) Time (s) Figure 4: (Left) Typical relaxation of the normal force measured by the rheometer when the packing is initially compressed with a normal force F z = 1 N. (Middle) Triangular oscillations of torque T with T max = 64 µn.m (Right) Triangular oscillation of vertical displacement z We use the following protocol to probe the elasticity of a packing: the first step is to compress the packing to our target pressure. After that, we have a shear stage and a compression stage. During the first one, we apply a tiny triangular oscillatory torque while holding vertical position constant in order to probe the shear elasticity of the packing (Figure 4-(middle)). Finally, we perform slight triangular oscillations in the vertical direction z to probe the compression elasticity of our packing (Figure 4- (right)). The period of the oscillations are long enough to consider the process to be quasi-static. It is important to notice that the whole protocol takes a considerable time to execute (about 30 min), and during this time the normal stress relaxes because of the poro-elasticity process described above (Section 2.1.1). This relaxation is very reproducible: it is very sharp at the beginning and then more flat (see Figure 4-(left)). Consequently, we perform measurements only after about 10 minutes, when the sharp part is finished measured stress applied strain fitted model 0.10 Deflection Angle (mrad) σ N (Pa) γ (%) Torque (mnm) t (s) Figure 5: (Left) Deflection angle θ as a function of T during the shear stage. One can extract G from this measurement. (Right) σ N and γ N as a function of time during the compression stage. After correction, one can fit σ N = Mγ N. Figure 5 shows two typical responses measured in compression and shear. In Figure 5-(left), we 5

10 can see the linear relationship between the applied torque T (T max = 64 µn.m) and the deflection angle θ during the fifth oscillation. Figure 5-(right) shows the applied strain γ N and the measured normal stress σ N as a function of time. Because of the relaxation, the normal stress oscillates but also slowly decreases. Over short times like here, this can be correctly fitted by a linear decay. By removing this decaying component, one can extract the relevant variation of the normal stress. We will see in the next section how to obtain the elastic moduli of the packing from these measurements Linear elasticity of a confined cylinder Our goal is to obtain the shear modulus (G) and bulk modulus (K) of the packing of spheres. In the following, we will assume that the packings are homogeneous, isotropic, elastic cylinders confined within a cylindrical container and an upper disk. Moreover, we use the linear elasticity theory which supposes only small displacements from equilibrium. The packing is described with the stress tensor σ ij and the deformation tensor γ ij in cylindrical coordinates (i, j) (r, θ, z). To obtain a relationship between the applied torque T, the angle of deflection θ and the shear modulus G, we use the Hooke s law relating the shear stress σ S = σ rθ to the shear strain γ S = γ rθ i.e. σ S = Gγ S. In the situation described in Figure 6-(B), the shear strain of a cylinder at a radius r and height L is γ S = rθ L. Then, the torque is given by T = σ S.r ds, where we integrate over the external surface S = πr 2. Finally, we have: T = G πr4 2L θ (5) (A) F z (B) T Δz θ R z L θ r L F z Figure 6: Schematics of a cylinder of height L and radius R (A) compressed at the top with a force F z and confined in the sideways, (B) sheared at the top by a torque T. When compressing the column of spheres we effectively restrict its movement in the sideways directions, so the response is necessarily stiffer than without this restriction. The modulus that we measure by indenting the cylinder this way is uniaxial compression modulus M and not the bulk K or Young s modulus E. A sketch of the situation is given in Figure 6-(A): a vertical force F z compresses the cylinder over a distance l z. Then, the Hooke s law gives the relationship between the normal stress σ N = σ zz = Fz S and the normal strain γ N = γ zz = z L : F z πr 2 = M z L (6) 6

11 One can finally obtain the bulk modulus which is a combination of M and G using [22]: K = M 4 3 G (7) 2.2 Previous results Here we review measurements obtained by Dieleman [16], for a packing consisting of approximately 350 beads, which corresponds to a cylinder of size L = 2.3 cm and R = 4.3 cm. Figure 7 shows the evolution of the elastic moduli G, K and M with respect to the applied normal stress σ N (each dot corresponds to an average over approximatively 80 measurements). As predicted by theory, G, M and K increase with pressure. Moreover, they become really close to zero for low pressures, which is consistent with the continuous evolution at the jamming point. However, one notice major problems: K < 0 and K < G. K < 0 is impossible since we are dealing with a regular material. K < G is possible only if the material has a small Poisson ratio, but because the hydrogel spheres are incompressible, this is quite unlikely Shear Modulus G Bulk Modulus K Uniaxial Modulus M Elastic Moduli (Pa) Normal Stress (Pa) Figure 7: Evolution of M, G and K with respect to the normal stress σ N (results from [16]). They show two major problems: K < 0 and K < G. The fact that these measurements (even with significant error-bars) do not follow the relationships K > 0 and K > G can have different reasons. Non-linearities, inhomogeneous material and anisotropy will be discussed later, because it seems to us they do not have high effects. On the contrary, we believe that friction plays a major role in our system, and is the major reason for these results. Indeed beadbead and wall-bead contacts are not perfectly frictionless and boundaries could have a strong effect on our measurements. Without friction, the relationship between T and θ is T = G πr4 2L θ. But if there is friction, the resistance will be higher and the angle of deflection will be lower : θ friction < θ. Then, the measured quantity is not G but G 2LT = θ friction > 2LT = G, which means G has been overestimated. πr 4 θπr 4 This will be studied further in Section 4.1. The effect of friction during compression is difficult to establish. Indeed in this case, the height z is imposed and not the force. Thus, it is hard to know how the resistance of the packing will change if the beads at the walls are highly deformed (friction) or only slightly compressed (no friction). However, it is clear that we are not only measuring the compression response of the packing. This will be studied further in Section

12 Experimental study of hydrogel sphere packings close to jamming 3 Clues about anisotropy and friction As explained in Section 2.1.3, we assume the packing of beads to behave like an homogeneous, linear, isotropic elastic material, without friction. Our goal in this section is to prove some these assumptions are actually wrong. We focus on the response of the packing under shear or compression by using the same protocol as described before, but we change the number of oscillations and the amplitudes of torque and vertical displacement. Coupling Between Compression & Rotation We first assumed our packing to be isotropic, which supposes the two modes "compression" and "rotation" to be decoupled. This is not trivial in our case because our material is a packing of randomly organised beads. We studied a packing of approximatively 350 beads, with the previously described setup. First, we imposed three successive stages in the protocol, for a packing under pressure σn = 150 Pa: high amplitude torque (γs = %), small amplitude torque (γs = %) and vertical compression (γn = 0.2 %). Figure 8-(left) shows one possible evolution of the deflection angle θ (θ < 0 is as likely as θ > 0). The mean angle decreases during the first stage, and then seems to approach a plateau during the second one. However, this does not mean the packing is in a stable state i.e. small perturbations induce small response. The third stage shows the angle sharply decreases while the packing is subject to a small compression-decompression. It is important to notice that the preferred direction of the angle is totally random. These observations prove there is a coupling between rotation and compression. It means we are not measuring only M and G but a linear combination of them and extra diagonal coefficients of the elasticity matrix. In order to quantify this coupling, one can study another measurement with the same setup where the angle is stable (the mean angle is constant). Figure 8-(right) shows the 4 measured normal stress σn,measured and the induced shear stress σs,induced = G πr 2L θmeasured during the compression stage in this case. Here, these quantities evolve exactly in phase (other measurements show they can be opposed in phase, or sometimes dephased, but always with the same period). However, this coupling effect is relatively small: the induced shear stress is between 0.1% and 1% of the normal stress. Therefore, such a weak anisotropic behaviour cannot explain the measurements discussed in Section Normal Stress (Pa) 0.00 Angle (mrad) 150 High amplitude Shear Small amplitude Shear Compression Shear Stress (Pa) Imposed Normal Stress Induced Shear Stress Time (s) t (s) Figure 8: (Left) Evolution of the deflection angle θ during three successive stages: high amplitude torque, small amplitude torque and vertical compression. (Right) Comparison between the normal and the shear stresses during a compression/decompression stage. Friction In section we show an almost prefect linear response between the torque and the angle during the shear stage. However, this is not always the case, depending on the amplitude of the applied torque. Indeed, a larger deformation can lead to a totally different behaviour as shown 8

13 Experimental study of hydrogel sphere packings close to jamming in Figure 9. For Tmax = 5 µn.m, the packing clearly has a preferred direction but behaves linearly: the mean angle decreases and the packing is stiffer when θ increases than when it decreases. If we assume there is friction, it means the particles are stuck to the walls and only deform. For Tmax = 500 µn.m, elasticity dominated friction because particles now slide on the walls. The hysteresis is due to the Hertzian behaviour of the particles. In a mid-regime (Tmax = 32 µn.m), the packing is linear and isotropic: we believe that friction and elasticity stresses are now of the same order of magnitude. The measurements showed in section 2.2 have been obtained in this regime. Consequently, they are not only dealing with the elasticity of the packing: friction might dramatically influence the values of G and M, and so K Deflection Angle (mrad) Deflection Angle (mrad) Deflection Angle (mrad) Torque (mn.m) Torque (mn.m) Torque (mn.m) Figure 9: Evolution of the angle θ as a function of the oscillating torque T for three different amplitudes of torque: Tmax = 5, 32 and 500 µn.m (colours show the evolution with time). The presence of friction can also be confirmed by studying the influence of boundaries. We perform several measurements of G and M at a fixed pressure σn = 150 Pa, with the same beaker R = 4.3 cm, but by varying the height L of the cylinder i.e. the number of particles. G was measured with a mid-range torque, so that the response is linear. M was measured in the same way as described in Section 2.2. Figure 10 shows the strong dependence of our measurements on the size of the system: G increases of 300 % when the system is two times bigger. A finite size effect can be expected because of the influence of the boundaries, but it should not be so large. If there is friction at the boundaries, our measurements include the elasticity of the packing but also the resistance of the beads stuck at the walls. Indeed, if the beads touching the walls do not slide, they will be highly sheared and deformed. This could dramatically change the values we measured for both M and G. The linear dependence of M and G with L could even prove that we are mainly measuring the resistance of the sticking beads and not the elasticity of the packing: this will be discussed further in Section M-modulus G-modulus G & M (Pa) Height (mm) Figure 10: Evolution of M and G with respect to the height L of the packing, at a fixed normal stress σn = 150 Pa. 9

14 4 Quantitative study of friction The goal of this section is to more precisely study the friction between the beads and the glass walls. We focus on the friction threshold, which determines the limit between the sticking and the sliding regime. This way, we can know more about surface properties of hydrogel beads and the resistance of deformed beads touching walls. We study packings of approximatively 350 beads (cylinder of size L = 2.3 cm and R = 4.3 cm) in both shear and compression situations. 4.1 Shear experiment In order to know more about the friction threshold, we perform shearing experiments with the same setup (Section 2.1.2) but without the rough bottom. This way, the packing is only in contact with glass (except for the upper probe which imposes the torque) and we can study the bead-glass contact in a more quantitative way. In the absence of friction, the packing should always rotate with a small but not null torque. But if there is friction, the packing will only rotate if the torque is higher than a certain critical torque T c. Instantaneous Response We first perform an experiment where the packing is compressed under a certain pressure σ N, and then impose a step in the torque: at a fixed time, the torque goes instantaneously from 0 to T. Figure 11-(left) shows the two possible responses of the packing: if the torque is small enough, it deforms and the angle θ reaches a plateau (blue curve). If the torque is high enough, static friction is overcome and the whole packing will rotate (the angle θ diverges (red curve)). We perform several measurements for different values of parameters (T,σ N ) which are summarized in Figure 11-(right) T=1.7 mn.m T=1.9 mn.m Deflection Angle (mrad) Torque (mn.m) time (s) Critical Torque Stick Slip N-Stress (Pa) Figure 11: (Left) Response angle measured for two different values of torque applied, for a packing under compression σ N = 170 Pa. For T = 1.9 mn.m the whole packing rotates, while for T = 1.8 mn.m it deforms but do not slides. (Right) Sticking or sliding behaviour for different points of measurements (T,σ N ). The limit defines the critical torque which corresponds to the friction threshold. First, we have a confirmation there is friction because the packing does not always slide. There is a critical value of T, T c (σ N ), between the sliding regime and the sticking regime. In order to have quantitative results, we have to make assumptions about the behaviour of our material. We use the Newton s friction law which defines the static friction coefficient µ as the following: on a flat surface, the threshold between sticking/sliding regimes is defined by σ T = µσ N where σ T and σ N are the tangential and normal stresses. This law is usually given with forces, but since we assumed our packing to be a bulk material, we are only dealing with pressures/stresses. 10

15 We assume a linear relationship between the radial and the vertical stresses in the packing σ rr = Kσ zz. We arbitrarily choose K = 1. One important thing could be the variation of the normal stress σ zz (z) in the packing. Using the assumptions we made, and writing the force balance on a slice of material like Janssen did [23] dσ but without gravity, one obtains the following differential equation: zz dz + σzz h = 0, with z going downwards and the characteristic length h = R 2µK. This leads to σ zz exp( z h ). In our case h 100 cm L 2 4 cm. Consequently, we can assume σ zz = cst = σ N in the whole packing, where σ N is the normal stress imposed by the probe. Friction occurs at the bottom and on the walls of the container, so one can integrate the critical stresses over those surfaces to get the torque needed to reach the sliding regime: At the bottom: σ rz = µσ zz and T c = σ rz.r ds. At the walls: σ rr = σ zz, σ rθ = µσ rr and T c = RF θ = R [(2πRL) σ rθ ]. Summing the two contributions, one obtains the following equation: T c = µ 2πR 2 ( R 3 + L ) σ N (8) Using our experimental results, we fit the critical torque to be T c = σ N. Using this linear relationship, one obtains the static friction coefficient µ 0 = ± This value is close to the friction coefficient previously measured for a single hydrogel particle µ = 0.03 (see Appendix), and is relatively small. A better way to obtain µ would have been to also vary L to check the linear dependence. However, it would require a lot of measurements, which is very long. It is important to notice that we are dealing with high values of torque compared to previous experiments. Consequently, beads are in this case highly deformed before they actually slide. However, this gives us no clue about the frictional behaviour of beads before they slide, since we are imposing a step of torque. Creep Response In order to study the behaviour of the packing under shear close to equilibrium, we decide to impose not a step but a ramp of torque. This way, the packing has time to deform for low enough slopes of torque, and one can see if the angle of deflection behaves differently after a certain point. The evolution of torque with time is described by its slope α and the starting time t 0, so that T (t) = 0 for t < t 0, and T (t) = α(t t 0 ) for t > t 0 (t = 0 corresponds to the end of the compression of the packing). We stop the torque when the packing has made one full rotation (2π radians). We first perform series of measurements for different packings compressed with σ N = 110 Pa imposing different values of α from 0.1 to 60 µn.m/s and t 0 = 0 s. Figure 12-(upper left) shows the typical measurements we obtain: when the torque starts increasing, the angle first increases linearly, then its behaviour is highly non linear and finally diverges. By imaging the evolution of the packing with a camera, we clearly see a critical point when the whole packing slides at high rotation speed. We define the critical torque T c and the critical time t c at the point when the rotating speed exceeds 0.5 mrad/s. Then, using equation (8), each value of T c corresponds to a friction coefficient µ. We plot in Figure 12-(upper right) the evolution of the corresponding values of µ as a function of log(t c ). The value of µ obtained for very short times is consistent with µ 0, obtained with a step of torque. The friction coefficient increases in a significant way with time and this evolution seems logarithmic. This could be characteristic of ageing effects [24], but in our case, we are not only studying the effect of time. Indeed by changing the slope α, the packing is not allowed to deform in the same way: for high slopes, the packing does not have enough time to deform and overcomes friction rapidly. For very low slopes, beads are gently deformed and we observed using a camera, the beads in contact with glass slowly sliding. Consequently, they overcome friction but for some reason the whole packing 11

16 Torque imposed Measured angle T c Torque (mn.m) Angle (mrad) µ Time (s) t c t c (s) Torque imposed Measured angle T c Torque (mn.m) Angle (mrad) µ Time (s) t c t c (s) 10 4 Figure 12: (Upper Left) Typical measurement when the packing is sheared with a linearly increasing torque. At the critical time t c, the whole packing rotates. (Upper Right) µ deduced from the critical torque T c and equation (8) as a function of t c (logarithmic scale). (Lower Left) Typical measurement when the packing stays at rest and then is sheared with a linearly increasing torque. (Upper Right) µ as a function of t c (logarithmic scale). does not. It might be caused by a difference between local and global threshold which is due to the poro-elasticty process (previously described). When water is going out of the beads because they are compressed, their contact area and its properties are changing. This process can be speeded up or slowed down depending on the strain applied to the packing. Consequently the friction is the same for all the beads in contact with the walls. Finally, we cannot clearly conclude an ageing effect because we are acting on different parameters at the same time. It also means there is no clear friction threshold (stick/slip) as we first thought. We perform a second series of measurements focusing on the effect of time. We choose a midrange slope (α = 2.4 µn.m/s) but wait for a certain time before increasing the torque (t 0 from 0 to s). The range of values of µ is lower than before, but here it is only an ageing effect because α is constant: the longer the packing stays at rest, the higher the friction is. The trend of µ(t c ) looks different especially because µ seems to reach a plateau for very long times. This is consistent with the poro-elasticity effect: we showed in Figure 5-(left) that the relaxation finally saturates. Then, for very long time, the strained area does not evolve any more and the friction becomes constant. However, we still do not explain the fact that beads at the walls slowly slide before the whole packing overcome friction. To understand what happens at the critical moment when the packing fully rotates, the evolution of the normal stress during shear can give us clues. Figure 13 shows the normal stress and the 12

17 Measured angle Normal Stress Measured angle Normal Stress Angle (mrad) σ N (Pa) Angle (mrad) σ N (Pa) t (s) t (s) Figure 13: (Left) Normal stress and deflection angle as a function of time during shear. (Right) Zoom on the critical point when the packing fully rotates. deflection angle as a function of time during shear for a long critical time t c = 6120 s. The normal stress first decreases because of the poro-elasticty process, as described before. But as we shear the packing, dilatancy makes it expand in volume, and the normal stress increases. We also noticed a relatively significant peak occurring at t c. For shorter experiments, the dilatancy is not visible, but the peak remains. A more precise graph (Figure 13-(right)) shows this peak begins when the packing starts to rotate and finishes when the torque is zero again. Consequently, the friction threshold might correspond to the moment when the beads are almost blocked due to dilatancy, so they are highly deformed for a short time and they finally slide. To conclude, we have seen in this section that our beads are not frictionless. The friction between beads and glass is relatively small (µ ) but definitively changes the behaviour of the packing under shear. We also noticed a significant ageing effect, that is to be related to the poroelasticity of hydrogel. 4.2 Compression experiment Another way to investigate friction effects is to study compression/decompression cycles for high vertical displacements z. Indeed, the direction of the friction force will change if one compresses or decompresses the packing. Experimental Results We study packings under different normal stresses σ N and 10 successive compression/decompression cycles (of period 80 seconds and amplitude z = 2 mm). Figure 14-(left) shows a typical σ N z curve for one compression/decompression cycle. We first notice the non-linear behaviour, due to Hertzian particles: the elasticity of the packing is proportional to z j z where z j corresponds to the jamming point. Thus, the linear elasticity assumption is only valid for small strains. Moreover, the curve shows hysteresis which proves the process is not reversible: we believe that the beads at the walls are sliding. The hysteresis proves the existence of friction. However, we do not know the effect of friction on the slope of the σ N z curve. The results shown in Section 2.2 have been obtained with tiny oscillations of z, after compression of the packing. Consequently, the beads at the walls are highly deformed, and we only impose small perturbations to this state. Here on the contrary, we impose large amplitudes of z, so that the beads touching the walls behave differently when they are compressed or decompressed. In order to see the difference between those two situations, we extract M i.e. the slope of σ N ( z L ). Figure 14-(right) shows M as a function σ N in both compression (red curves) and decompression (blue curves) stages for about 10 different measurements. First, one can see there is no clear difference between the compression and the decompression stages. Second, the values of M are about 50% 13

18 power law exp=0.60 Normal Stress (Pa) σ N σ N σn M (Pa) z (mm) σ N (Pa) Figure 14: (Left) Typical σ N z hysteresis curve for one compression/decompression cycle. We defined σ N as the distance between the upper and lower branches. (Right) M as a function σ N obtained with high amplitude compression and decompression cycles. Red curves are compressions and blue one decompressions. smaller than the ones showed in Figure 7. It might be due to the difference of protocol: Dieleman [16] measured mainly the resistance of beads highly deformed and stuck at the walls, whereas we are measuring the resistance of beads sliding on the walls. This difference could explain so different numerical values of M. Finally, we obtain a scaling law M σ N 0.60, but this value cannot be compared to any prediction µ=0.035 µ=0.047 µ=0.057 At rest k After compression F Δz δ 0 25 k k σ N (Pa) k L z δ 1 k f 10 5 δ N Normal Stress (Pa) k Figure 15: (Left) σ N as a function of σ N for three different positions in the σ N z curve. One can deduce µ form the slope. (Right) Schematic of the model we used : several springs of strength k in series and coupled with friction springs of strength k f in parallel. One can also obtain the static friction coefficient µ using these measurements. Assuming the packing slides on the walls in both compression and decompression regimes and using the assumptions made in Section 4.1, the distance between the upper and the lower branches of the σ N z curve (Figure 14-(left)) is σ N = 2σ friction = 2µσ N (9) We measure σ N at three different points of the cycle (see Figure 14-(left)) because this quantity 14

19 is not constant in a cycle, since the curve is closed. These three points are chosen not too close to the ends of the cycle, so that the transitional regime is not involved. Figure 15-(left) shows the average values over 10 cycles as a function of the normal stress σ N. Using equation (9), we extract three friction coefficients µ = 0.035, and for the three different points. Those values are consistent with results obtained in the case of shear (Section 4.1). Having three different values in a cycle might be due to non-linearities, or difference between static and dynamic friction. Toy Model In order to more precisely understand our experimental observations, we build a toy-model which describes our packing under compression. Since we are only working in compression, our model is in one dimension. We modelled the beads with springs of strength k. We consider N points (i [ 0 ; N 1 ]) related by N+1 springs in series in the vertical direction with a beam touching the wall which can be deformed. The friction force is equivalent to an additional spring of strength k f (see schematic in Figure 15-(right)) and is modelled by F friction = k f δ i (δ i is the displacement of point i). We work in a quasi-static process, so that each contact point is at equilibrium. The displacement z = L z of the upper point is imposed to deform the system, with a top-force F. The goal here is obtain the law F ( z). The force balance of point i is k(δ i δ i 1 ) + k(δ i δ i+1 ) + k f δ i = 0. One can go from a discrete description to a continuous one in order to understand more easily the behaviour of the system. However, this is only an approximation. Introducing the rest length of one spring l so that L = (N + 1)l, one can use the continuous coordinate x li (x is oriented downwards), so that δ i δ(x = li). δ(x) is the vertical displacement of a point at distance x from the top of the system. Using the force balance and using a Taylor expansion of δ, one obtain the following differential equation: d 2 δ dx 2 δ λ 2 = 0 with λ = l Using boundaries conditions δ(0) = z and δ(l) = 0, one can find the solution δ(x) = z Then, the top force is F = k( z δ 0 ) kl dδ dx (0), kf k F ( z) = ( tanh Equation (10) can be transformed: for low frictional forces i.e. k f for high frictional forces i.e. k f L l kf k k ( l L) 2 k, F ( z) = kl L z k f ( l L) 2 k, F ( z) = k f k z sinh( L x λ ) sinh( L λ ). ) z (10) A 3-D equivalent of this model would be several columns of beads of diameter l in parallel, the number of columns depending on the radius R of the container. Then, one can extract M from this relation using equation (6), and interpret the evolution of M with L at σ N = 150 Pa, previously shown in Figure 10 (Section 3). Our packing is made of approximatively N total S l 90 columns of beads. N walls of them are touching the walls where friction is dominant and N bulk are in the bulk of the packing without friction. Consequently, the total response of the packing would be: σ N = F S = ( Nbulk S kl + N ) walls z k f k L S L so M = ( Nbulk S kl + N ) walls k f k L S The dependence of M with L observed in Figure 10 is completely consistent with the presence of friction at the walls. Moreover, one can obtain the approximative values of k and k f using equation 15 (11)

20 (11): l corresponds to the size of one bead (8 mm), N walls 2πR l 30 and N bulk = N total N walls 60. With an affine fit of M(L), one can extract the coefficients of equation (11). We obtain k 16 N.m 1 ( ) 2 and k f 80 N.m 1 (consistent with the assumption k f l L k). The value of k f is consistent with previous measurements performed by Dieleman with a single bead (see Appendix). He measured k f 40 N.m 1, which is the same order as our calculated value. However, it is difficult to extrapolate this result to our case since we are working at different pressures. The value of k leads to the real value of M, the intrinsic uniaxial modulus of the packing : M = N total S kl 2000 Pa for a packing under pressure σ N = 150 Pa. This value seems to make more sense since we removed the contribution of friction, but it is only an approximative value and cannot be interpreted further. Our model shows the major role played by friction, but does not predict the hysteresis experimentally observed for high amplitudes of z in σ N z curves (Figure 14). Indeed, hysteresis appears when the process is not reversible. This could be modelled if we assume the particles to slide when they cannot deform more. We define a mobilization length l m, which tunes the friction force introduced previously: F friction = k f δ i if δ i < l m : the bead deforms; the friction force increases linearly. F friction = k f l m if δ i > l m : after l m, the bead cannot deform more and slides; the friction force is constant. Introducing this behaviour when the packing is subject to compression/decompression cycles might explain the hysteresis. The situation is described by schematics for a single bead in Figure 16: During compression: when a force F is applied, the bead first deforms (δ > 0), and F friction = k f δ > 0 (B). Then, the bead slides (C), and F friction = k f l m. During decompression: the bead is deformed in the opposite direction form δ > 0 i.e. F friction > 0 (D), to δ < 0 i.e. F friction < 0 (F). Finally, the bead slides again and F friction = k f l m (G). COMPRESSION DECOMPRESSION F δ δ A B C D E F G Figure 16: Successive stages of a bead under the action of force F, touching a wall with friction such that F friction = k f δ if δ < l m and F friction = k f l m if δ > l m. One can try to solve the N equations k(δ i δ i 1 ) + k(δ i δ i+1 ) + F friction,i = 0, for i [ 1 ; N ] and boundary conditions δ 0 = z (at the top) and δ N+1 = 0 (at the bottom). Using F = k( z δ 0 ), one can obtain the relation between F and z. However, we did not manage to find a way to solve it analytically because the friction force has to be re-defined after each compression/decompression stage (in Figure 16, δ is defined differently in compression and decompression stages). We believe that this model can be solved numerically, and it is likely to reproduce the behaviour observed experimentally in Figure 14-(left). 16

21 Experimental study of hydrogel sphere packings close to jamming 5 Couette geometry experiment In previous sections, we show friction has a major contribution in the response we measure in both cases of compression and shear. Consequently, the values of G, M and K we obtain for different pressures cannot be compared to the theoretical predictions (see Section 1). Therefore, our goal here is to use a new geometry, where the contact surface between moving beads and glass is as small as possible. 5.1 Setup and motivations Our choice is to use a Taylor-Couette geometry: the material is confined in a hollow cylinder of inner radius R1, outer radius R2 and height L. Then, the material is sheared by imposing a torque on the inner cylinder, whereas the external container is motionless. Using this geometry, the only beads in contact with boundaries that have to slide are the ones at the top and the bottom. Therefore, the effect of friction should be less significant. It is important to notice that one can only get the shear modulus G using this geometry. However, we can compare these measurements to the previous ones (see Section 2.2). Setup We use a beaker of radius R2 = 6.73 cm with vertical strips glued to the walls, and a PVC vane of radius R1 = 2.27 cm made of four blades. To confine the packing, we use a disk placed at the top with a central hole to let the vane in the packing. By adding rings of stainless steel of weight m = 40 g, we are able to control the pressure applied to the packing. We calculate the pressure using the corrected weight (gravity and buoyant force) and the surface of the disk (S = π(r22 R12 )), and assume it is homogeneous. The gap R2 R1 is large enough so that we are actually measuring the elasticity of several layers of beads. Using the rheometer, one can impose a triangular oscillating torque T and measure the angle of deflection θ. We choose a low frequency to make sure the evolution of the packing is quasi-static. R1 σ (r+dr) r+dr rθ R2 r u(r) L σrθ(r) θ R2 R1 Vane Packing Figure 17: (Left) Schematic and picture of the Couette geometry setup. The inner cylinder (a vane here) rotates due to a torque T. Picture of the Couette geometry setup. (Right) Top view of the packing when sheared ; The rotation of the vane creates a displacement field u(r) in the material, and tangential stresses σrθ (r). Derivation of G Our aim is to extract the shear modulus G from measurements performed with this geometry. We derive here the relationship between the torque T applied to the vane and its angle of deflection θ [22]. To describe the behaviour of the material, we used the displacement field in the tangential direction u(r), with r [R1 ; R2 ], the stress tensor σij and the deformation tensor γij with cylindrical coordinates (r, θ, z) (see Figure 17-(right)). None of those quantities will depend on 17

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