aËœaËœcontinuous Shear Thickening and Colloid Surfacesaa Phys Re
The popular interest in cornstarch and water mixtures known as "oobleck" after the
in one of Dr. Seuss's classic children's books arises from their transition from fluid-like to
behavior when stressed. The
that emerges from a roughly 2-to-1 (by volume) combination of starch to water can be poured into one's hand. When squeezed, the
morphs into a doughy
that can be formed into shapes, only to "melt" into a puddle when the applied stress is relieved.
videos show people running across a large pool of the stuff, only to sink once they stop in place, and "monsters" that grow out of the mixture when it's acoustically vibrated (for an example, see the online version of this article).
fluids certainly entertain and
our curiosity, but their
can also vex
by fouling pipes and spraying equipment, for instance. And yet, when engineered into
STFs can be controlled and harnessed for such exotic applications as shock-absorptive skis and the soft body armor discussed in box 1
and
scientists have wrestled with the scientific and practical problems of
dispersions—typically composed of condensed
or oxides suspended in a
more than a century. More recently, the physics community has explored the highly nonlinear
in the context of jamming 1 (see the article by Anita Mehta, Gary Barker, and Jean-Marc Luck in PHYSICS TODAY, May 2009, page 40) and the more general study of
as
for understanding soft condensed matter.
Hard-sphere
are the "hydrogen atom" of
Because of their greater size and
times compared with atomic and molecular
are often well suited for
and
experiments using light,
and
That makes the
beyond their own intrinsic technological importance, ideal
for exploring equilibrium and near-equilibrium phenomena of interest in atomic and molecular physics—for example, phase behavior and "dynamical arrest," in which particles stop moving collectively at the
The relevance of
to atomic and molecular
breaks down, though, for highly nonequilibrium phenomena. Indeed,
in strongly flowing
may be among the most spectacular, and elucidating, examples of the differences between the
Figure 1 . illustrates the
The addition of
particles to a
such as water results in an increase in the
and, with further addition, the onset of non-Newtonian behavior—the dependence of its
on an applied shear stress or
At high particle concentrations, the fluid behaves as if it has an apparent
That is, it must be squeezed, like ketchup, before it can actually
At such concentrations, the
fit into the general paradigm for jamming in soft matter: 2 At high particle densities and low stresses (and low temperatures, usually), the
dynamically arrests, just as atomic, molecular,
and
do. But once the
is exceeded, the fluid's
a response known as
That rheology is engineered into a range of consumer products, from shampoos and paints to
detergents, to make them gel-like at rest but still able to
easily under a weak stress. Again, the
fits the general paradigm for how matter behaves: It
when sheared strongly enough.
Figure 1. The viscosity of colloidal latex dispersions, as a function of applied shear stress. The volume fraction φ of latex particles in each dispersion distinguishes the curves. A critical yield stress must be applied to induce flow in a dispersion with high particle concentration. Beyond that critical stress, the fluid's viscosity decreases (shear thinning). At yet higher stress, its viscosity increases (shear thickening), at least for latex dispersions above some concentration threshold.
(Adapted from ref. 12)
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At higher stresses,
occurs:
rises abruptly, sometimes discontinuously, once a critical shear stress is reached. The rise is counterintuitive and inconsistent with our usual experience. Experiments and simulations on atomic and small-molecule
predict only
at least until the eventual onset of
at
rates that vastly exceed those of interest here.
The ubiquity of the phenomenon in the
of suspended
is a serious limitation for
especially when it involves high
operations. In a 1989 review, Howard Barnes writes,
Concentrated
of nonaggregating
particles, if
in the appropriate
range, will always show (reversible)
The actual nature of the
will depend on the parameters of the suspended phase: phase volume, particle size (distribution), particle shape, as well as those of the suspending phase
and the details of the
i.e., shear or
steady or transient, time and rate of
3
Inks,
binders for paints,
alumina casting
blood, and clays are all known to shear thicken. But the earliest searches for the root cause came from industrial laboratories that coated paper at high speeds
typically up to 106 Hz), a process in which the coating's increasing
would either tear the paper or ruin the equipment. Industrial labs remain intensely interested in the science. Hundreds of millions of metric tons of cement are used globally each year, for example, and production
are careful to formulate modern high-strength cements and concretes that don't suffer from the effect—at least in a range of
important for processing and construction. 4
In pioneering work in the 1970s, Monsanto's Richard Hoffman developed novel light-scattering experiments to probe the underlying microstructural transitions that accompanied
in concentrated latex
5 The transition was observed to correlate with a loss of Bragg peaks in the
On that basis, Hoffman developed a micromechanical
of
as a flow-induced
In the 1980s and early 1990s BASF's Martin Laun and others interested in products such as paper coatings and emulsion-polymerized
used then emerging small-angle neutron-scattering techniques to demonstrate that an
was neither necessary nor alone sufficient to induce significant
6 Because
is a highly nonequilibrium, dissipative state, though, a full understanding had to await the development of new
and experimental tools.
Box 1. Soft armor and other applications
The unique
of increased energy dissipation combined with increased
make
fluids (STFs) ideal for damping and shock-absorption applications. For example, so-called EFiRST fluids can be switched between shear-thickened and flowing states using an applied
which controls the damping.
have also explored the STF response in sporting equipment 14 and automotive applications, 15 such as skis and tennis rackets that efficiently dissipate vibrations without losing stiffness or STFs embedded in a passenger compartment liner designed to protect passengers in a car accident.
One commercial application of STF composites is expected to be protective clothing. 16 The fabric imaged in these scanning electron micrographs has STFs intercalated into its woven yarns. Initial applications are anticipated in flexible vests for correctional officers. Longer-term research is being performed in one of our laboratories (Wagner's), in conjunction with the
Army Research Laboratory, to use STF fabrics for
puncture, and blast protection for the military, police, and first responders.
Tests of the
demonstrate a marked enhancement in performance. Consider this comparison between two STF-based fabrics: The velocity at which a quarter-inch steel ball is likely to penetrate a single layer of Kevlar is
at about 100 m/s. The velocity required to penetrate Kevlar formulated with
(polymethyl-methacrylate) is about 150 m/s, and that for Kevlar formulated with
is 250 m/s, 2.5 times that for the Kevlar alone. High-speed video demonstrations and further test details are available at http://www.ccm.udel.edu/STF. Many other composites are now under investigation for armor applications. (Images courtesy of Eric Wetzel,
Army Research Laboratory.)
Section:
The
of
is inherently a many-body, multiphase fluid-mechanics problem. But first consider the case of a single particle. Fluid drag on the particle leads to the Stokes-Einstein- Sutherland
relationship:
The diffusivity D 0 scales with the thermal energy kT divided by the suspending medium's
µ and the particle's
radius a. That diffusivity sets the characteristic time scale for the particles'
it takes the particle a 2/D 0 seconds to diffuse a distance equal to its radius. The time scale defines high and low
A dimensionless number known as the Péclet number, Pe, relates the
of a
to the particle's
rate; alternatively, the Péclet number can be defined in terms of the applied shear stress τ:
The number is useful because
rheology is often
by applied
or shear stresses. Low Pe is close enough to equilibrium that
can largely restore the equilibrium microstructure on the time scale of slow
At sufficiently high
or stresses, though,
of the
microstructure by the
occurs faster than
can restore it.
is already evident around Pe ≈ 1. And higher
or stresses (higher Pe) trigger the onset of
The presence of two or more particles in the
fundamentally alters the
due to the inherent coupling, or
between the
of the particles and the displacement of the suspending fluid. In a series of seminal articles in the 1970s, Cambridge University's George Batchelor laid a firm foundation for understanding the
7 In essence, because any particle
must displace incompressible fluid, a long-ranged—and inherently many-body—force is transmitted from one particle through the intervening fluid to neighboring particles; the result is that all particles collectively disturb the local
field through
Such
are absent in atomic and molecular fluids, where the intervening medium is vacuum.
Batchelor's calculation of the trajectories of non-Brownian particles under
identified the critical importance of what's known as
which describes the behavior of particles
via the suspending medium at very close range. Those
were already well known in the
of journal bearings, which Osborne Reynolds investigated in the late 1800s and which remain of great importance to the workings of modern machines. As box 2 explains, the force required to push two particles together in a fluid diverges inversely with their separation distance. Of particular significance is that at close range, the trajectories that describe their relative
become correlated. That is, the particles effectively orbit each other—indefinitely if they are undisturbed.
Batchelor's work also led to a formal understanding of how
coupling alters the
relationship, which, in turn, enabled him to calculate the
coefficient and
of dilute
of Brownian
at equilibrium. 7 Although it was not fully appreciated at the time, the
of
on
is the basis for understanding the
Beyond two particles
Section:
in real
require numerical methods to solve. The method of Stokesian
outlined in box 3 calculates the
of ensembles of
and noncolloidal spheres under
A great advantage of the simulations is their ability to resolve which forces contribute to the
Moreover, they demonstrate that the ubiquitous
in hard-sphere
is a direct consequence of particle rearrangement due to the applied shear.
The equilibrium microstructure is set by the balance of stochastic and interparticle forces at play—including
and van der Waals forces—but is not affected by hydro-dynamic
The low-shear (Pe ≪ 1)
has two components, one due to direct interparticle forces, which dominate, and one due to
7 Under weak but increasing
(Pe ~ 1), the fluid structure becomes
as particles rearrange to reduce their
so as to
with less resistance. Figure 2 illustrates the evolution schematically. Near equilibrium, the resistance to
is naturally high because shearing the random distribution of particles causes them to frequently collide, like cars would if careening haphazardly along a road. With increasing
though, particles behave as if merging into highway traffic: The
becomes streamlined and the increasingly efficient
of
particles reduces the system's
Figure 2. The change in microstructure of a colloidal dispersion explains the transitions to shear thinning and shear thickening. In equilibrium, random collisions among particles make them naturally resistant to flow. But as the shear stress (or, equivalently, the shear rate) increases, particles become organized in the flow, which lowers their viscosity. At yet higher shear rates, hydrodynamic interactions between particles dominate over stochastic ones, a change that spawns hydroclusters (red)—transient fluctuations in particle concentration. The difficulty of particles flowing around each other in a strong flow leads to a higher rate of energy dissipation and an abrupt increase in viscosity.
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Simulations that ignore
coupling between particles show that the ordered, low-viscosity state persists even as the Péclet number approaches infinity. Think of particles sliding by in layers orthogonal to the shear-gradient direction. Stokesian
simulations, however, demonstrate that
forces become larger at high
(Pe ≫ 1) than do interparticle forces that drive
So when the particles are driven close together by applied shear stresses,
strongly couple the particles' relative
The result is a
that has a microstructure significantly different from the one near equilibrium, and hence, the energy dissipation increases. In hindsight, that should not be surprising given Batchelor's calculation of closed trajectories.
In both semidilute and concentrated
the strong
coupling between particles leads to the formation of hydroclusters—transient concentration
that are driven and sustained by the applied shear field. Here again, the analogy to traffic collisions disrupting organized, low-dissipation
may be helpful. Unlike the seemingly random microstructure observed close to equilibrium, however, this microstructure is highly organized and
The transient hydroclusters are the defining feature of the
state.
Referring back to figure 1 , one can see that a
volume fraction φ = 0.50 produces a latex
whose
is 1 Pa·s at a low shear stress and again at one more than four orders of magnitude higher. The same
emerges for very different reasons, though. Changes in the particles' size, shape, surface chemistry, and ionic strength and in
of the suspending medium all affect the interparticle forces, which dominate the
at low shear stress.
forces, in contrast, dominate at high shear stress. Understanding the difference is critical to formulating a
that behaves as needed for specific processes or applications.
As shown in figure 3 ,
on
experimentally confirm the predictions of simulations that the shear-thickened state is driven by dissipative
The
generates strong
in the nearest-neighbor distributions (see box 3 ). The
give rise to clusters of particles and concomitant large stress
8 that, in turn, lead to high dissipation rates and thus a high shear
The formation of hydroclusters is generally reversible, though, so reducing the
returns the
to a stable, flowing
with lower
Moreover, even very dilute
will shear thicken, although the
is hard to observe. 9
Figure 3. The measured viscosity of a concentrated colloidal suspension (squares) can be resolved into two components—a thermodynamic component (circles) associated with the stochastic motion of particles and a hydrodynamic component (triangles) associated with forces acting between particles due to motion through the suspending fluid. Light-scattering experiments combined with numerical simulations determine which forces dominate in different stress regimes.
(Adapted from ref. 13)
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Controlling
requires different strategies from those typically employed to control the low-shear
The addition, for example, of a
"brush" grafted or adsorbed onto the particles' surface can prevent particles from getting close together. With the right selection of graft density, molecular weight, and
the onset of
moves out of the desired processing regime. 10 The strategy is often used to reduce the
at high processing rates but could increase the suspension's low-shear
Indeed, because the separation between hydroclustered particles is predicted to be on the order of nanometers for typical
behavior directly reflects the particles'
and any short-range interparticle forces at play. Fluid slip, adsorbed ions,
and
all significantly influence the onset of
Simple
based on the hydrocluster mechanism have proven valuable in predicting the onset of
and its dependence on those stabilizing forces. 11
Figure 4 shows a toy-model calculation in which
is suppressed by imposing a purely repulsive force field—akin to the
of a
brush—around each particle that prevents the particles from getting too close to each other. 9 When the range of the repulsive force approaches 10% of the particle radius, the
is effectively eliminated and the
with low
Manipulating those nanoscale forces, the particles' composition and shape, and
of the suspending fluid so as to control the sheer thickening, however, remains a challenge for the
formulator.
Figure 4. Shear thickening can be suppressed by reducing the interactions between particles, as shown here based on numerical calculations. The extent of the reduction affects whether an increasing Péclet number (a measure of shear rate) leads to a shear-thickening state or a shear-thinning one. The effect is evident experimentally when a polymer layer, or brush, is grated onto the particles: The dispersion becomes progressively less viscous as the brush thickness on each particle increases.
(Adapted from ref. 9)
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Box 2.
When two
particles approach each other, rising
pressure between them squeezes fluid from the gap. At close range, the
force increases inversely with the distance between the particles' surfaces and diverges to a singularity. The graph at right plots the normalized force required to drive two particles together (along a line through their centers) at constant velocity. The
that govern the
behavior between particles are time reversible, so the force is the same one required to separate two particles.
In simple
are strongly coupled by the hydro-dynamic
if the particles are close together. The inset of the plot shows a test particle's trajectories, sketched as paths as it moves in a
relative to a reference particle (gray sphere). The trajectories are reversible and can be divided into two classes: those that come and go to infinity and those that lead to correlated orbits—so-called closed trajectories—around the reference particle.
Simulations and
of concentrated
that account for those short-range
show that at high
particles that are driven into close proximity remain strongly correlated and are reminiscent of the closed trajectories observed in dilute
The flow-induced density
are known as hydroclusters. Because the particle concentration is higher in the clusters, the fluid is under greater stress, which leads to an increase in energy dissipation and thus a higher
The illustration at right during a stage of the Stokesian
simulation shows
particles in hydroclusters. 8
Box 3. Stokesian
The
of particles suspended in an incompressible Newtonian fluid is a challenging fluid-mechanics problem that can be handled analytically for a single sphere and semianalytically for two spheres. For three or more particles, though, it requires a
to the Stokes equation—the
without inertia. Solution strategies range from brute-force
calculations, to more elegant
to coarse-grained methods, such as smoothedparticle
or
techniques, for representing the fluid. The method of Stokesian
17 starts with the
for N-particle
in which the
M is a generalized mass, a 6N × 6N mass and moment-of-inertia matrix; U is the 6N-dimensional particle translational and rotational velocity vector; and the 6N-dimensional force and
vectors represent the interparticle and external forces F P (such as gravity),
forces F H acting on the particles due to their
relative to the fluid, and stochastic forces F B that give rise to
The stochastic forces are related to the
through the
In
the
forces and torques are linearly related to the particle translational and rotational velocities as F H = −R · U, where R is the configuration-dependent
resistance matrix. In the Stokesian
method, the necessary matrices are computed by taking advantage of the linearity of the Stokes
and their integral solutions. Long-range many-body hydro-dynamic
are accurately computed by a force-multipole expansion and combined with the exact, analytic
to calculate the forces.
Armed with that method, one can predict the
microstructure associated with a particular shear
Take, for instance, a concentrated
whose particles occupy nearly half the volume. If the positions of those particles are represented as dots, the figure illustrates how the hydro-dynamic forces affect their probable locations around some arbitrary test particle (black). The three panels differ only in the
represented by the Péclet number Pe, the ratio of the shear and
rates.
At low Péclet number (0.1), the distribution of neighboring particles is isotropic, which is typical of a concentrated
Red indicates the most probable particle positions as nearest neighbors and green the least probable. At Pe = 1, significant shear distortion appears in neighbor distributions, such that particles are convected together along the compression axes (135° and −45°) relative to the
At high Péclet numbers, the
regime, particles aggregate into closely connected clusters, which is manifest as yet greater
in the microstructure. Particles are more closely packed and thus occupy a narrower region (red) around the test particle than at lower Péclet numbers, indicative of being trapped by the
forces. 18
Beyond hard spheres
Section:
Although the basic
of shear behavior in
are understood, many aspects of the fascinating and
remain active research problems. At very high particle densities,
can undergo discontinuous
whereby the
will not shear at any higher rate. Rather, increasing the power to a rheometer, for example, leads to such dramatic increases in
and large
in stress that the
either refuses to
any faster or
Samples that exhibit strong
are particularly interesting as candidates for soft body armor (see box 1 ), and that application has prompted investigations of transient
at microsecond time scales and at stresses that approach the ideal strength of the particles.
Another active research topic concerns jamming transitions under
As figure 1 suggests, concentrated
could be jammed at low and high shear stresses but
in between. Evidence also exists, as the figure more subtly suggests, that
may exhibit a second regime of
at the highest stresses rather than continuing to resist the increasing
The
can be understood as a manifestation of the finite
of the particles—relatively soft plastic in this case. At very high stresses, particles stop behaving like billiard balls and elastically
which alters their rheology. The same forces that drive the hydrocluster formation, which is reversible as the
is reduced, can also lead to irreversible
That is, particles forced into contact remain in contact even as the
weakens. Such shear-sensitive
irreversibly thicken and are often undesirable in practice.
Conversely, in
composed of particle aggregates or fillers such as fumed
or
black, the extreme forces can lead to particle breakage and
(time-dependent
Indeed, propagating those forces into the
may be key to splitting the
into
It's thought, for instance, that the extreme mechanical stress required to grind up and pulverize particles is more effectively transferred to the particles when they are in a shear-thickened state in the
of a mill.
Interesting questions arise in the role of
in chemical mechanical planarization, a critical step in
processing. Concentrated
are useful for other
operations as well, and the control of their
can be critical to performance.
Although it's impossible to completely survey the science surrounding
in
and its applications, we hope the highly counterintuitive rheology has piqued your interest. A wealth of fascinating challenges and applications awaits.
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