^{1,a)}and Heinrich M. Jaeger

^{1}

### Abstract

Many densely packed suspensions and colloids exhibit a behavior known as Discontinuous Shear Thickening in which the shear stress jumps dramatically and reversibly as the shear rate is increased. We performed rheometry and video microscopy measurements on a variety of suspensions to determine the mechanism for this behavior. We distinguish Discontinuous Shear Thickening from inertial effects by showing that the latter are characterized by a Reynolds number but are only found for lower packing fractions and higher shear rates than the former. Shear profiles and normal stress measurements indicate that, in the shear thickening regime, stresses are transmitted through frictional rather than viscous interactions. We come to the surprising conclusion that for concentrated suspensions such as cornstarch in water which exhibit the phenomenon of Discontinuous Shear Thickening, the local constitutive relation between stress and shear rate is not necessarily shear thickening. If the suspended particles are heavy enough to settle, we find the onset stress of shear thickening corresponds to a hydrostatic pressure from the weight of the particle packing where neighboring particles begin to shear relative to each other. Above , dilation is seen to cause particles to penetrate the liquid–air interface of the sheared sample. The upper stress boundary of the shear thickening regime is shown to roughly match the ratio of surface tension divided by a radius of curvature on the order of the particle size. These results suggest a new model in which the increased dissipation in the shear thickening regime comes from frictional stresses that emerge as dilation is frustrated by a confining stress from surface tension at the liquid–air interface. We generalize this shear thickening mechanism to other sources of a confining stress by showing that, when instead the suspensions are confined by solid walls and have no liquid–air interface, is set by the stiffness of the most compliant boundary which frustrates dilation. All of this rheology can be described by a nonlocal constitutive relation in which the local relation between stress and shear rate is shear thinning, but where the stress increase comes from a normal stress term which depends on the global dilation.

The authors thank S. Nagel, T. Witten, and W. Zhang for thoughtful discussions. The authors thank Marc Miskin for designing the optical setup used in the shear profile videos, Franco Tapia Uribe for taking preliminary shear profile measurements, and Trevor Martin for machining the solid wall setup for the rheometer. This work was supported by DARPA through Army Grant No. W911NF-08-1-0209 and by the NSF MRSEC program under DMR-0820054.

I. INTRODUCTION

II. BACKGROUND

III. MATERIALS AND METHODS

A. Suspensions

B. Tools

C. Measurement procedure

IV. INERTIAL SCALINGS

V. GRAVITY AND THE ONSET STRESS

VI. SHEAR PROFILE

A. Local constitutive relations

B. Localized viscosity curves

VII. NORMAL FORCES AND THE BOUNDARY CONDITION

A. Frictional scaling

B. Transient normal-force-control measurements

VIII. DILATION

IX. CAPILLARY FORCES

A. Relating dilation to confining stress

B. Surface tension scaling

X. SOLID BOUNDARIES

XI. DISCUSSION

A. Nonlocal constitutive relation

B. Dominant stress scales for the onset

C. Connection to confining stresses in other systems

D. Summary of the mechanism for discontinuous shear thickening

XII. CONCLUSIONS

### Key Topics

- Shear thickening
- 305.0
- Suspensions
- 138.0
- Shear rate dependent viscosity
- 116.0
- Shear rate dependent structure
- 64.0
- Surface tension
- 52.0

## Figures

(a) A standard parallel plate rheometer setup. The suspension is confined between the plates by the surface tension of the suspending liquid. (b) A modified parallel plate setup with solid walls around the sample. In this setup, the use of a suspending liquid is optional, and the wall stiffness can be modified by inserting layers of different stiffness between the suspension and plate.

(a) A standard parallel plate rheometer setup. The suspension is confined between the plates by the surface tension of the suspending liquid. (b) A modified parallel plate setup with solid walls around the sample. In this setup, the use of a suspending liquid is optional, and the wall stiffness can be modified by inserting layers of different stiffness between the suspension and plate.

(a) Apparent viscosity vs stress for 100 μm glass spheres in mineral oil taken with different measurement durations to show hysteresis loops. Open symbols: measurement duration of 40 s per decade of the stress ramp. Solid symbols: 500 s per decade. Up-pointing triangles correspond to increasing stress ramps, while down-pointing triangles correspond to decreasing stress ramps. (b) Characterization of the hysteresis as the geometric mean of the viscosity ratio between the decreasing and increasing ramps of the hysteresis loop, plotted for different ramp durations per decade of stress. Dashed line: a ratio of 1 between increasing and decreasing ramps corresponding to no hysteresis. Dotted line: ramp rate used for later steady state measurements.

(a) Apparent viscosity vs stress for 100 μm glass spheres in mineral oil taken with different measurement durations to show hysteresis loops. Open symbols: measurement duration of 40 s per decade of the stress ramp. Solid symbols: 500 s per decade. Up-pointing triangles correspond to increasing stress ramps, while down-pointing triangles correspond to decreasing stress ramps. (b) Characterization of the hysteresis as the geometric mean of the viscosity ratio between the decreasing and increasing ramps of the hysteresis loop, plotted for different ramp durations per decade of stress. Dashed line: a ratio of 1 between increasing and decreasing ramps corresponding to no hysteresis. Dotted line: ramp rate used for later steady state measurements.

(a) Viscosity vs stress for suspensions of cornstarch in water. Mass fractions are shown in the key; higher curves correspond to larger and . The solid line corresponds to a constant shear rate. (b) A rescaling of the data as vs Reynolds number which should result in a data collapse for hydrodynamic flows. The line corresponds to a scaling in the range .

(a) Viscosity vs stress for suspensions of cornstarch in water. Mass fractions are shown in the key; higher curves correspond to larger and . The solid line corresponds to a constant shear rate. (b) A rescaling of the data as vs Reynolds number which should result in a data collapse for hydrodynamic flows. The line corresponds to a scaling in the range .

Viscosity vs stress for cornstarch in a glycerol–water mixture in which the Reynolds number remains small. Mass fractions are shown in the key; higher curves correspond to larger , and . The solid line of slope 1 corresponds to a constant shear rate and the steepest possible steady state viscosity curve. The vertical dashed lines define the stress scales and that bound the shear thickening regime.

Viscosity vs stress for cornstarch in a glycerol–water mixture in which the Reynolds number remains small. Mass fractions are shown in the key; higher curves correspond to larger , and . The solid line of slope 1 corresponds to a constant shear rate and the steepest possible steady state viscosity curve. The vertical dashed lines define the stress scales and that bound the shear thickening regime.

The stress at the onset of shear thickening for glass spheres of different diameters *a* in mineral oil (solid circles, ) or water (open circles, ). Open diamond: polyethylene in silicone oil (). The solid line is the shear stress required to lift particles off the top layer of the packing against friction and gravity . Dashed line: representative curve for data where gravity is not the dominant interparticle interaction. Dotted line: bound above which larger particles did not exhibit any shear thickening regime.

The stress at the onset of shear thickening for glass spheres of different diameters *a* in mineral oil (solid circles, ) or water (open circles, ). Open diamond: polyethylene in silicone oil (). The solid line is the shear stress required to lift particles off the top layer of the packing against friction and gravity . Dashed line: representative curve for data where gravity is not the dominant interparticle interaction. Dotted line: bound above which larger particles did not exhibit any shear thickening regime.

(a) Shear profiles at the plate edge for settling particles of ZrO_{2} in mineral oil (). The mean velocity is normalized by the plate edge velocity , and the depth *h* is normalized by the gap *d*. Shear stress for each profile is shown (in Pa) in the key; higher curves correspond to larger . Dashed line: upper bound of for a measurement at . Dotted black line: depth equal to 1 particle diameter. Solid lines: fits of Eq. (5) to the data for . Inset: same data on log-linear scale. (b) Local viscosity curves based on the local shear rate from the shear profile. Open circles: local viscosity in bulk region. Solid circles: local viscosity in the shear band near the top plate. Solid line: global viscosity curve.

(a) Shear profiles at the plate edge for settling particles of ZrO_{2} in mineral oil (). The mean velocity is normalized by the plate edge velocity , and the depth *h* is normalized by the gap *d*. Shear stress for each profile is shown (in Pa) in the key; higher curves correspond to larger . Dashed line: upper bound of for a measurement at . Dotted black line: depth equal to 1 particle diameter. Solid lines: fits of Eq. (5) to the data for . Inset: same data on log-linear scale. (b) Local viscosity curves based on the local shear rate from the shear profile. Open circles: local viscosity in bulk region. Solid circles: local viscosity in the shear band near the top plate. Solid line: global viscosity curve.

(a) Shear profiles at the plate edge for density matched polyethylene in silicone oil (). The mean velocity is normalized by the plate edge velocity , and the depth *h* is normalized by the gap *d*. Shear stress for each profile shown in the key; lower curves correspond to larger . Dotted black line: depth equal to 1 particle diameter. Solid lines: fits of Eq. (6) to the data for with the substitution since the particles are lighter than the liquid. (b) Local viscosity curves based on the local shear rate from the shear profile. Open circles: local viscosity in bulk region. Solid circles: local viscosity in the shear band near the bottom plate. Solid line: global viscosity curve.

(a) Shear profiles at the plate edge for density matched polyethylene in silicone oil (). The mean velocity is normalized by the plate edge velocity , and the depth *h* is normalized by the gap *d*. Shear stress for each profile shown in the key; lower curves correspond to larger . Dotted black line: depth equal to 1 particle diameter. Solid lines: fits of Eq. (6) to the data for with the substitution since the particles are lighter than the liquid. (b) Local viscosity curves based on the local shear rate from the shear profile. Open circles: local viscosity in bulk region. Solid circles: local viscosity in the shear band near the bottom plate. Solid line: global viscosity curve.

Quadratic curvature obtained from fit of Eq. (7) to velocity profiles. The curvature is normalized by the model prediction from Eq. (6) with used as an estimate for the viscous stress . Data are fit for different normalized shear stresses for glass in mineral oil (solid triangles, ), polyethylene in silicone oil (open circles, ), and ZrO_{2} in mineral oil (solid circles, ). The data collapse close to a value of 1 for suggests that the curvature of the shear profile is due to the weight of the particles on deeper layers which is transferred via frictional contacts, and that the contribution of viscous stresses to the viscosity does not increase significantly in the shear thickening regime.

Quadratic curvature obtained from fit of Eq. (7) to velocity profiles. The curvature is normalized by the model prediction from Eq. (6) with used as an estimate for the viscous stress . Data are fit for different normalized shear stresses for glass in mineral oil (solid triangles, ), polyethylene in silicone oil (open circles, ), and ZrO_{2} in mineral oil (solid circles, ). The data collapse close to a value of 1 for suggests that the curvature of the shear profile is due to the weight of the particles on deeper layers which is transferred via frictional contacts, and that the contribution of viscous stresses to the viscosity does not increase significantly in the shear thickening regime.

(a) Comparison of flow curves measured with different boundary conditions for glass spheres in water at (). Solid circles: shear stress for 100 μm spheres in a fixed gap measurement with the standard parallel plate setup. Open triangles: normal stress from the same measurement. The absolute uncertainty on the normal stress is 2 Pa, so the normal stress cannot be resolved at the low end. Open circles: for 500 μm spheres with a fixed normal stress of 2040 Pa (solid line) in the modified parallel plate setup with a hard wall. Dashed line: slope 1 corresponding to a Newtonian scaling for reference. (b) Circles: same data with the shear stress normalized by normal stress vs . Open triangles: constant shear rate measurements in the standard parallel plate setup in which the normal force was recalibrated before each measurement. Solid line: indicating a frictional scaling. Dashed line: corresponding to a viscous scaling.

(a) Comparison of flow curves measured with different boundary conditions for glass spheres in water at (). Solid circles: shear stress for 100 μm spheres in a fixed gap measurement with the standard parallel plate setup. Open triangles: normal stress from the same measurement. The absolute uncertainty on the normal stress is 2 Pa, so the normal stress cannot be resolved at the low end. Open circles: for 500 μm spheres with a fixed normal stress of 2040 Pa (solid line) in the modified parallel plate setup with a hard wall. Dashed line: slope 1 corresponding to a Newtonian scaling for reference. (b) Circles: same data with the shear stress normalized by normal stress vs . Open triangles: constant shear rate measurements in the standard parallel plate setup in which the normal force was recalibrated before each measurement. Solid line: indicating a frictional scaling. Dashed line: corresponding to a viscous scaling.

(a) Transient time series of shear stress (circles) and normal stress (triangles) in normal-force-control measurements for cornstarch in water at . The sample starts at rest, then the shear is switched on at time . Solid symbols: shear rate (above ). Open symbols: (below ). (b) Same data as panel (a) but extended to longer times to see the steady state behavior. Right axis: change in gap size (squares). (c) Effective viscosity curves obtained from transient measurements. Solid circles: transient shear stress averaged between 0.4 and 1.0 s after shear starts. Solid triangles: transient normal stress averaged over the same time. Open circles: steady state shear stress at the end of the time series where was below the resolution limit for each shear rate. Discontinuous Shear Thickening is suppressed when the normal stress at the boundary is removed. Solid line: stress as a function of shear rate obtained from a steady-state viscosity curve for the same sample with fixed gap size. Dashed line: Newtonian scaling. Dotted line: normal stress resolution limit.

(a) Transient time series of shear stress (circles) and normal stress (triangles) in normal-force-control measurements for cornstarch in water at . The sample starts at rest, then the shear is switched on at time . Solid symbols: shear rate (above ). Open symbols: (below ). (b) Same data as panel (a) but extended to longer times to see the steady state behavior. Right axis: change in gap size (squares). (c) Effective viscosity curves obtained from transient measurements. Solid circles: transient shear stress averaged between 0.4 and 1.0 s after shear starts. Solid triangles: transient normal stress averaged over the same time. Open circles: steady state shear stress at the end of the time series where was below the resolution limit for each shear rate. Discontinuous Shear Thickening is suppressed when the normal stress at the boundary is removed. Solid line: stress as a function of shear rate obtained from a steady-state viscosity curve for the same sample with fixed gap size. Dashed line: Newtonian scaling. Dotted line: normal stress resolution limit.

Top views of a 2.4 mm deep layer of cornstarch in water. (a) Below in a shear cell at rest. (b) At a shear rate above , taken after a shear displacement of relative to panel a. Dilation can be observed as an increase in surface roughness in the sheared region near the wall.

Top views of a 2.4 mm deep layer of cornstarch in water. (a) Below in a shear cell at rest. (b) At a shear rate above , taken after a shear displacement of relative to panel a. Dilation can be observed as an increase in surface roughness in the sheared region near the wall.

Images a suspension of 150 μm ZrO_{2} particles in mineral oil in the standard parallel plate setup with a gap of 890 μm. The camera is focused at a point on the edge of the suspension, with the line of sight tangent to the surface to view radial variations in the boundary position. The rest of the image is out of focus because of the large amount of depth in the image. (a) The suspension at rest. (b) The suspension is sheared at constant shear rate of 3 Hz corresponding to . It can be seen that shear results in both radial dilation of the suspension and increased local curvature at the surface on the particle scale. Vertical lines: reference lines indicating the plate edge in each image.

Images a suspension of 150 μm ZrO_{2} particles in mineral oil in the standard parallel plate setup with a gap of 890 μm. The camera is focused at a point on the edge of the suspension, with the line of sight tangent to the surface to view radial variations in the boundary position. The rest of the image is out of focus because of the large amount of depth in the image. (a) The suspension at rest. (b) The suspension is sheared at constant shear rate of 3 Hz corresponding to . It can be seen that shear results in both radial dilation of the suspension and increased local curvature at the surface on the particle scale. Vertical lines: reference lines indicating the plate edge in each image.

Radial dilation measured as a function of shear stress for 135 μm polyethylene spheres in silicone oil. Solid line: predicted relationship between and for a model in which there is a confining stress from surface tension , where the local radius of curvature is calculated geometrically. A proportionality coefficient of 0.14 shifting the curve horizontally is used to fit the data. Dashed line: dilation value where the contact line is expected to reach the second layer of particles, resulting in a dramatic increase in confining stress with dilation.

Radial dilation measured as a function of shear stress for 135 μm polyethylene spheres in silicone oil. Solid line: predicted relationship between and for a model in which there is a confining stress from surface tension , where the local radius of curvature is calculated geometrically. A proportionality coefficient of 0.14 shifting the curve horizontally is used to fit the data. Dashed line: dilation value where the contact line is expected to reach the second layer of particles, resulting in a dramatic increase in confining stress with dilation.

Viscosity curves for 100 μm glass spheres in liquids with different values of surface tension. Solid symbols: particles were suspended in water. Open symbols: particles were suspended in water with surfactant (above the critical micelle concentration). Triangles: . Circles: . Solid line: without surfactant. Dotted line: with surfactant. Both and the yield stress above decreased when the surface tension was reduced.

Viscosity curves for 100 μm glass spheres in liquids with different values of surface tension. Solid symbols: particles were suspended in water. Open symbols: particles were suspended in water with surfactant (above the critical micelle concentration). Triangles: . Circles: . Solid line: without surfactant. Dotted line: with surfactant. Both and the yield stress above decreased when the surface tension was reduced.

The stress at the upper bound of the shear thickening regime for a variety of suspensions plotted against the confining stress scale from surface tension . Particle materials are listed in the key. Solid symbols: measured by us. Open symbols: polyvinyl chloride [PVC, circles (Hoffman (1972))], polystyrene-acrylonite [PSAN, down-pointing triangles (Hoffman (1972))], polystyrene [up-pointing triangles (Boersma *et al.* (1991))], glass [square (Boersma *et al.* (1990))], silica [diamond (Bender and Wagner (1996))], CaCO_{3} [diagonal crosses (Egres and Wagner (2005))], PMMA [crossed square (Kalman *et al.* (2009))], BiOCl [cross (Bertrand *et al.* (2002))], latex [diagonally crossed square (Laun *et al.* (1991))]. The solid line corresponds to a scaling . Dotted line: lower bound on for measurements in which was not reached [Maranzano and Wagner (2001a)], which often occurs in colloid measurements.

The stress at the upper bound of the shear thickening regime for a variety of suspensions plotted against the confining stress scale from surface tension . Particle materials are listed in the key. Solid symbols: measured by us. Open symbols: polyvinyl chloride [PVC, circles (Hoffman (1972))], polystyrene-acrylonite [PSAN, down-pointing triangles (Hoffman (1972))], polystyrene [up-pointing triangles (Boersma *et al.* (1991))], glass [square (Boersma *et al.* (1990))], silica [diamond (Bender and Wagner (1996))], CaCO_{3} [diagonal crosses (Egres and Wagner (2005))], PMMA [crossed square (Kalman *et al.* (2009))], BiOCl [cross (Bertrand *et al.* (2002))], latex [diagonally crossed square (Laun *et al.* (1991))]. The solid line corresponds to a scaling . Dotted line: lower bound on for measurements in which was not reached [Maranzano and Wagner (2001a)], which often occurs in colloid measurements.

Stress vs shear rate for 500 μm diameter glass spheres in a solid-walled rhoemeter with *no liquid*. Packing fractions are shown in the key; higher curves correspond to larger . Discontinuous Shear Thickening is still seen, confirming that viscous interactions are not necessary.

Stress vs shear rate for 500 μm diameter glass spheres in a solid-walled rhoemeter with *no liquid*. Packing fractions are shown in the key; higher curves correspond to larger . Discontinuous Shear Thickening is still seen, confirming that viscous interactions are not necessary.

Shear stress (open triangles) and normal stress (solid circles) vs gap size *d* for a sample of 500 μm glass spheres with no liquid under slow compression with solid walls. The sample is compressed at a rate of 0.25 μm/s and sheared at a rate of 1 Hz (3 mm/s). The shear stress is close to the zero shear rate limit, so the measured is a good proxy for the yield stress. The solid line is a linear fit used to obtain the per-particle stiffness *k* of the system of sheared grains and solid wall in series.

Shear stress (open triangles) and normal stress (solid circles) vs gap size *d* for a sample of 500 μm glass spheres with no liquid under slow compression with solid walls. The sample is compressed at a rate of 0.25 μm/s and sheared at a rate of 1 Hz (3 mm/s). The shear stress is close to the zero shear rate limit, so the measured is a good proxy for the yield stress. The solid line is a linear fit used to obtain the per-particle stiffness *k* of the system of sheared grains and solid wall in series.

Maximum stress of the shear thickening regime vs the confining stress scale due to the restoring force of the boundary with per-particle stiffness *k*. Data are for 500 μm glass spheres under several different boundary conditions. Solid triangle: hard wall rheometer setup, with particles suspended in water. Open triangle: hard wall, no liquid. Solid square: hard wall with a soft foam rubber insert, with particles suspended in water. Open circle: in a standard parallel plate setup with a liquid–air interface, where we use the surface tension to represent the per-particle stiffness (i.e., ). Solid circle: polyethylene in mineral oil from Fig. 13 where we calculate . The solid line has a slope of 1, corresponding to a stress response proportional to the restoring force of the boundary against a typical dilation of the sample by .

Maximum stress of the shear thickening regime vs the confining stress scale due to the restoring force of the boundary with per-particle stiffness *k*. Data are for 500 μm glass spheres under several different boundary conditions. Solid triangle: hard wall rheometer setup, with particles suspended in water. Open triangle: hard wall, no liquid. Solid square: hard wall with a soft foam rubber insert, with particles suspended in water. Open circle: in a standard parallel plate setup with a liquid–air interface, where we use the surface tension to represent the per-particle stiffness (i.e., ). Solid circle: polyethylene in mineral oil from Fig. 13 where we calculate . The solid line has a slope of 1, corresponding to a stress response proportional to the restoring force of the boundary against a typical dilation of the sample by .

Dimensions used for calculation of relationship between dilation and confining stress due to surface tension at a liquid–air interface with radius of curvature *r*.

Dimensions used for calculation of relationship between dilation and confining stress due to surface tension at a liquid–air interface with radius of curvature *r*.

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