Shear thickening in colloidal dispersions - Wagner2009p6165Phys

The popular interest

in cornstarch and water mixtures

known as “oobleck” after the complex fluid in one of Dr.

Seuss’s classic children’s books arises from their transition

from fluid-like to solid-like behavior when stressed. The vis-

cous liquid 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 liquid morphs into a doughy paste that

can be formed into shapes, only to “melt” into a puddle when

the applied stress is relieved. Internet 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).

Shear-thickening fluids certainly entertain and spark our

curiosity, but their effect can also vex industrial processes by

fouling pipes and spraying equipment, for instance. And yet,

when engineered into composite materials, STFs can be con-

trolled and harnessed for such exotic applications as shock-

absorptive skis and the soft body armor discussed in box 1.

Engineers and colloid scientists have wrestled with the

scientific and practical problems of shear-thickening col-

loidal dispersions—typically composed of condensed poly-

mers, metals, or oxides suspended in a liquid—for more than

a century. More recently, the physics community has ex-

plored the highly nonlinear materials in the context of jam-

ming

(see the article by Anita Mehta, Gary Barker, and Jean-

Marc Luck in P

HYSICS

ODAY

, May 2009, page 40) and the

more general study of colloids as model systems for under-

standing soft condensed matter.

Hard-sphere colloids are the “hydrogen atom” of col-

loidal dispersions. Because of their greater size and interaction

times compared with atomic and molecular systems, colloidal

dispersions are often well suited for optical microscopy and

scattering experiments using light, x rays, and neutrons. That

makes the dispersions, beyond their own intrinsic technolog-

ical importance, ideal models for exploring equilibrium and

near-equilibrium phenomena of interest in atomic and molec-

ular physics—for example, phase behavior and “dynamical ar-

rest,” in which particles stop moving collectively at the glass

transition. The relevance of colloids to atomic and molecular

systems breaks down, though, for highly nonequilibrium phe-

nomena. Indeed, shear thickening in strongly flowing colloidal

dispersions may be among the most spectacular, and elucidat-

ing, examples of the differences between the systems.

2009 American Institute of Physics, S-0031-9228-0910-010-8

October 2009 Physics Today

Shear thickening in

colloidal dispersions

Norman J. Wagner and John F. Brady

Shampoos, paints, cements, and soft body armor that stiffens under impact

are just a few of the materials whose rheology is due to the change in

viscosity that occurs when colloidal fluids experience shear stress.

Norm Wagner

is the Alvin B. and Julia O. Stiles Professor and chairperson of the department of chemical engineering at the University of

Delaware in Newark.

John Brady

is the Chevron Professor of Chemical Engineering at the California Institute of Technology in Pasadena.

The unique material properties of increased energy dissipation

combined with increased elastic modulus make shear-

thickening 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 electric field, which controls the damping. Researchers

have also explored the STF response in sporting equipment

and automotive applications,

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.

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 (Wag-

ner’s), in conjunction with the US Army

Research Laboratory, to use STF fabrics for

ballistic, puncture, and blast protection for

the military, police, and first responders.

Tests of the materials 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 measured at about 100 m/s.

The velocity required to penetrate Kevlar formulated with poly-

meric colloids (polymethyl-methacrylate) is about 150 m/s, and

that for Kevlar formulated with silica colloids is 250 m/s,

2.5 times that for the Kevlar alone. High-speed video dem -

onstrations 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, US Army Research Laboratory.)

1mm

50μm

5μm

Box 1. Soft armor and other applications

Figure 1 illustrates the effect. The addition of colloidal

particles to a liquid such as water results in an increase in the

liquid’s viscosity and, with further addition, the onset of non-

Newtonian behavior—the dependence of its viscosity on an

applied shear stress or shear rate. At high particle concentra-

tions, the fluid behaves as if it has an apparent yield stress.

That is, it must be squeezed, like ketchup, before it can actu-

ally flow. At such concentrations, the colloidal dispersions fit

into the general paradigm for jamming in soft matter:

high particle densities and low stresses (and low tempera-

tures, usually), the system dynamically arrests, just as atomic,

molecular, polymeric, and granular systems do. But once the

yield stress is exceeded, the fluid’s viscosity drops, a response

known as shear thinning. That rheology is engineered into a

range of consumer products, from shampoos and paints to

liquid detergents, to make them gel-like at rest but still able

to flow easily under a weak stress. Again, the colloid model

fits the general paradigm for how matter behaves: It flows

when sheared strongly enough.

At higher stresses, shear thickening occurs: Viscosity

rises abruptly, sometimes discontinuously, once a critical

shear stress is reached. The rise is counterintuitive and incon-

sistent with our usual experience. Experiments and simula-

tions on atomic and small-molecule liquids predict only

shear thinning, at least until the eventual onset of turbulence

at flow rates that vastly exceed those of interest here.

The ubiquity of the phenomenon in the flow of sus-

pended solids is a serious limitation for materials processing,

especially when it involves high shear-rate operations. In a

1989 review, Howard Barnes writes,

Concentrated suspensions of nonaggregating

solid particles, if measured in the appropriate

shear rate range, will always show (reversible)

shear thickening. The actual nature of the shear

thickening will depend on the parameters of the

suspended phase: phase volume, particle size

(distribution), particle shape, as well as those of

the suspending phase (viscosity and the details of

the deformation, i.e., shear or extensional flow,

steady or transient, time and rate of deformation).

Inks, polymeric binders for paints, pastes, alumina casting

slurries, 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 (shear rates typ-

ically up to 10

Hz), a process in which the coating’s increasing

viscosity would either tear the paper or ruin the equipment.

Industrial labs remain intensely interested in the science. Hun-

dreds of millions of metric tons of cement are used globally

each year, for example, and production engineers are careful

to formulate modern high-strength cements and concretes that

don’t suffer from the effect—at least in a range of shear rates

important for processing and construction.

In pioneering work in the 1970s, Monsanto’s Richard

Hoffman developed novel light-scattering experiments to

probe the underlying microstructural transitions that accom-

panied shear thickening in concentrated latex dispersions.

The transition was observed to correlate with a loss of Bragg

peaks in the scattering measurement. On that basis, Hoffman

developed a micromechanical model of shear thickening as

a flow- induced order–disorder transition.

In the 1980s and early 1990s BASF’s Martin Laun and

others interested in products such as paper coatings and

emulsion-polymerized materials used then emerging small-

angle neutron-scattering techniques to demonstrate that an

order–disorder transition was neither necessary nor alone

sufficient to induce significant shear thickening.

Because

shear thickening is a highly nonequilibrium, dissipative state,

though, a full understanding had to await the development

of new theoretical and experimental tools.

Hydrodynamics

The dynamics of colloidal dispersions 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 fluctuation–dissipation

relationship:

(1)

The diffusivity

scales with the thermal energy

divided

by the suspending medium’s viscosity

and the particle’s hy-

drodynamic radius

. That diffusivity sets the characteristic

time scale for the particles’ Brownian motion; it takes the par-

ticle

seconds to diffuse a distance equal to its radius. The

time scale defines high and low shear rates

.

A dimensionless number known as the Péclet number,

Pe, relates the shear rate of a flow to the particle’s diffusion

rate; alternatively, the Péclet number can be defined in terms

of the applied shear stress

(2)

The number is useful because dispersion rheology is often

measured by applied shear rates or shear stresses. Low Pe is

close enough to equilibrium that Brownian motion can

largely restore the equilibrium microstructure on the time

scale of slow shear flow. At sufficiently high shear rates or

stresses, though, deformation of the colloidal microstructure

by the flow occurs faster than Brownian motion can restore

it. Shear thinning is already evident around Pe ≈ 1. And

higher shear rates or stresses (higher Pe) trigger the onset of

shear thickening.

Pe =

γa

τa

October 2009 Physics Today

www.physicstoday.org

= 0.47

= 0.43

= 0.34

= 0.28

= 0

= 0.18

= 0.09

= 0.50

SHEAR STRESS (Pa)

VISCOSITY (Pa·s)

−3

−2

−1

Figure 1. The viscosity of colloidal latex dispersions,

as a

function of applied shear stress. The volume fraction

latex particles in each dispersion distinguishes the curves. A

critical yield stress must be applied to induce flow in a dis-

persion with high particle concentration. Beyond that criti-

cal 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 thresh-

old. (Adapted from ref. 12.)

The presence of two or more particles in the suspension

fundamentally alters the Brownian motion due to the inher-

ent coupling, or hydrodynamic interaction, between the mo-

tion 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 un-

derstanding the colloidal dynamics.

In essence, because any

particle motion must displace incompressible fluid, a long-

ranged—and inherently many-body—force is transmitted

from one particle through the intervening fluid to neighbor-

ing particles; the result is that all particles collectively disturb

the local flow field through hydrodynamic interactions. Such

interactions are absent in atomic and molecular fluids, where

the intervening medium is vacuum.

Batchelor’s calculation of the trajectories of non-

Brownian particles under shear flow identified the critical im-

portance of what’s known as lubrication hydrodynamics,

which describes the behavior of particles interacting via the

suspending medium at very close range. Those hydrodynam-

ics were already well known in the fluid mechanics 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 motion

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 hydrodynamic coupling alters the fluctuation–

dissipation relationship, which, in turn, enabled him to cal-

culate the diffusion coefficient and viscosity of dilute disper-

sions of Brownian colloids at equilibrium.

Although it was

not fully appreciated at the time, the effect of hydrodynamic

interactions on particle trajectories is the basis for under-

standing the shear-thickening effect.

Beyond two particles

Hydrodynamic interactions in real colloidal suspensions re-

quire numerical methods to solve. The method of Stokesian

dynamics outlined in box 3 calculates the properties of ensem-

bles of colloidal and noncolloidal spheres under flow. A great

advantage of the simulations is their ability to resolve which

forces contribute to the viscosity. Moreover, they demonstrate

that the ubiquitous shear thinning in hard-sphere colloidal

dispersions 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 electro-

static and van der Waals forces—but is not affected by hydro-

dynamic interactions. The low-shear (Pe ≪ 1) viscosity has

two components, one due to direct interparticle forces, which

dominate, and one due to hydrodynamic interactions.

Under weak but increasing shear flow (Pe ~ 1), the fluid

structure becomes anisotropic as particles rearrange to re-

duce their interactions so as to flow with less resistance. Fig-

ure 2 illustrates the evolution schematically. Near equilib-

rium, the resistance to flow is naturally high because shearing

the random distribution of particles causes them to fre-

quently collide, like cars would if careening haphazardly

along a road. With increasing shear rates, though, particles

behave as if merging into highway traffic: The flow becomes

streamlined and the increasingly efficient transport of col-

loidal particles reduces the system’s viscosity.

Simulations that ignore hydrodynamic coupling be-

tween 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-

www.physicstoday.org

October 2009 Physics Today

When two colloidal particles approach

each other, rising hydrodynamic pressure

between them squeezes fluid from the

gap. At close range, the hydrodynamic

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 Navier–Stokes equations that govern

the flow behavior between particles are

time reversible, so the force is the same

one required to separate two particles.

In simple shear flow, particle trajecto-

ries are strongly coupled by the hydro -

dynamic interactions 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 shear flow 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 theory of concentrated dispersions that

account for those short-range hydrodynamics show that at

high shear rates, particles that are driven into close proximity

remain strongly correlated and are reminiscent of the closed

trajectories observed in dilute suspensions. The flow-induced

density fluctuations 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 viscosity. The illustration at right during a

stage of the Stokesian dynamics simulation shows colloidal par-

ticles in hydro clusters.

Box 2. Lubrication hydrodynamics and hydroclusters

NORMALI

ZED LUBRI

CATION FORCE

DISTANCE BETWEEN PARTICLE SURFACES

1000

100

0.1

0.2

0.3

0.4

0.5

FLOW DIRECTION

SHEAR GRADI

ENT

−4

−3

−2

−1

−2

−1

Closed

SHEAR DI

RECTION