Abstract
A remarkable property of dense suspensions is that they can transform from liquidlike at rest to solidlike under sudden impact. Previous work showed that this impactinduced solidification involves rapidly moving jamming fronts; however, details of this process have remained unresolved. Here we use highspeed ultrasound imaging to probe noninvasively how the interior of a dense suspension responds to impact. Measuring the speed of sound we demonstrate that the solidification proceeds without a detectable increase in packing fraction, and imaging the evolving flow field we find that the shear intensity is maximized right at the jamming front. Taken together, this provides direct experimental evidence for jamming by shear, rather than densification, as driving the transformation to solidlike behaviour. On the basis of these findings we propose a new model to explain the anisotropy in the propagation speed of the fronts and delineate the onset conditions for dynamic shear jamming in suspensions.
Introduction
Dense suspensions are complex fluids that can exhibit strong, discontinuous shear thickening, where the viscosity jumps up orders of magnitude when a critical shear stress is exceeded^{1,2,3,4}. Under a wide range of dynamic conditions, dense suspensions can also undergo a transformation to solidlike behaviour, for example, during sudden impact at their free surface^{5,6,7}, ahead of quickly sinking objects^{8,9}, under shear^{10} or during rapid extension^{11}. Detailed investigation of the dynamics during impact has shown how such solidification is associated with a propagating front that converts fluidlike, unjammed suspension into rigidly jammed material in its wake^{5,12}. This dynamic jamming front moves through the suspension with a speed much greater than the impactor itself.
To explain this solidification, a model was proposed^{5} that assumed the impact pushes the particles closer together until they jam. This densification scenario was based on the standard jamming phase diagram for frictionless hard particles, where entry into a jammed state requires an increase in particle packing fraction ϕ (ref. 13). Since the volume of particles is conserved, the front propagation speed v_{f} along the direction of impact is then related to the impactor speed v_{p} via^{14}
where ϕ_{J} is the packing fraction at which jamming occurs and ϕ_{0}<ϕ_{J} is the packing fraction of the initially unjammed suspension at rest. The closer the initial packing fraction is to jamming, the faster the front will propagate, in principle diverging at ϕ_{J}. This model shows excellent agreement with measurements of v_{f} in systems where the local packing fraction can change easily, such as dry granular particle layers that are being compacted snowploughlike from one end^{14}.
In suspensions, the presence of an interstitial liquid makes it possible to prepare threedimensional (3D) systems at packing fractions ϕ well below ϕ_{J} by density matching the particles to the liquid. Given that such systems can still exhibit impactinduced solidification, jamming by densification would imply significant particle packing fraction changes Δϕ=ϕ_{J}−ϕ. However, unless the impact speed is so high that the liquid becomes compressible^{15}, viscous drag will counteract any densification of the particle subphase. This calls into question the mechanism underlying equation (1), even though there is experimental evidence for the basic outcome, namely that the ratio v_{f}/v_{p} increases dramatically as Δϕ approaches zero^{5,12}.
One intriguing alternative mechanism has recently emerged with the concept of jamming by shear^{3,16}. In this extension of frictionless, standard jamming, the presence of frictional interactions between particles makes it possible to start from initially isotropic, unjammed configurations at ϕ=ϕ_{0}<ϕ_{J} and, without changing ϕ, rearrange the particles into anisotropic fragile or jammed configurations by applying shear. Shear jamming is also possible in frictionless systems, albeit over a much smaller range in Δϕ (ref. 17). So far, such shear jamming has been observed experimentally in twodimensional (2D) dry granular systems under quasistatic conditions, where there is direct visual access to particle positions and stresses by imaging perpendicular to the 2D plane^{16}. Investigating the role of shear jamming in dynamic impactinduced solidification of 3D suspensions requires the capability of noninvasively tracking the jamming fronts and the associated, quickly evolving flow field in the interior of an optically opaque system.
Here we achieve this with ultrasound. Related methods have been applied to studying dry granular materials^{18,19} and steadystate flow in suspensions^{20,21}. Measuring the speed of sound c we obtain an upper bound on the change of packing fraction Δϕ as the suspension jams. We find that at v_{p}<<c, Δϕ is much smaller than required if densification was the primary driver for impactactivated solidification. To investigate the crossover as v_{p} increases, we use highspeed ultrasound imaging to track the emergence of concentrated shear bands at the location of the propagating jamming fronts. In the regime of small v_{p}, the suspension responds to stress like a fluid, and in the regime of large v_{p}, the suspension develops solidlike characteristics, which we identify by investigating the flow fields. The invariant packing fraction and existence of shear bands provides direct evidence of dynamic shear jamming in 3D suspensions. Furthermore, access to the full flow field allows us to extract the local shear rates and identify the origin of a key, but so far unexplained, feature of the response to impact: the longitudinal front propagation speed exceeds the transverse propagation speed by a factor very close to two^{12}.
Results
Experimental setup and extraction of the flow field
The experiments were performed with a model suspension: cornstarch particles dispersed in water–glycerol CsCl solutions. The experimental setup is illustrated in Fig. 1. In the impact experiments the impactor was driven vertically downward with constant velocity v_{p} by a linear actuator. A representative flow field (u_{r}, u_{z}) inside the suspension during an impact is shown in Fig. 2a–c. The vertical and horizontal axes in the image correspond to the z and r directions in cylindrical coordinates. The flow field shows a jammed solidlike plug in the centre, as evidenced by the fact that all points move vertically with speed close to v_{p}. Also evident is a strong velocity gradient around the periphery of the jammed region. To show this more explicitly, we calculate the local shear rate from the velocity field (u_{r}, u_{z}). Given rotational symmetry, the shear rate tensor becomes
Figure 2d,e shows the two components and . Underneath the jammed region, that is, in longitudinal direction, dominates. This corresponds to pure shear that compresses the suspension in the z direction and expands it radially. By contrast, along the sides of the jammed plug dominates, and here the main contribution arises from the term . As a result, the velocity gradient is mainly perpendicular to v_{p}. This is analogous to simple shear as seen, for example, in parallel plate setups. We will return below to the implications of having both types of shear.
Speed of sound
In an unjammed suspension of solid particles in a Newtonian liquid, the shear modulus vanishes. In the limit that the solid particles are much smaller than the wavelength, the speed of sound is then given by , where K is the average bulk modulus and ρ the average material density of the suspension^{22,23}. In our experiments, the particles and suspending solvent are density matched (see Methods section), but the average K still increases with ϕ since cornstarch particles have a bulk modulus larger than that of the liquid^{24}. As shown in Fig. 3a, the resulting dependence of c on ϕ is, to a good approximation, linear across the regime of packing fractions probed by our experiments. This allows us to obtain Δϕ straightforwardly by detecting changes in c with ultrasound.
A schematic illustration of the suspension under impact is shown in Fig. 3b, indicating two regions: a jammed region (orange) directly underneath the impactor and an unjammed region (yellow) ahead of the jamming front. Our measurements provide the average speed of sound as determined from the time it takes sound pulses to traverse both regions (see Methods section). The process of transforming unjammed suspension to jammed suspension could be expected to increase the speed of sound via three possible mechanisms: (1) an increase Δϕ in packing fraction; (2) an increase in effective bulk modulus K; (3) the development of a finite shear modulus G as the suspension jams^{13}. Currently, we cannot determine how much they each contribute, but we can estimate the upper limit of each individual term by assuming the other two zero.
First, we estimated the maximum possible increase in ϕ. Figure 3c shows the measured change in sound speed Δc=−c(ϕ_{0}) during impact for a suspension prepared at ϕ_{0}=0.48. The impactor hits the suspension surface at time t=0, generating a jamming front that reaches the bottom at t≈0.035 s. We can identify this point by the dramatic increase in force on the impactor, as established by prior investigations of quasi2D (ref. 12) and 3D (ref. 5) systems. Until the jamming front interacts with the bottom wall Δc=−c(ϕ_{0}) is less than the measurement uncertainty of about 5 m s^{−1}. Neglecting any increases in K and G, we find from Fig. 3a that our noise floor Δc≈5 m s^{−1} implies Δϕ≈0.006 during the free propagation of the fronts. This means that ϕ could have increased to 0.49 at best. On the other hand, even at the highest packing fraction in Fig. 3a ϕ=0.52, the suspension can still flow when sheared slowly, implying an isotropic jamming threshold ϕ_{J}>0.52. This means that the increase in packing fraction due to impact is much less than required for jamming by the densification model.
If instead we assume Δϕ=0 and use , as appropriate for solids^{25}, to describe the dependence of the speed of sound on K and G within the jammed region behind the front, the same noise floor Δc≈5 m s^{−1} implies that the net increase in the sum of the moduli could not have been larger than MPa. This is very small compared with the bulk modulus K_{0} of the quiescent suspension at ϕ_{0}=0.48: .
Once the front has reached the bottom, Δc increases to ≈16 m s^{−1}. While this is significantly above the noise floor, it limits any packing fraction changes to Δϕ≈0.02, still less than necessary to reach ϕ_{J}. Nevertheless, the interaction with the solid boundary increases the stress in the dynamically jammed region significantly and this drives the suspension deeper into the shearjammed state. We can therefore expect concomitant increases in bulk and shear moduli and thus in sound speed. From the sound speed data in Fig. 3, the upper limit of the net increase in the sum of the moduli is MPa.
Propagation of fronts
From the evolution of the flow fields as shown in Fig. 2, we extract the position of the moving jamming front, defining the front position as the line of points where the vertical component of the impact velocity has dropped to v_{p}/2. In the following we focus on the points in the flow field that propagate the furthest in z and r directions, that is, on the maximum longitudinal and transverse front speeds. As shown in Fig. 4a, after an initial stage the fronts in both directions propagate essentially linearly as function of time before slowing down when they start to interact with boundaries and the incipient jammed region gets compressed by the impactor; further compression causes the motion to stop quickly (Supplementary Fig. 1). Here we investigate this linear regime, where the front propagates freely. To compare how quickly the front propagates relative to v_{p}, we define two dimensionless front propagation factors, or normalized front speeds, k as
where the subscripts t and l represent transverse and longitudinal directions, respectively. The ‘−1’ in k_{l} compensates for the vertical motion of the impactor itself.
Our experiments show that the parameters that affect k_{l} and k_{t} include ϕ, v_{p} and the suspending liquid’s viscosity η_{0}. For a suspension with given η_{0} and ϕ_{0} that is impacted very slowly, the response is fluid like and both k_{t} and k_{l} are close to zero. However, beyond a threshold value v_{p0} jamming fronts start to appear. Their normalized speeds initially increase quickly with impactor speed v_{p} but eventually asymptote to a fixed k*. The relation between k_{l} and v_{p} in suspensions with the same η_{0} but different ϕ_{0} is shown in Fig. 4b; the behaviour of k_{t} is similar. As ϕ increases, the curves shift towards lower v_{p0} and higher k*. For comparison, in suspensions with the same ϕ_{0}, larger solvent viscosity η_{0} leads to lower threshold v_{p0} (Supplementary Fig. 2). To extract k* and v_{p0} we fit the data to
Equation (4) is not derived but a phenomenological relation that captures the key aspects of the data discussed above: (1) k=0 at small v_{p}; (2) k increases when v_{p}>v_{p0}; (3) k approaches k* as v_{p}→∞. Plotting the data in terms of normalized variables k/k* and v_{p}/v_{p0} scales out the dependencies on ϕ_{0} and η_{0}. The resulting data collapse for the longitudinal front speed ratio is plotted in Fig. 4c. A similar result is obtained for the transverse speed ratio .
To quantify the anisotropy in front propagation in longitudinal and transverse directions, we plot versus in Fig. 4d, using the data from experiments varying ϕ (larger packing fraction increases both and ) and η_{0}. To a good approximation, all the data follow . Comparison of our data with the data obtained for quasi2D suspensions^{12} shows excellent agreement as well, except that for higher k_{t} the ratio k_{l}/k_{t} slightly exceeds 2.
Discussion
Our data in Fig. 3c demonstrate that impactactivated jamming of dense suspensions proceeds without significant increase of ϕ, and certainly without increasing ϕ to values close to ϕ_{J}. This rules out earlier explanations based on entering the jammed state via densification of the particle subphase^{5,14}. Instead, analysis of the flow field shows that the jamming fronts initiated by the impact coincide with the location of the maxima in local shear rate (Fig. 2d,e). Altogether, these two findings provide strong evidence for dynamic shear jamming: the impact triggers propagating fronts that locally create sufficient shear to reorganize particles into (anisotropic) jammed configurations without changing the average packing fraction. There are several implications of the shear jamming scenario for suspensions and several differences from dry granular systems, both of which we discuss next.
To start, we examine the stress. In a dry granular system, stress is sustained only via direct contact between particles. By contrast, in a dense suspension stress can also be transmitted without contact via lubrication forces. Thus, while in dry granular systems there is only a single characteristic stress scale for entry into the shearjammed regime^{16}, for a suspension the situation can be more complex. A number of theoretical models^{3}, simulations^{2,26} and experiments^{27,28,29} have recently suggested that lubrication breaks down and particles start to experience frictional interactions beyond a local stress threshold σ_{1}. Thus, for stress levels below σ_{1} the suspension behaves fluidlike, while above σ_{1} the system can be thought to behave more like a frictional granular system, that is, enter a fragile regime before crossing over into the shearjammed regime at a second characteristic stress level σ_{2} (ref. 16). Within this picture, we associate the transition at v_{p0} with the situation where the stress levels at the leading edge of the jamming front have reached σ_{1} and are large enough for frictional interactions to occur. Thus, when v_{p}<v_{p0} the suspension is in the lubrication regime, but when v_{p}>v_{p0} it transitions into a fragile state with behaviour intermediate between solid and fluid^{16,30,31}, as frictional contacts start percolating through the system to form a loadbearing network, eventually reaching a shearjammed state as v_{p} increases further. We point out that a nonzero shear modulus is not strictly necessary for the front to propagate (with k>0). The front will propagate as long as it transforms the initially liquidlike suspension into a state with sufficiently large viscosity. However, we know that this state behaves solidlike, while interacting with a system boundary^{5,6,10,12}. Thus, how the shear modulus evolves in the region behind the jamming front before a systemspanning jammed state has been established remains an open question.
The stressbased argument also provides an explanation of the relaxation or ‘melting’ of the jammed region when the impact stops. During front propagation the stress inside the jammed region is sustained by the inertia of the suspension in the shear zone ahead of the jammed region. When the motion of the impactor stops, the shear zone disappears and the stress applied on the boundary of the jammed suspension falls below σ_{1}, insufficient to sustain frictional interactions between particles and therefore any network of force chains that could generate a yield stress and support a load. As a result, the suspension returns to the lubrication regime.
However, while necessary, the existence of threshold stress levels is not sufficient to explain the asymptotic front speed k* at high v_{p} and the seemingly universal anisotropy in front propagation, expressed by the ratio ≈2. Particles also need to move out of an initial uniform isotropic distribution and reorganize under shear into anisotropic structures (force chains) that can support the stress. Such reorganization requires a minimum shear strain ɛ_{c} to engage neighbouring particle layers. As a result, shear jamming happens only when stress and strain both reach their threshold values. In a quasistatic granular system^{16,17} the threshold strain only matters when the shearjammed state is prepared or when the shear is reversed. In the dynamic system considered here, the front continues to propagate into unperturbed suspension, and therefore, the front advances by applying strain ɛ_{c} locally during the whole process of front propagation.
For dense suspensions in the high v_{p} regime, where the stress threshold is clearly exceeded, we can show that k* is governed by ɛ_{c} and that, in fact, the front speed anisotropy is a direct consequence of the existence of a strain threshold. As described above, the suspension experiences pure shear in the longitudinal direction and simple shear in the transverse direction. In 2D, we can directly compare the two types of shear using the positive eigenvalues of the shear rate tensors, treating the propagation of the front in the longitudinal and transverse directions as two effectively 1D problems. We now assume that a suspension element jams when the shear strain it experiences reaches ɛ_{c}, irrespective of propagation direction. This leads to the following relations between , and ɛ_{c} (see Methods section):
and
Equation (6) is plotted in Fig. 4d. For small ɛ_{c} we find from equation (5) that . In other words, the anisotropy ratio of 2 in the normalized front speeds originates from the factor 1/2 in the nondiagonal terms of the shear rate tensor, which in turn arises because simple shear can be decomposed into a combination of pure shear and solid body rotation. In 3D, it is not possible to quantify the effects of pure shear and simple shear via the same approach (see Methods section). However, one possible solution is to use the ‘strain intensity’ suggested by Ramsay and Huber^{32}, which provides a scalar measure of the relative strength of the two types of shear. As we show in the Methods section, this leads to a ratio , very close to the value for the 2D case. Therefore, an anisotropy ratio ≈2 agrees very well with the experimental data for both quasi2D (ref. 12) and 3D systems within our measurement precision.
With increasing packing fraction ϕ the average distance between particles decreases and we expect the strain threshold ɛ_{c} to decrease as well, which agrees qualitatively with the measurements of ɛ_{c} in dry granular systems^{16}. Via equation (5) this explains the increase in k with ϕ seen in Fig. 4b: since it takes less strain to reorganize the particles into a shearjammed network the front will propagate faster for given impactor velocity. We point out that equation (1), which formalizes such relationship between packing fraction and front speed, appears to capture the overall trend qualitatively. However, this seems fortuitous, since equation (1) was based on the assumption that the moving front significantly increases the packing fraction, in fact driving it up all the way to ϕ_{J}, which we now can rule out in our system. In addition, equation (1) cannot predict the observed propagation anisotropy. One of the outstanding tasks therefore is to develop a model for k*(ϕ) that is based on jamming by shear rather than densification.
An interesting aspect of the data in Fig. 4d is the deviation from the anisotropy ratio ≈2 at large k* values or, equivalently, large packing densities. This is most apparent in the data available for the quasi2D system and it indicates that the longitudinal speed becomes faster. We speculate that this may be connected to a breakdown of the assumption of an isotropic threshold ɛ_{c}. For example, if the impact were to introduce a small amount of compression of the particle subphase in longitudinal direction, ɛ_{c} would be reduced in that direction. This effect would become increasingly significant at large ϕ. We can model this by introducing a correction δ so that
Using and δ≈0.01 we can reproduce the trend in Fig. 4d well (dotted red line). However, we point out that this is just the simplest way to account for the trend in the data and there might be other reasons for the deviation.
Taken together, these results provide important insights into the mechanism responsible for impactinduced solidification of dense suspensions. The finding that the packing fraction does not increase measurably during impact, together with the observation of strong shear at the leading edge of the propagating solidification fronts, rules out jamming via densification as the dominant mechanism and points to jamming by shear (densification is likely to play a significant role at much larger impact velocities, when the interstitial liquid’s compressibility can no longer be neglected^{15}). In dense suspensions this introduces a new stress scale or, equivalently, an impact velocity threshold, which we associate with the breakdown of lubrication films between particles and the onset of frictional interactions^{2,3,27,28,29}. Behind the front, these frictional interactions create a dynamically shearjammed region (corresponding to fragile and/or shearjammed states in refs 3, 16). Further support for the shearjamming scenario comes from the observation of anisotropic front propagation, where we can relate the fact that the fronts propagate longitudinally twice as fast as in transverse direction to an equivalent factor in the ability to transmit shear strain. We point out that for dynamic shear jamming, both shear stress and strain need to exceed threshold values, and the critical shear strain determines the front propagation speed.
Methods
Experimental setup
The ultrasound measurements and imaging were performed with a Verasonics Vantage 128 system. The sample container was 3Dprinted from ultravioletcured resin (‘Vero White Plus’, Objet Geometries Inc.) whose acoustic impedance matched the suspensions we studied. The inner diameter of the container was 100 mm and the typical depth H of the suspension was 25 to 35 mm. This insured that the front reached the bottom before it interacted with the side wall. The impactor was a cylinder of diameter of 6 or 10 mm, driven by a computer controlled linear actuator (SCN5, Dyadic Systems) and equipped with a force sensor (DLC101, Omega). The ultrasound transducer (Philips L74; 128 independent elements; total width of 38 mm) contacted the bottom of the container through a thin layer of ultrasound gel. The ultrasound system was triggered as the impactor approached the surface; images were taken at a frame rate of 10 to 10,000 frames per second, adjusted according to v_{p}. The spatial resolution of the ultrasound was limited by the wavelength, about 0.4 mm in our experiments.
Suspensions
The cornstarch (Ingredion) was stored in the lab environment at 22.5±0.5 °C and 51±2% relative humidity. Individual particles had a diameter of 5–30 μm (refs 5, 33). The suspending solvent was a solution of CsCl, glycerol and water. Its viscosity η_{0} was adjusted by the mass ratio of glycerol and water, and its density was matched to that of the cornstarch particles by adjusting the mass ratio of CsCl. The density of the particles was ρ_{cs}=(1.63±0.01) × 10^{3} kg m^{−3} as measured by density matching. To calculate the packing fraction ϕ, an accurate determination of the volume occupied by particles and interstitial liquid is required. For cornstarch this is difficult because the particles are porous and they already contain some moisture before they are dispersed in the suspending solvent. Therefore, often a value ϕ_{m} based on the mass fractions before mixing is quoted^{4,5,7}, which is proportional to ϕ. For example, in Fig. 4b, the ϕ_{m} values were 0.390, 0.402, 0.409, 0.425 and 0.444. To obtain the actual packing fraction, we account for a small fraction α of suspending liquid that is wicked up by the porous particles and write ϕ=(1+α)ϕ_{v}, where ϕ_{v} is the material volume packing fraction^{34,35}. When calculating ϕ_{v}, we considered the moisture in the cornstarch particles. We assume the moisture is pure water and its mass ratio in cornstarch is β in our lab environment. This leads to
where m_{cs} is the mass of cornstarch particles, m_{l} is the mass of the solvent liquid, ρ_{l} is the density of the solvent and ρ_{w} is the density of pure water. In this paper we use α=0.3 and β=0.11 by estimation. Changing of these numbers will not affect the conclusions we make.
Since air bubbles mixed in the suspension scatter ultrasound signals significantly, the suspensions were debubbled before using. To keep ϕ fixed during debubbling, we filled the suspensions into sealed syringes and then lowered the pressure by pulling the plungers. The syringe walls were tapped gently to help bubbles separate from the suspension. After debubbling a small amount of suspension was used to measure the speed of sound c as required for image reconstruction. For imaging the flow field, a small amount of air bubbles were added back to the debubbled suspensions to act as tracer particles. This was done by slowly stirring the suspension, then tilting and slowly rotating the container till the bubbles were uniformly distributed. We verified that the amount of bubbles did not suppress the penetration of the ultrasound in the suspensions and produced a negligible change in the speed of sound. We also determined that the effect of the bubbles on the suspension viscosity is limited (Supplementary Fig. 4). Between successive impact experiments the suspension was relaxed by gently shaking and rotating the container.
Data acquisition and analysis
Once triggered, the ultrasound system made several hundred acquisitions consecutively. In one acquisition each of the 128 transducer elements transmitted the same ultrasonic pulse at the same time and received an individual reflected time series. The pulse was a 5MHz sinusoidal wave modulated by a Gaussian profile (Gaussian envelope) for a pulse length of 6 periods. From the time series received by the transducer and using the previously measured speed of sound c we reconstructed Bmode images (using brightness to represent the echo signal amplitude)^{25} that captured the positions of the tracer particles (air bubbles) in the suspension. Given our finding that c does not change measurably during the impact, the image reconstruction process does not need to account for spatial or temporal variations in c. By tracking the tracer bubble displacements with a particle imaging velocimetry (PIV) algorithm, we obtained a 2D flow field from within the bulk of the suspension.
Change of packing fraction measurements
The experimental setup was identical to the one illustrated in Fig. 1 and a schematic illustration is shown in Fig. 3b. The impactor started from the surface of the suspension and pushed down a distance z_{p}. The position of the impactor was measured with a highspeed camera (Phantom V9, Vision Research). The ultrasound measured the time of flight T of the signal transmitted from the bottom, reflected by the impactor and sent back to the bottom. Thus the average speed of sound along this path is
We started with experiments at a low v_{p} (5 mm· s^{−1}) to measure the speed of sound in the liquidlike, unjammed suspension, where =c(ϕ_{0}). Define T_{0} as the initial time of flight when z_{p}=0 mm, we have H=c(ϕ_{0})T_{0}/2, and from this
The initial packing fraction ϕ_{0} in these experiments was 0.48. The liquid was a mixture of 44.3% CsCl, 27.8% glycerol and 27.8% water by mass, with η_{0}=4.6 mPa·s and ρ=1.63 × 10^{3} kg·m^{−3}. From six measurements we obtain c(ϕ_{0})=1939.2±4.6 m s^{−1} and H=c(ϕ_{0})T_{0}/2=34.1±0.1 mm.
For the high v_{p} (200 mm s^{−1}) experiments we used the value of H measured above and equation (9) to calculate . The time of flight now becomes
For Δϕ=0, T_{Δϕ=0}=2(H−z_{p})/c (ϕ_{0}). If Δϕ>0, there will be a difference between T and T_{Δϕ=0}, and the difference becomes increasingly large as z_{f} increases, which leads to an increase in according to equation (9).
Derivation of equation 5 in 2D
For an idealized 2D system we define a Cartesian coordinate with x axis in the transverse and y axis in the longitudinal direction. To obtain the relation between the strain threshold ɛ_{c} and the normalized front speeds k we consider how much shear strain a suspension element experiences when it accelerates from u_{y}=0 to u_{y}=v_{p}. We consider the propagation in the transverse and longitudinal directions separately as two quasi1D problems. Exemplary sketches of the velocity profiles are provided in Supplementary Fig. 5. The experimental data did not show a significant change in front width, so here we assume the shape of the front does not change during propagation. In this case the velocity profiles can be expressed as
in the transverse direction and
in the longitudinal direction. In both the equations t is time, v_{ft} and v_{fl} are front propagation speeds. f_{t}(X) and f_{l}(X) are functions that satisfy f_{t}=f_{l}=v_{p} as X→−∞ and f_{t}=f_{l}=0 as X→+∞.
On either side of the impactor the front propagates in transverse direction and the front speed v_{ft}=k_{t}v_{p}, while the suspension itself is sheared longitudinally by the advancing front. The acceleration of a suspension element is then
where D/Dt is the material derivative and . Below the impactor there are two differences: one is that the suspension element now moves along the propagation direction of the front and the other is that v_{fl}=(k_{l}+1)v_{p} as defined in equation (3). The acceleration then becomes
Now we look at the relation between the local shear rate and the velocity gradient. In general, for an incompressible 2D fluid the shear rate tensor is
where . From experimental observation we have . For the transverse direction, where simple shear dominates, the diagonal terms vanish and the shear rate tensor becomes
while for pure shear in the longitudinal direction the offdiagonal terms vanish and we have
In either case the matrix has two eigenvalues with the same magnitude but opposite sign, and the eigenvalues represent the shear rate on the principal axes. Thus, we can represent the shear intensities by the tensors’ positive eigenvalues: and .
Using the velocity gradient, we relate the local shear rate with the acceleration of the element:
Consequently, the total shear strain ɛ a suspension element experiences before jamming is:
and
Equation (19) gives ɛ_{l}≈1/k_{l} for k_{1}>>1. If we assume the strain threshold to jamming ɛ_{c} is isotropic, then k_{t}=1/(2ɛ_{c}) and .
Relation between k _{l} and k _{t} in 3D
In 3D, the shear rate tensor is shown in equation (2). In the longitudinal direction, pure shear dominates and the shear rate tensor is
where and . In the transverse direction simple shear dominates. This gives
where we have used . Note that though the system is 3D simple shear only operates in the rz plane while leaving the azimuthal direction invariant. The eigenvalues become and . Unlike the 2D case, we cannot simply use a positive eigenvalue to represent the shear intensity. However, we can define infinitesimal strains e_{i} (i=1, 2, 3) along the three principal axes and rankorder them according to e_{1}>e_{2}>e_{3}. Following the definition given in ref. 32, the ‘strain intensity’ is
For pure shear in the longitudinal direction e_{1}=e_{2}=−e_{3}/2 and , so , which leads to . For simple shear in the transverse direction e_{1}=−e_{3}, e_{2}=0 and . This leads to , and therefore . Following the procedure for the 2D case we have
Integration then leads to
Now we again assume that the system shearjams when reaches a threshold strain value , independent of the type of shear it experiences. From this we find
and for large k.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Additional information
How to cite this article: Han, E. et al. Highspeed ultrasound imaging in dense suspensions reveals impactactivated solidification due to dynamic shear jamming. Nat. Commun. 7:12243 doi: 10.1038/ncomms12243 (2016).
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Acknowledgements
We thank Patrick La Riviere for help with the ultrasound system. We thank Mengfei He, Nicole James, Victor Lee, Sayantan Majumdar, Daniel Szyld, Scott Waitukaitis, Tom Witten, Matthieu Wyart and Qin Xu for insightful discussions. This work was supported by the US Army Research Office through grants W911NF1210182 and W911NF1610078. Additional support was provided by the Chicago MRSEC, which is funded by NSF through grant DMR1420709.
Author information
Affiliations
James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA
 Endao Han
 , Ivo R. Peters
 & Heinrich M. Jaeger
Department of Physics, The University of Chicago, Chicago, Illinois 60637, USA
 Endao Han
 & Heinrich M. Jaeger
Engineering and the Environment, University of Southampton, Highfield, Southampton SO17 1BJ, UK
 Ivo R. Peters
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Contributions
All authors designed the experiments and wrote the paper. E.H. performed the experiments and analysed the data. I.R.P. contributed the PIV code.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Heinrich M. Jaeger.
Supplementary information
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Supplementary Information
Supplementary Figures 15
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Supplementary Movie 1
Visualization of the flow field with ultrasound. The movie shows the velocity field of the flow and the shear rate distribution in the suspension under impact.
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Further reading

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