Discontinuous shear thickening in concentrated suspensions

Abstract

The flow of concentrated suspensions of solid particles can be suddenly blocked by the formation of a percolated network of frictional contacts above a critical value of the applied stress. Suspensions of magnetic particles coated with a superplastifier molecule were shown to produce a strong jamming transition. We find that, for these suspensions with an abrupt discontinuous shear thickening, a model using the divergence of the viscosity at a volume fraction that depends on the applied stress does not well describe the observed behaviour both below and above the critical stress. At a constant applied stress above the critical one, we have a stick–slip behaviour of the shear rate whose period can be predicted and scaled as the square root of the relaxation time of the frictional contacts. The application of a small magnetic field allows us to continuously decrease the critical shear rate, and it appears that the yield stress induced by the magnetic field does not contribute to the jamming transition. Finally, it is shown that this jamming transition also appears in the extrusion of a suspension through a die, but with a much slower dynamics than in the case of stress imposed on a rotational geometry.

This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

1. Introduction

A shear thickening fluid is one in which there is an increase in viscosity with shear rate, in contrast to a Newtonian fluid where the viscosity remains constant whatever the shear rate. In colloidal suspensions, this shear thickening is frequently observed and is due to the formation of aggregated flocs that result from the increase in the shear force between particles which overcomes the potential barrier due to the coating polymer or ionic layer. The suspending fluid being trapped inside these flocs, they behave as quasi solid particles. The result is an increase in the effective volume fraction and thus in the viscosity. A model of this mechanism was described in [1,2]. In a hard-sphere suspension where there are only hydrodynamic interactions, the viscosity does not depend on the shear rate and diverges at a maximum volume fraction Φ_max that is close to the maximum random close packing, Φ_RCP = 0.64:

$$\eta (\mathit{\Phi })={\eta }_0{\left(1-\frac{\mathit{\Phi }}{{\mathit{\Phi }}_{max}}\right)}^{-p},$$ 1.1

where p is an exponent close to 2. Experiments designed to find the value of the exponent p and of Φ_max suffer from inaccuracies related to the polydispersity, non-negligible short-range interactions, uncertainty in the measurement of volume fraction, etc.

In colloidal suspensions, the thermodynamic stresses, which scale as kT/a³, where a is the radius of the particles, are far from being negligible and are responsible for a first shear thinning behaviour before a shear thickening one due to the formation of hydroclusters when hydrodynamic forces dominate at large Péclet number [3]. Non-Brownian suspensions are obtained with particles whose radius is typically larger than one micrometre. Still, in these suspensions, sedimentation and van der Waals forces are present and together they provoke the aggregation of the particles if they are not coated with a polymer layer which generates an entropic repulsive force between the adsorbed layers on the surface of the particles. In the suspensions formed with non-Brownian particles, a shear thickening is observed when the volume fraction is increased (typically above 50%) and, above a critical volume fraction, this continuous shear thickening is replaced by a discontinuous shear thickening (DST), which appears as a decrease in the shear-rate in a stress-imposed ramp, or as a jump in stress in a shear-rate-imposed ramp. In a well monodisperse suspension, this transition was shown to be due to the disappearance of an ordered structure, formed of hexagonal planes sliding over each other, above a critical stress that is able to destroy it [4,5]. Nevertheless, this DST behaviour also happens in suspensions where the distribution of particles is far from being monodisperse and can also have an irregular shape, as is the case with the well-known corn-starch suspension [6,7]. Many other systems based on silica [8,9], gypsum [10], poly(methylmethacrylate) (PMMA) [11] and calcium carbonate [12,13] also show this jamming transition. There is a large body of evidence, supported by numerical simulation [14] and also by atomic force microscopy measurement of forces between particles [15], that this transition is related to the passage from lubricated contacts to frictional ones when the shear force increases enough to overcome the potential barrier formed by a stabilizing layer. The critical shear force needed to produce this transition depends on the characteristics of the adsorbed layer of polymer. For instance, we have shown that, for the same particles, we can observe a difference in critical stress by a factor of three by changing the superplastifier, and we have related this difference to the amplitude of the repulsive force generated by different polymers [13]. In the industrial processing of pastes, it is of interest to delay this jamming transition until the highest possible value of stress by the proper choice of the coating polymer; it would be still more interesting to be able to control the critical stress by external means. We have succeeded in doing so by using suspensions of magnetic particles whose interaction forces can be modulated by the application of a magnetic field [16]. Indeed, the critical shear rate is shown to decrease strongly with the amplitude of the magnetic field and, contrarily to expectation, the critical stress is not constant but increases with the magnetic field.

In the next section, we shall present some typical rheological behaviour associated with the jamming transition and we shall adapt the model of Wyart & Cates [17] in the presence of a yield stress and discuss the ability of this model to represent this transition. In §3, we shall more specifically address the control of the jamming transition with a magnetic field and we shall try to understand why the critical stress increases in the presence of a magnetic field. The fourth section is devoted to the study of the instabilities that occur in the regime of flow in the jammed state and the last section (§5) is devoted to a preliminary study of the jamming transition during the extrusion of a suspension through a die.

2. Rheological characteristics of the jamming transition

(a). Experimental signature of the jamming transition

When the volume fraction of a suspension of non-Brownian particles is increased, the rheogram, as represented by the shear stress versus the shear rate, will change progressively from a quasi-Newtonian behaviour to a shear thickening behaviour and will end up with the jamming transition, as represented in figure 1.

Figure 1.

Evolution of the rheology of a suspension of carbonyl iron particles in the presence of the PPP44 coating molecule. The curves shift to the left with increasing volume fraction. (Online version in colour.)

These rheograms are obtained on a rheometer MCR 501 (Anton Paar) in a plate–plate geometry, and the ramp of stress is made at typically 50 Pa min⁻¹, which is slow enough to guarantee the absence of thixotropy. The suspension is made of carbonyl iron particles in a mixture of ethylene glycol and water to which we have added 0.2% by mass of a superplastifier molecule, which is a polyoxyethylene polyphosphonate used in the cement industry. This molecule strongly adsorbs on the calcium sites of the particles through the $\text{P}{{\text{O}}_3}^{-}$ phosphonate ionic groups of the molecule. The polyoxyethylene tail is composed on average of 44 oxyethylene groups, CH₂–CH₂–O, which has a quite good affinity with water; in the following, we shall call this molecule PPP44 [13]. We see in figure 1 that even at a volume fraction as high as 57%, the yield stress τ_y is quite low (around 15 Pa) and that the dynamic viscosity is constant, so the whole rheogram is well represented by the Bingham law: $\tau ={\tau }_{\mathrm{y}}+{\eta }_{\mathrm{d}}\dot{\gamma }$. At Φ = 59%, the behaviour is already quite different, with a strong increase in the viscosity above a shear rate of 60 s⁻¹, followed by a domain with small oscillations. Then at Φ = 61%, there is a strong jamming transition with a sudden decrease in the shear rate followed by a regime of strong oscillations around a mean value of 35 s⁻¹. Finally, at 62%, there is also a jamming transition but at a lower shear rate. If we increase the volume fraction still further, at some given value close to 66% the system is completely jammed and there is no flow at all when the stress is increased. In fact, the complete blockage of the system whatever the imposed stress will depend a lot on the plasticity of the particles, the roughness of the walls of the cell and the non-dilatancy of the gap under large applied shear stress.

The DST transition can be more or less abrupt. For instance, in figure 2 we have reported the jamming transition for carbonyl iron particles (average diameter 1.2 µm), for calcium carbonate particles (average diameter 4 µm) and for silica particles (average diameter 5 µm). The three suspensions are in water, but the first two are stabilized by 0.2% by mass of the molecule PPP44, whereas there is no surfactant with silica whose stabilization is due to its natural ionization in water. The iron particles present the strongest transition with, above the critical shear stress τ_c = 100 Pa, large fluctuations of the shear rate—we shall come back to this point in §4. The calcium carbonate at Φ = 68% has also an abrupt transition for τ_c = 147 Pa, but followed by some much smaller oscillations of the shear rate. Finally, for the suspension of silica particles, there is a continuous decrease in the shear rate above τ_c = 12 Pa but without a clear regime of oscillation. Note also that the abrupt increase in shear rate at τ = 165 Pa is just due to the expulsion of the suspension out of the gap between the two plates. It is worth noting that some other systems do not show a decrease in the shear rate but only a halt to its increase, which corresponds to a vertical line on the stress–shear rate graph; this is the case for corn-starch suspension in water or silica particles in mineral oil [18]. These different regimes probably depend on the deformability of the particles and on the softness of the repulsive barrier formed by the coating polymer or ionic layer.

Figure 2.

Jamming transition for suspensions composed of iron particles (red symbols, thick line), calcium carbonate particles (solid blue parabolic line) and silica particles (solid green lower line). (Online version in colour.)

In figures 1 and 2, the rheograms correspond to a ramp of shear stress, that is to say to a continuous increase in the torque on the upper plate. In this situation, the transformation of the suspension in a jammed state can produce a decrease in the rotation speed or even stop it completely. But, if the motor imposes a ramp of velocity, when the shear rate exceeds the critical one, we expect a jump in the shear stress which will recover, if it exists, the branch where the shear rate begins again to increase in the regime of imposed stress. In fact, in the suspensions with an abrupt jamming transition, the jump of stress above the critical shear rate is very high and well above the rheometer maximum torque. Furthermore, the stress generated is transmitted radially and ejects the particles outside the suspension [19]. In a plate–plate geometry, the suspension goes out of the gap, while in cylindrical Couette geometry, particles are pushed outside the free surface and air enters the suspension, producing a foam at the surface. To get rid of a free interface and also to have a high enough torque capacity, we have used a high-torque viscosimeter developed by the CAD company, mainly to measure the viscosity of concrete. We have designed a cell where the suspension is confined by a Teflon seal (figure 3a). To prevent slipping, we have used either a vane or a double helix as the rotating tool; also the outside wall is serrated with stripes of depth 0.2 mm.

Figure 3.

(a) Sketch of the confined cell. (b) Ramp of shear rate. Iron suspension, Φ = 62%. (Online version in colour.)

This rheometer works in the imposed velocity mode and, in figure 3b, we have reported the stress measured during a ramp of shear rate. We see that there is an abrupt jump of stress which reaches 80 kPa; nevertheless, this stress does not remain at this high level but oscillates between the flowing and jamming state and its amplitude increases with the shear rate. In this case, the structure of the suspension manages to come back transiently to the flowing state, despite the fact that the shear rate is above the critical one. We shall discuss this point in the section (§3) devoted to the application of a magnetic field.

(b). Rheological model of the jamming transition

It is now quite well established that the DST is generated by the transition from lubricated to frictional contacts as shown by numerical simulations [20]. The key parameter is the fraction of frictional contacts f(σ), which is expected to be a growing function of the stress. It is necessary to relate the viscosity to this function; this can be done directly by assuming that the viscosity should diverge when a given fraction of frictional contacts is reached [13] or indirectly by means of the volume fraction as proposed in [17]. In this last approach, two limiting volume fractions are considered corresponding to f(σ) = 0 and f(σ) = 1. The first one Φ₀ corresponds to the maximum packing fraction of the suspension in the absence of frictional contacts. It would be the random close packing Φ₀ = 0.64 for monodisperse spheres, although the structure formed by the flow can change this diverging volume fraction, which can reach 0.7 and may correspond to the stacking of face-centred cubic multilayer rafts [21]. For polydisperse suspensions, it can be much higher, around Φ = 0.8 [22]. The second critical volume fraction is that corresponding to the jamming of the actual suspension, Φ_m, where all the contacts are frictional and the suspension can no longer flow whatever the stress. Then, for a given stress, the authors assume that the viscosity will follow the usual law of divergence of the viscosity, but with a maximum volume fraction Φ_m < Φ_j(σ) < Φ₀:

$$\eta (\sigma )={\eta }_0\frac{1}{{(1-\mathit{\Phi }/{\mathit{\Phi }}_j(\sigma ))}^p}\text{with}\hspace{0.17em}p=2.$$ 2.1

The diverging volume fraction Φ_j(σ) must be related to the function f(σ) because its two limits correspond to f = 0 and f = 1 and the simplest hypothesis is a linear dependence:

$${\mathit{\Phi }}_j(\sigma )={\mathit{\Phi }}_{\mathrm{m}}\hspace{0.17em}f(\sigma )+{\mathit{\Phi }}_0(1-f(\sigma )).$$ 2.2

As f(σ) is a growing function of the stress, the maximum packing fraction will decrease with the stress and so the viscosity will increase (cf. equation (2.1)). If the viscosity η(σ) grows faster than σ, then the shear rate $\dot{\gamma }(\sigma )=\sigma /\eta (\sigma )$ will begin to decrease, corresponding to the beginning of the DST transition. We are going to see how this model can be used to reproduce our experimental data. In figure 4, we have reported the rheograms obtained on a suspension of carbonyl iron with 0.2% by weight of superplastifier and different volume fractions. At Φ = 56%, we have a shear thickening behaviour followed, at high shear rates, by a slight ‘S-shaped’ characteristic of the DST transition. At Φ = 58%, the S-shape is more pronounced, and at Φ = 62% and Φ = 64%, the S-shape has been replaced by an abrupt decrease in the shear rate followed by an oscillation regime.

Figure 4.

Rheograms of a suspension of carbonyl iron with 0.2% by weight of PPP44 for different volume fractions. The curves are shifted to the left with the increase of volume fraction.

We have fitted the curve presenting the characteristic S-shape at Φ = 58% with the help of equations (2.1) and (2.2). The maximum volume fraction at which the suspension is completely jammed from the beginning is Φ_m = 67%. As already discussed, the maximum packing fraction without friction can be taken as Φ₀ = 0.8 for a polydisperse suspension. The shape of the function f(σ) is unknown; it is just a monotonic growing function between 0 and 1. The choice of the simple function 1 − exp(−σ/σ_c), where σ_c is the critical shear stress, was made in [17], but it is convenient to introduce a parameter λ which will determine the steepness of the function and will help to fit the experimental result. In this case, we can take f(σ) = 1 − exp(−λσ/σ_c). The two remaining parameters λ and η₀ can be obtained by requiring first that the curve passes through the point $({\sigma }_{\mathrm{c}},{\dot{\gamma }}_{\mathrm{c}})$ and second that the derivative $\text{d}\dot{\gamma }\text{/d}\sigma {|}_{\sigma ={\sigma }_{\mathrm{c}}}=0$ at this point. This last condition links λ to p and it is expressed by the following equation:

$$({\sigma }_{\mathrm{c}}-{\tau }_{\mathrm{y}}){\frac{\text{d}\text{Log}(\eta )}{\text{d}\sigma }|}_{\sigma ={\sigma }_{\mathrm{c}}}=1\to \frac{p\mathit{\Phi }({\mathit{\Phi }}_0-{\mathit{\Phi }}_{\mathrm{m}})}{{\mathit{\Phi }}_{j\mathrm{c}}[{\mathit{\Phi }}_{j\mathrm{c}}-\mathit{\Phi }]}{\frac{\text{d}f(\sigma )}{\text{d}\sigma }|}_{\sigma ={\sigma }_{\mathrm{c}}}=\frac{1}{{\sigma }_{\mathrm{c}}-{\tau }_{\mathrm{y}}}.$$ 2.3

We have introduced the yield stress τ_y which is often present at high volume fraction. Together with equation (2.1), it imposes the condition that the viscosity η(σ_c) at the critical point satisfies the Bingham law. In this expression, Φ_jc is the value of Φ_j for σ = σ_c. It appears that with the value p = 2 for the divergence of the viscosity and f(σ) = 1 − exp(−λσ/σ_c), there is no value of λ that can fulfil equation (2.3) to represent the S-shape at Φ = 0.58 with the corresponding critical point σ_c = 927 Pa and ${\dot{\gamma }}_{\mathrm{c}}=536\hspace{0.17em}{\text{s}}^{-1}$. A stronger divergence of the viscosity would be needed (p = 4) to satisfy equation (2.3). Instead, we have used a sigmoid function for f(σ) which allows us to satisfy this condition with p = 2:

$$f(\sigma )=\frac{1}{1+\exp [-\mathrm{\lambda }(\sigma /{\sigma }_{\mathrm{c}}-1)]}.$$ 2.4

In equation (2.1), the value of η₀ is fixed by the following condition:

$${\eta }_0=\frac{{\sigma }_{\mathrm{c}}-{\tau }_{\mathrm{y}}}{{\dot{\gamma }}_{\mathrm{c}}}{\left(1-\frac{\mathit{\Phi }}{{\mathit{\Phi }}_j({\sigma }_{\mathrm{c}})}\right)}^p,$$ 2.5

but does not correspond to the viscosity of the suspending fluid. In figure 5, we have plotted the experimental curves of figure 4 for Φ = 0.58 and Φ = 0.62 together with the theoretical expressions given by equations (2.1), (2.2) and (2.4). The values of λ and η₀ obtained from equations (2.3) and (2.5) are, for Φ = 0.58, η₀ = 0.076 Pa s, λ = 3.12, and for Φ = 0.62, η₀ = 0.077 Pa s, λ = 2.51, respectively. The yield stress was taken, respectively, as 7 Pa and 20 Pa. We first note that the model reproduces qualitatively the S-shape at Φ = 0.58 but not at all the abrupt decrease at Φ = 0.62. To have this abrupt decrease, the viscosity should increase abruptly, but the divergence of the viscosity (equation (2.1)) with Φ_j(σ) > Φ_m = 0.67 cannot reproduce it because it will smoothly tend towards η_max = (1 − Φ/Φ_m)⁻² and then end up with a Newtonian behaviour at high stress, as we see on figure 5 (black solid line). A possible explanation of this failure would be the onset of instability at the turning point. We shall discuss this hypothesis in §4. Concerning the parameter η₀, it is interesting to note that it remains almost the same for the two volume fractions.

Figure 5.

Fit of the experimental rheograms with equations (2.1), (2.2) and (2.4) for Φ = 0.58 (thin blue solid line) and Φ = 0.62 (thin black solid line). (Online version in colour.)

3. Effect of a magnetic field on the jamming transition

The jamming transition is provoked by the failure of the repulsive coating layer adsorbed on the particles to prevent a direct contact between the surfaces, above a given applied stress, the applied stress is a key parameter because we expect that the force which pushes two particles against each other will behave like F_a = πa²σ_a, where σ_a is the applied stress. Above a given applied force, we assume that the compression of the polymer layer is large enough to be able to push away the molecules adsorbed on the surface and then to induce the transition from a lubricated to a frictional regime. So, for σ_a > σ_c, the transition should occur. If we add an external attractive force between the particles, such as a magnetic force, we could expect that the transition will take place at a lower hydrodynamic stress, σ_h, because we have now the condition σ_h + τ_y > σ_c, where τ_y is the magnetic stress which is approximately equivalent to the yield stress due to the application of the magnetic field [23,24]. Then, the condition

$${\eta }_0\frac{1}{{(1-\mathit{\Phi }/{\mathit{\Phi }}_j({\sigma }_{\mathrm{c}}))}^2}{\dot{\gamma }}_{\mathrm{c}}+{\tau }_{\mathrm{y}}={\sigma }_{\mathrm{c}}$$ 3.1

states that if τ_y increases, the critical shear rate should decrease, at least if Φ_j(σ_c) does not vary too much with the magnetic field. This is what we observe in figure 6 where we present a complete sweep of amplitudes of magnetic field for a volume fraction of carbonyl iron particles with Φ = 63% suspended in a mixture of water and ethylene glycol. In these experiments, we have used a plate–plate geometry and the field is perpendicular to the surface of the rotating disc.

Figure 6.

Ramp of stress for different amplitudes of magnetic field at Φ = 63%. A magnetic field of 10 kA m⁻¹ is equivalent to an induction of 12.5 × 10⁻³ T.

We actually observe the decrease in the critical shear rate as the amplitude of the magnetic field increases and also a regular increase in the critical stress σ_c. In figure 7, we have reported the critical stress (figure 7a) and the difference between the critical stress and the yield stress (figure 7b). We observe in figure 7a that the final critical stress is about two times the initial one. But, if we subtract the yield stress from the critical one to keep only the hydrodynamic component of the stress, then this component does not increase with the magnetic field (figure 7b). This indicates that it is the hydrodynamic component which drives the jamming transition; the magnetic one just modifies the yield stress and the dynamic viscosity of the suspension (because the slope of the curve increases steadily with the field), but does not contribute directly to the force which will sweep out of the coating polymer.

Figure 7.

(a) Critical stress of the jamming transition versus the applied magnetic field for Φ = 63%. (b) The critical stress minus the yield stress for the same experiment as in (a). (Online version in colour.)

In other words, this means that it is not so much the compression of the polymer layer that we need to get rid of the polymer but rather the shear force that is provided by the hydrodynamic stress. If it was the compression force that was required to remove the polymer, then, because both the magnetic and the hydrodynamic stresses contribute to the compression, the critical stress should remain constant instead of increasing, as shown in figure 7a.

Another interesting insight is obtained if we impose a given shear rate and raise the magnetic field. The result of this experiment is shown in figure 8. A shear rate of 30 s⁻¹ was applied from the beginning and we progressively raised the amplitude of the magnetic field. The jump of stress occurs a few seconds after the step of 21 kA m⁻¹. This jump of stress corresponds to two orders of magnitude and is actually limited by the maximum torque of the rheometer (here 0.3 N m). Using the homemade rheometer with the cell shown in figure 3a, we found that the yield stress can jump up to 150 kPa for a field of only 8 kA m⁻¹. Furthermore, instead of oscillating between a jammed state and a flowing state as is the case in the absence of the field (cf. figure 3b), it remains in the jammed state [25], which allows us to use this field-induced jamming transition for applications in force or torque transmission.

Figure 8.

Increase of the field step by step at a constant shear rate of 30 s⁻¹. Volume fraction Φ = 0.61.

We have seen that, above the jamming transition, we observe large fluctuations of the shear rate during a ramp of stress (figure 2) or of the stress during a ramp of shear rate (figure 3b). We are now going to analyse this instability more specifically, which occurs above the critical point both in the absence and in the presence of a magnetic field.

4. Analysis of the instabilities above the jamming transition

(a). No magnetic field

In contrast to what is observed with corn-starch suspensions or some silica suspensions, where we have a soft transition manifested by an S-shape, as is the case also at the lowest volume fraction (cf. figure 4), we have here an immediate decrease in the shear rate followed by a regime of strong oscillations (cf. figures 1 and 2). These oscillations were previously observed at imposed shear rate [8] and imposed stress [26,27] in shear thickening suspensions. If we conduct an experiment at a fixed stress above the critical one in a plate–plate geometry, as we see in figure 9, we have some regular oscillations with a sawtooth shape. At the lowest stress, just above the transition, the shear rate can even change sign, as we previously observed in calcium carbonate suspensions, meaning that the rotational velocity of the upper disc is inverted during a short interval of time. The second observation is that, at higher imposed stress, the oscillations are of smaller amplitude and higher frequency. We previously [13] gave an explanation for these oscillations with a sawtooth shape by introducing the inertia of the tool in the equation of motion in the presence of a rheological model presenting an S-shape but with a simple equilibrium model where the viscosity was assumed to diverge as η( f) = η₀ ( f − f_M)^−p, where f_M was a parameter. We are now reconsidering this model with the viscosity represented by equations (2.1) and (2.2). We are also including the yield stress, τ_y, in the two dynamical equations for the stress and the fraction of frictional contacts:

$$\frac{I}{C}\ddot{\gamma }(t)={\sigma }_{\mathrm{a}}(t)-\eta (\hspace{0.17em}f(t))\dot{\gamma }(t)-{\tau }_{\mathrm{y}}\text{or}{\sigma }_{\mathrm{s}}(t)={\sigma }_{\mathrm{a}}(t)-\frac{I}{C}\ddot{\gamma }(t)=\eta (\hspace{0.17em}f(t))\dot{\gamma }(t)+{\tau }_{\mathrm{y}}$$ 4.1

and

$$\frac{\mathrm{\partial }f}{\mathrm{\partial }t}=-\frac{1}{\tau }(\hspace{0.17em}f-f_{\mathrm{e}}({\sigma }_{\mathrm{s}})).$$ 4.2

Figure 9.

Shear rate versus time for a constant applied stress of 120 Pa (upper, blue, curve) and then 150 Pa (lower, red, curve). Carbonyl iron suspension in water at Φ = 0.62 with 0.2% by weight of PPP44. (Online version in colour.)

In equation (4.1), I = 9.36 × 10⁻⁵ is the inertia of the tool plus that of the motor, which is attached on the same axis, and C = πR⁴/2h = 2.51 × 10⁻⁴ is a constant specific to the plate–plate geometry having a gap h = 1 mm and a radius R = 20 mm. The term ρ dv/dt expressing the inertia of the suspension does not appear because it is negligible compared to the mechanical inertia. Nevertheless, this term, which gives a spatially variable stress, is responsible also for an instability but at much higher frequencies [28] than those appearing in figure 9.

An important point is that the real stress, σ_s(t), acting on the suspension is not the applied one but the applied one plus the stress coming from the inertia of the tool, as was demonstrated in [27]. Consequently, the fraction of frictional contacts should depend on the actual stress, σ_s, which is different from the applied stress, σ_a, because of the inertia term (see equation (4.1)). The yield stress τ_y introduced in this equation (4.1) can arise from any attractive force, including the magnetic one. We can apply a linear stability analysis to equations (4.1) and (4.2) to predict the frequency of the oscillations.

The perturbations are given in the usual way, with respect to the equilibrium values:

$$f=f_{\mathrm{e}}({\sigma }_{\mathrm{a}})+\delta f,\dot{\gamma }={\dot{\gamma }}_{\mathrm{a}}+\delta \dot{\gamma }\text{with}\delta f=A{\text{e}}^{\mathit{\Omega }t}\text{and}\delta \dot{\gamma }=B{\text{e}}^{\mathit{\Omega }t}.$$ 4.3

The viscosity is a function of f(σ) through equations (2.1) and (2.2) and needs to be developed as follows:

$$\eta (\hspace{0.17em}f)=\eta (\hspace{0.17em}{f}_{\mathrm{e}})+{\frac{\mathrm{\partial }\eta }{\mathrm{\partial }f}|}_{f_{\mathrm{e}}}\delta f\text{and we have also}{f}_{\mathrm{e}}({\sigma }_{\mathrm{s}})=f_{\mathrm{e}}({\sigma }_{\mathrm{a}})+({\sigma }_{\mathrm{s}}-{\sigma }_{\mathrm{a}}){\frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}.$$ 4.4

Taking into account that (cf. equation (4.1)) ${\sigma }_{\mathrm{s}}-{\sigma }_{\mathrm{a}}=-(I/C)\ddot{\gamma }$ and that $\ddot{\gamma }=\delta \ddot{\gamma }\hspace{0.17em}\text{since}\hspace{0.17em}{\ddot{\gamma }}_{\mathrm{e}}=0$ at equilibrium, and inserting equations (4.3) and (4.4) into equations (4.1) and (4.2), we end up, after keeping only the linear terms in the perturbation, with

$$\left[\frac{I}{C}\mathit{\Omega }+\eta ({\sigma }_{\mathrm{a}})\right]\delta \dot{\gamma }+{\dot{\gamma }}_{\mathrm{a}}{\frac{\mathrm{\partial }\eta }{\mathrm{\partial }f}|}_{{\sigma }_{\mathrm{a}}}\delta f=0\text{and}\frac{I}{C}{\frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}\mathit{\Omega }\delta \dot{\gamma }+(1+\tau \mathit{\Omega })\delta f=0.$$ 4.5

The condition of a zero discriminant of these two equations gives the following equation for Ω:

$${\mathit{\Omega }}^2+\mathit{\Omega }\left[\frac{1}{{\tau }_{\mathrm{I}}({\sigma }_{\mathrm{a}})}+\frac{\eta ({\sigma }_{\mathrm{a}}^{\hspace{0.17em}})}{\tau }{\frac{\mathrm{\partial }\dot{\gamma }}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}\right]+\frac{1}{\tau {\tau }_{\mathrm{I}}}=0\text{with}{\tau }_{\mathrm{I}}({\sigma }_{\mathrm{a}})=\frac{I}{C\eta ({\sigma }_{\mathrm{a}})}=\frac{I}{\eta ({\sigma }_{\mathrm{a}})}\frac{2h}{\pi R^4},$$ 4.6

where τ_I is the inertial time, which depends strongly on the radius, R, of the upper disc. To derive equation (4.6), we have used the following equation:

$${\frac{\mathrm{\partial }\eta }{\mathrm{\partial }f}|}_{{\sigma }_{\mathrm{a}}}{\frac{\mathrm{\partial }f}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}={\frac{\mathrm{\partial }\eta }{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}=\frac{1}{{\dot{\gamma }}_{\mathrm{a}}}\left[1-\frac{\eta ({\sigma }_{\mathrm{a}})}{\tau }{\frac{\mathrm{\partial }\dot{\gamma }}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}\right].$$

In these equations, σ_a is the applied stress and ${\dot{\gamma }}_{\mathrm{a}}$ the corresponding shear rate in the absence of instability. The result for the angular frequency is then

$$\mathit{\Omega }=0.5\left[-\left(\frac{1}{{\tau }_{\mathrm{I}}({\sigma }_{\mathrm{a}})}+\frac{\eta ({\sigma }_{\mathrm{a}}^{\hspace{0.17em}})}{\tau }{\frac{\mathrm{\partial }\dot{\gamma }}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}\right)\pm \sqrt{{\left(\frac{1}{{\tau }_{\mathrm{I}}({\sigma }_{\mathrm{a}})}+\frac{\eta ({\sigma }_{\mathrm{a}}^{\hspace{0.17em}})}{\tau }{\frac{\mathrm{\partial }\dot{\gamma }}{\mathrm{\partial }\sigma }|}_{{\sigma }_{\mathrm{a}}}\right)}^2-\frac{4}{\tau {\tau }_{\mathrm{I}}({\sigma }_{\mathrm{a}})}}\right].$$ 4.7

If the first term becomes negative, then the term exp(Ωt) diverges and we have the growth of the instability whose period is given by the following equation:

$$T=\frac{2\pi }{{\mathit{\Omega }}_{\mathrm{c}}^{\hspace{0.17em}}}=2\pi \sqrt{\tau {\tau }_{\mathrm{I}}({\sigma }_{\mathrm{c}})}.$$ 4.8

In principle, we see that the parameter τ describing the rate at which the percolation fraction returns to its equilibrium value can be deduced from the experimental period of the oscillations. We have compared, in figures 10 and 11, the predictions of the dynamical model (equations (4.1) and (4.2)) both for the ramp of stress and for the two stationary values of 120 and 150 Pa, which are above the critical stress σ_c = 97 Pa. In figures 9–11, the experimental curves correspond to the same suspension of carbonyl iron in water at Φ = 62% with 0.2% of superplastifier.

Figure 10.

Results for Φ = 62% iron particles in water with 0.2% PPP44. Purple (upper, left) curve, experimental stress versus applied shear rate. Blue (lower, right) curve, model with equations (4.1) and (4.2). (Online version in colour.)

Figure 11.

Oscillations of shear rate at constant applied stress. Blue diamond: experiment at 120 Pa. Solid blue line: model at 120 Pa. Black dotted line: model at 150 Pa. (Online version in colour.)

In figure 10, the theoretical curve (in blue) was obtained with τ_y = 15 Pa, λ = 2.48 for the sigmoid (equation (2.4)) and η₀ = 0.059 for the viscosity (equation (2.5)). These two last values are those which match the critical point to the experimental one obtained with equations (2.3) and (2.5). We can see that, given these constraints at $\dot{\gamma }=0$ and at $\dot{\gamma }={\dot{\gamma }}_{\mathrm{c}}=34\hspace{0.17em}{\text{s}}^{-1}$, the theoretical curve is quite different from the experimental one. In fact, the theoretical model is based on a progressive shear thickening before the DST transition because the jamming volume fraction Φ_j(σ) decreases continuously when σ increases, which results in an increasing viscosity with σ. On the contrary, the experiment (purple curve in figure 10) can be represented by the Bingham law up to the transition point, meaning that the fraction of percolating contacts remains low and suddenly increases at the critical stress. As in the experiment, above the critical stress, we obtain strong oscillations of the shear rate, but with two differences: the amplitude of these oscillations is smaller than in the experiment, and their maximum value remains close to or slightly above ${\dot{\gamma }}_{\mathrm{c}}$. This is more visible in figure 11 where we have plotted the experimental oscillations of the shear rate for a constant applied stress of 120 Pa (blue diamonds) together with the prediction of the model (solid blue line). We see that the theoretical oscillations are well above the experimental ones. To get the same period as the experimental one, T_exp = 0.16 s, we have taken τ = 2 ms. The theoretical period given by equation (4.8) with τ = 2 ms is slightly smaller, T_th = 0.11 s; this is not surprising because it comes from a linear approximation valid when the instability just begins to develop.

Finally, we see, in figure 11, that for a higher stress σ = 150 Pa, the period of the oscillation has slightly decreased (black dotted line) with T_th = 0.12 s. This is coherent with the experimental observation (cf. figure 9), although the decrease is more pronounced with T_exp = 0.083 s. Qualitatively, this decrease of the period is related to the decrease of τ_I(σ) (equations (4.6) and (4.8)) because the viscosity increases with the stress.

To reproduce the large value of the oscillations observed in figure 9, it is necessary to use a different model for the divergence of the viscosity. For instance, in figure 12, we have used a stronger divergence of the viscosity with p = 4 instead of p = 2. Comparing the black curve with the red one, we see that the amplitude of the oscillations is increased and also that the drop of the shear rate is more abrupt like that of the experimental curve (blue triangles). Nevertheless, the major difference from the experimental curve is that the upper value of the shear rate remains close to the critical one whereas in the experiments, after the abrupt recoil of the shear rate, the oscillations never again reach the critical value (cf. figure 10 or 1). The use of p = 4 in equation (2.1) for the viscosity has no theoretical justification, and it just emphasizes the need for a more abrupt increase in the viscosity with the stress or, in other words, to generate a stronger negative slope in the S-shape of the stress versus shear rate curve. We believe that, because the jamming transition generates a frictional stress, there is no reason to stick to a dependence of the viscosity that is only justified in the presence of lubrication and soft repulsive forces.

Figure 12.

Comparison of two divergences of the viscosity (equation (2.1)) with either p = 2 (black triangles) or p = 4 (red triangles). The blue triangles (lowest curve) are from the experiment.

In [13], we took η( f) = η₀ ( f(σ) − f_M)^−p, which is somewhat arbitrary but takes into account that the divergence of the viscosity is directly driven by a critical fraction of frictional contacts rather than by a critical volume fraction, even if this critical fraction should be a function of the volume fraction of the suspension.

(b). Influence of magnetic field on the oscillation regime

As we have seen previously in figure 6, the application of a magnetic field allows us to decrease the critical shear rate of the jamming transition. We are going to investigate the effect of a magnetic field on the oscillation regime above the transition. This effect is shown in figure 13 where we have increased the magnetic field step by step, keeping the applied stress constant, always at a volume fraction of 62%. The oscillations of the shear rate quickly decrease when we increase the field and their frequency increases. Then, their shape becomes more irregular (black line with crosses) and finally, above 10 kA m⁻¹, they have totally disappeared (solid horizontal green line).

Figure 13.

Oscillating regime for the shear rate at constant applied stress, σ_a = 120 Pa, for different magnetic fields. Volume fraction of carbonyl iron Φ = 62%. (Online version in colour.)

In fact, as shown in figure 6, when we increase the magnetic field, we also increase the critical stress above which the instability appears. It is then understandable that, if we apply a constant stress (here 120 Pa) and increase the magnetic field, the critical stress will go above the applied one and we shall fall inside the stable region, so the instability will disappear. Nevertheless, if we apply the model described above by increasing progressively the critical stress between 100 Pa (the critical stress in the absence of the field) and 120 Pa to represent the effect of the magnetic field, we do not observe the same behaviour as that described in figure 13: the amplitude and the frequency of oscillations remain practically constant until a critical stress of 119.5 Pa is reached, at which the oscillations suddenly disappear. Once again, it appears that the description of the evolution of the viscosity with stress (equations (2.1), (2.2) and (2.4)) is not well adapted to systems where the fraction of frictional contacts varies abruptly with the stress.

5. Jamming transition in a capillary

Until now, we have considered rotational geometries, but in many industrial processes, like the injection of pastes or ceramics inside moulds, the suspension is pushed inside a capillary, so it is important to see how the jamming transition behaves in a capillary. To achieve this aim, we have used a Malvern RH7 capillary rheometer whose diameter of the barrel was 9.5 mm and its length 28 cm. The diameter of the capillary was d = 2R = 0.75 mm and its length L = 3 mm. A pressure sensor was placed inside a hole perpendicular to the axis of the barrel and close to its end where the capillary is screwed.

For a viscous medium, the shear rate at the wall of the capillary and the pressure drop are, respectively, given by the following equations:

$$\dot{\gamma }=\frac{4Q}{\pi R^3}\text{and}{P}_{\mathrm{vis}}=\frac{8L\eta Q}{\pi R^3},$$ 5.1

where η is the viscosity of the suspension, Q the volume flow rate and L the length of the capillary. The experiment was done with a volume fraction of iron Φ = 64% in a mixture of ethylene glycol and water always with 0.2% of superplastifier. The viscosity of the suspension was η = 5.4 Pa s. The pressure versus the shear rate is represented in figure 14a for two different ramps of shear rate.

Figure 14.

(a) Pressure (MPa) versus shear rate in a capillary rheometer. Suspension of iron Φ = 64%. Red curve: slow increase of shear rate (as in panel (b)). Black curve: fast increase. (b) Evolution of shear rate (curve) and of the pressure (red curve) with time. It corresponds to the red curve in panel (a).

The first one corresponds to a constant shear rate of 50 s⁻¹ during 300 s and then a ramp from 50 s⁻¹ to 130 s⁻¹ in 700 s; the second one to a ramp of shear rate from 0 to 180 s⁻¹ in 175 s. In figure 14b, we have plotted, for the slow ramp, the pressure and the shear rate versus time. We can observe that, when the shear rate is raised slowly, we have an abrupt transition at the last step of shear rate with the pressure rising from 0.02 to 0.3 MPa. This is the signature of a jamming transition; otherwise, the pressure should vary linearly with the shear rate (cf. equation (5.1)). After this peak, if the shear rate is kept constant, the pressure decreases slightly and fluctuates around 0.22 MPa (see figure 14b). If, on the contrary, we increase the shear rate more rapidly, then we still observe an increase in pressure, which is not at all linear with time but does not show this abrupt transition. The pressure steps observed in figure 14a (black line) correspond to the shear rate steps. If we calculate the pressure at the maximum shear rate of 130 s⁻¹ from equation (5.1), that is to say for the flow of a viscous suspension, we obtain a pressure drop of 0.011 MPa instead of 0.22 MPa. This means that we no longer have a suspension flowing in the capillary, but a less concentrated suspension flowing through a jammed skeleton of particles forming a porous medium. Although not directly related to the jamming transition, this phenomenon is well known in the extrusion of pastes where a plug of solid particles is formed and the liquid of the suspension filters through this porous plug [29,30]. It was also studied in suspensions of PMMA particles, with a suspending liquid of about the same refractive index to visualize the flow, which was sucked inside a syringe [31]. If we consider a flow of the suspending fluid which filters through a porous medium made of a network of the solid particles, instead of a viscous flow, the pressure drop can be obtained from the Carman–Kozeny equation:

$$P_{\mathrm{por}}=L\mu \frac{Q}{\pi R^2}\frac{1}{K}\text{with}K=\frac{{\epsilon }^3{d}^2}{180{(1-\epsilon )}^2}.$$ 5.2

Here K is the Carman–Kozeny constant for a bed of spherical particles of diameter d, ε = 1 − Φ is the porosity and μ is the viscosity of the suspending fluid. For the maximum shear rate, we have now P_por = 13.6 MPa. This is much bigger than the observed pressure. A possible explanation is that the jamming does not occur inside the capillary but rather before its entrance [29], so the superficial velocity Q/(πR²) and the filtration pressure would be much lower than predicted by equation (5.2) because R is no longer the radius of the capillary but rather the diameter of the barrel. It is also coherent with the high value of the critical shear rate $\dot{\gamma }\approx 130\hspace{0.17em}{\text{s}}^{-1}$ with respect to the one obtained in rotational geometry, which is around 30 s⁻¹, meaning that the jamming does not occur in the capillary. The fluctuations of pressure observed at constant shear rate are probably produced by the intermittent collapse of the jammed structure at the entrance of the die as observed in [26].

6. Conclusion

In this paper, we have presented experimental results showing a strong discontinuous shear thickening obtained with a suspension of carbonyl iron particles in the presence of a superplastifier molecule. The jump of stress at imposed velocity can reach more than 100 kPa. The model proposed in [17] can reproduce the S-shape observed at the lowest volume fraction, but in this model the transition is preceded by a shear thickening behaviour that is practically absent in our experiments at high volume fraction. Introducing a relaxation time, τ, for the fraction of frictional contacts and the inertia of the tool used in rotational rheometry has allowed us to recover the right frequency of the oscillations with only one supplementary parameter, τ. A linear stability analysis shows that the period of the oscillations is simply related to the square root of the product of τ and of the inertial time (equation (4.8)), allowing us to obtain directly this relaxation time from the experimental period. The amplitude of the oscillations obtained with the model of Wyart & Cates [17] appears too small. To reproduce both the amplitude and the shape of these oscillations, a stronger dependence of the viscosity with the stress is needed. The jamming transition of this suspension of iron particles is very sensitive to the application of a magnetic field and the critical shear rate decreases quickly with the amplitude of the magnetic field, but simultaneously the critical stress increases. Nevertheless, if we subtract the magnetic stress from the applied one, the difference remains constant, indicating that it is the hydrodynamic stress which drives the jamming transition. We have also found that this jamming transition occurs in capillaries and manifests itself by the building of a porous medium constituted of particles in frictional contact. The dynamics of this transition is much slower than in rotational geometry and is probably related to the formation of a plug at the entrance of the die. Further investigations, especially in the presence of a magnetic field, are needed to understand the dynamics of the formation of this plug and the influence of sedimentation on the time of its formation. Depending on these results, we can envisage several applications based on the control of the pressure at imposed flow rate or of the flow rate at imposed pressure, with a small magnetic field.

Data accessibility

This article has no additional data.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the Centre National d'Etudes Spatiales CNES.