Seminar
Shear thickening in colloidal dispersions
Author: Andraž Krajnc
Advisor: prof. dr. Rudolf Podgornik
Ljubljana, February 2011
Abstract
First of all, I will introduce the basics of Newtonian fluids, followed by differences in
comparison to non-Newtonian fluids. Following that, I will consider suspensions and
focus on the shear thickening phenomena. Different theories behind that lead us to an
order-disorder system and a hydro-clusters formulation. Finally, I will present its
applications and prospects for the future.
Contents
1
Introduction .............................................................................................................. 3
2
Newtonian Fluids ...................................................................................................... 3
3
Non-Newtonian Fluids ............................................................................................. 4
3.1
4
5
6
Rheology of Suspensions ................................................................................... 5
3.1.1
Zero-Shear Viscosity .................................................................................. 5
3.1.2
Viscosity under Shear ................................................................................. 7
Hydrodynamics ......................................................................................................... 7
4.1
Lubrication force................................................................................................ 8
4.2
Stokesian dynamics ............................................................................................ 9
Applications ............................................................................................................ 11
5.1
Liquid Armor ................................................................................................... 11
5.2
Protective Equipment ....................................................................................... 12
5.3
Other Applications ........................................................................................... 13
Conclusion .............................................................................................................. 13
Literature ........................................................................................................................ 14
2
1 Introduction
Although shampoos, paints, cements and blood are known for a very long time, there
has been only little interest for the exploration of their physics. The main liquid has
always been water and its simple physics, which obey Newton’s laws, satisfied many
scientists in history until 20th century. After digging deeper and deeper many different
phenomena were discovered. One of them, known as shear thickening, is the topic of
this seminar.
2 Newtonian Fluids
It would be useful to begin with the definition of a Newtonian fluid. Viscosity is the
main attribute of all liquids. It is the measure of the resistance of a fluid which is being
deformed by either shear stress or tensile stress. The less viscous a liquid is, the greater
its ease of movement. An ideal fluid has no resistance to shear stress and therefore its
viscosity is equal to zero. Viscosity is influenced only by temperature and atmospheric
pressure, so under stable conditions the stress versus strain rate curve is linear and
passes through the origin [1]. We define Newtonian fluids with
̇
(1)
where is the shear stress exerted by the fluid, the fluid viscosity and ̇ the velocity
gradient perpendicular to the direction of the shear, or shorter, the strain rate. A
schematic representation of a unidirectional shearing flow is presented in Figure 1a. In
equation (1) the simplest case is presented; it only includes one non-zero component of
velocity,
.
a)
b)
Figure 1 (a) A schematic representation of flow in shear. (b) Stress components in three dimensions [1].
In a three dimensional flow there are six shearing and three normal components of stress
tensor, (Figure 1b). We usually split the total stress into
(2)
where is an isotropic part (pressure) and the traceless deviatoric part of the shear
stress. Because of the physical requirement of symmetry
, there are only three
independent shear components. For Newtonian fluids in Cartesian coordinates, these
3
components are related linearly to the rate of deformation tensor components via scalar
viscosity. For instance, let us observe components acting on the x-plane
(
)
(3)
(
)
(4)
(5)
We can see that in a simple shear diagonal components of shear tensor are equal to zero,
because velocity
only varies in the y-direction. To satisfy Navier-Stokes equations
we require the complete definition of Newtonian fluids rather than simply exhibiting a
constant value of shear viscosity.
3 Non-Newtonian Fluids
There exist also some fluids whose viscosity changes with the applied stress and they
are called non-Newtonian. In such cases, stress versus strain rate curve is not straight
and/or does not pass through the origin (Figure 2). In other words, Navier-Stokes
equations do not have solutions for such fluids. But viscosity is not only a function of
flow conditions (geometry, rate of shear, etc.). It depends also on the kinematic history
of the fluid.
Figure 2 Graph shows shear stress versus shear rate for different fluids. Curve for Newtonian fluid is
straight and goes through the origin, whereas curve for non-Newtonian fluids is not straight and/or does
not pass through the origin [2].
According to the stress-rate curve, we differentiate different types of non-Newtonian
fluids (Table 1). One of them is the shear thickening fluid, which we will talk about in
detail later. Shear thickening itself is a time independent process, in which viscosity
increases with increased stress. Shear thinning usually happens in the same fluids but at
lower shear stress.
4
Kelvin material
Viscoelastic
Anelastic
Rheopectic
Time-dependent viscosity
Thixotropic
Shear thickening (dilatant)
Time-independent viscosity
Shear thinning (pseudoplastic)
"Parallel" linear combination of
elastic and viscous effects
Material returns to a welldefined "rest shape"
Apparent viscosity increases
with duration of stress
Apparent viscosity decreases
with duration of stress
Apparent viscosity increases
with increased stress
Apparent viscosity decreases
with increased stress
Table 1 Presentation of different types of non-Newtonian fluids. [2].
3.1 Rheology of Suspensions
The addition of dispersed solid phase to Newtonian fluid forms suspension, which can
lead to the introduction of all kinds of non-Newtonian behavior. If the solid phase is
made of small particles, such a suspension is often called colloidal suspension. The
paramount concern in any multiphase liquid is stability. Thermodynamics drives
clumping of dispersed particles, and this is sometimes enhanced by flow. First of all,
particles must be small enough to prevent settling under gravity, unless their density
matches that of the suspending medium, or the suspending medium is very viscous.
Secondly, we need to prevent particles to clump and therefore need to increase the
repulsive interactions between them. Brownian motion of small particle movement
promotes particle collisions, which could lead to aggregation and consequently to
gravitational settling [3]. Electrostatic and steric stabilization are the most common.
There is a possibility for another problem if the particles are big enough –
inhomogeneities due to its migration with a flow. Particle size should be kept below
about 1 µm.
3.1.1 Zero-Shear Viscosity
In 1906, Einstein in his PhD thesis came up with a simple formula for dilute
suspensions at very low volume fractions ( ≤ 0.03). It was the first analytical solution
for hydrodynamics around an isolated sphere. Relative viscosity is expressed as
(6)
where is the Einstein coefficient, and takes the value
. This coefficient is still
not incontrovertibly validated, and varies between 1.5 and 5 in different researches [4].
Equation (6) is only valid for suspensions dilute enough for the flow field around one
sphere not to be influenced by the presence of neighboring spheres. If two spheres are
close enough, they experience hydrodynamic interactions, which leads to a contribution
to viscosity that is proportional to
. When there are more neighboring inter-particle
interactions, higher orders of contribute to viscosity. Such a polynomial dependency
can suitably describe semi-dilute suspensions ( ≤ 0.25) with the equation
(7)
5
where we find values of
varying from 7.35 to 14.1 derived from the consideration of
particle-particle interactions and the value can be even lower when Brownian motion
and inertia have a big enough influence. For concentrated suspension ( > 0.25),
relationship (7) typically describes viscosity behavior poorly. At the densest possible
packing for monodisperse spherical particles, that is
, we expect viscosity to
be infinite. Therefore, a much more appropriate choice is the Krieger & Dougherty
equation
(
)
(8)
where B is the Einstein coefficient and
the maximum packing volume fraction.
Binomial expansion of equation (8) also recovers the polynomial given in equation (7).
The product
usually remains in the range 1.4-3 [5]. As the particle aspect ratio
increases, increases and
decreases. Viscosity is very sensitive to volume fraction
at large , thus small errors of could lead to large errors in the value of . When we
obtain a steric stabilization of particles, the hard-sphere radius and the volume fraction
must be adjusted to avoid these errors (Figure 3a). If the volume fraction is not too close
to the maximum packing fraction, the grafted layer can be considered as hard coating
and can be corrected with the formula
(
)
(9)
where
is the volume fraction of uncoated particles, the thickness of coating and
the particle radius. At high values, the particle-size distribution has a strong effect on
viscosity. Figure 3b shows how relative viscosity changes with the fraction of large
particles in bidisperse suspension with a 5:1 ratio of diameters. For a total volume
fraction greater than 0.60, the viscosity drops by more than a decade as the fraction of
large particles increases from 0 to 0.6. This is a consequence of the packing of smaller
particles into the interstices between the large ones.
a)
b)
Figure 3 (a) Hard sphere particle grafted with a layer of polymer can be approximated by a hard sphere
with bigger radius [6]. (b) Relative viscosity versus fraction of large particles in a bimodal distribution of
particle sizes with a 5:1 ratio of diameters is shown at different total volume particle fractions [5].
6
3.1.2 Viscosity under Shear
For colloidal dispersions with a volume fraction > 0.3, viscosity becomes sensitive to
shear rate (or equivalently shear stress). This occurs when the shear rate is high enough
to disturb the distribution of interparticle spacings from equilibrium [7]. At high particle
concentrations, suspension has a much greater value of viscosity and it noticeably flows
under yield stress. The region where viscosity drops with shear rate is called shear
thinning. Hard sphere colloids are similar to atoms, unless they are much bigger and
thus well suited for optical microscopy and scattering experiments. That makes the
dispersions ideal models for exploring equilibrium and near-equilibrium phenomena of
interest in atomic and molecular physics. However, its relevance to atomic scale breaks
down for highly non-equilibrium regime. In suspensions, at high shear rates, we witness
the growth of viscosity, which is known as shear thickening. Figure 4 illustrates both
effects.
Figure 4 Viscosity versus shear stress for colloidal latex dispersions at various volume fractions. At high
critical yield stress must be applied to flow and at higher stress viscosity increases (shear thickening)
[7].
4 Hydrodynamics
In zero-shear-rate suspension there are particles that move with Brownian motion.
Diffusivity in dilute solution is given by
(10)
where k is the Boltzmann constant, T the temperature, the solvent viscosity and r the
radius of the particle. It takes
seconds for the particle to diffuse the radius
distance. Instead of the shear rate, it is more appropriate to define the dimensionless
Pécklet number
̇
̇
7
(11)
This number is a direct indicator of the interparticle hydrodynamic forces in comparison
to the Brownian force, which tries to establish equilibrium. For low
, Brownian
motion dominates, but with large
numbers deformation occurs so fast that Brownian
motion cannot restore it [7]. Shear thinning is evident for Pécklet numbers around 1, but
on the other hand, a much higher
triggers the onset of shear thickening.
4.1 Lubrication force
The key mechanism that overcomes Brownian motion is lubrication hydrodynamics.
Every particle motion must displace an incompressible fluid, which results in a longrange transmission of forces via suspending medium. All particles collectively
contribute to local flow field disturbance through hydrodynamic interactions [8]. This is
not present in atomic fluids, where the intervening medium is a vacuum and from that
point of view it is understandable that we should not observe shear thickening at atomic
scale.
Let us, for example, take a look at hard-sphere colloids approaching each other in a
suspended fluid. If the gap between particle surfaces is much smaller than the particle
diameter, lubrication force comes into effect [9]. The average distance between
neighboring particles in concentrated suspension is given as follows
(
(12)
)
where h is the shortest distance between two particles, the radii of the particle and
the particle loading. For
the average distance between two particles is 0.046
times the radius. For small gaps (
) we can apply the lubrication approximation
⁄
to the Stokes equations. We take into account the leading singular terms
from the expansion in the narrow gap, which can be considered as different modes [10].
In Figure 5a, there are two modes with the greatest impact on lubrication force. Squeeze
mode can be determined by the leading order of the force on particle as
∑
(
)
(13)
where we sum over nearest neighbors ,
is the surface separation between particles,
and
are the particle velocities, and
is the unit vector along the line of the
centers between two particles. Equation (13) can also be derived from the Stefan force,
which is the interaction between two surfaces in a viscous fluid at small separation [11].
There are also other lubrication modes (shear, pump, and twist). Shear mode has the
second greatest impact on lubrication force. It diverges logarithmically with the
interparticle gap as we see in
∑
(
) (
8
) (
)
(14)
Here is the unit matrix. The lubrication force increases inversely with the distance
between the surfaces of the particles and diverges to a singularity as we see in Figure
5b. The Navier-Stokes equations are time reversible; consequentially the lubrication
hydrodynamics have the same effect on particles separation.
a)
b)
Figure 5 (a) In squeeze mode particles move along the line of the particle centers, rotation is not
included. Squeeze force acts so as to oppose the relative motion [10]. (b) Lubrication force increases
inversely with the distance between particle surfaces and diverge on contact [7].
Very common is the polymer stabilization of the particles, which changes lubrication
force and also affects shear thickening characteristics. A polymer coated particle feels
an enhanced dissipative force due to the flow of the solvent within the coat. By solving
the Brinkman equation (Stokes eqn. extension for porous medium) within the coat
(15)
where
is the flow penetration depth into the brush, we get (see [12] for details)
(16)
(17)
where U is the relative velocity of the particles;
the function of penetration
depth , length of polymer coat and the distance between hard spheres [12]. At high
shear rates, particles are driven into close proximity and remain strongly correlated. The
flow-induced density fluctuations are known as hydroclusters. Clustering leads to an
increase in energy dissipation and consequently to a higher viscosity [7]. Stokesian
dynamics gives the correct weight to the forces in a colloidal motion mechanism.
4.2 Stokesian dynamics
The simulation of a colloidal system presents a significant challenge. For a single sphere
it can be handled analytically, for two spheres semianalytically and for more particles it
requires numerical solution of the Navier-Stokes equation without inertia [7]. There are
9
many different methods for describing fluids, most of them start with the Langevin
equation for N-particle dynamics,
(18)
where
is 6N×6N mass and moment-of-inertia matrix (generalized mass),
is 6Ndimensional particle velocity vector,
represents the interparticle and external forces,
hydrodynamic forces and
Brownian forces. This equation must be solved at each
time step to determine the particle velocities. In Stokes flow the hydrodynamic forces
are linearly related to the particle velocities as
(19)
where is the hydrodynamic resistance matrix, the elements of which contain details
about the hydrodynamic interactions between particles. In a simulation we need
inversion of the resistance matrix which is
operation [10]. To overcome this, one
can approximately describe the evolution of the system by just including two body
terms with divergent lubrication interactions between close particle surfaces, within the
elements of the matrix, . Now inversion of the resistance matrix becomes an
process and the results of simulation reveal the colloidal microstructure associated with
a particular shear viscosity (Figure 6).
a)
b)
Figure 6 (a) Microstructure in hard-sphere colloidal suspensions and relationship between viscosity and
shear stress is shown. In equilibrium, random collision prevents flow, increasing shear rate forces
particles to organize into layers and viscosity lowers. At yet a higher shear rate interparticle interactions
dominate over stochastic ones and there occurs the formulation of hydroclusters, which increases
viscosity [7]. (b) The contribution of Brownian and hydrodynamic particle movement to total viscosity
shows that at low Pecklet number Brownian force is dominated and at high Pe the lubrication forces [5].
At equilibrium Brownian force prevails, colloids are not correlated. The resistance to
flow is naturally high, because of shearing, the random distribution of particles causes
them to frequently collide [8]. At low shear stress Brownian contribution disappears,
leaving only a hydrodynamic contribution (see Figure 6b). Simulations show the
formation of layers of particles parallel to the flow direction [13], but they are not
necessarily rigorously ordered, and layer thickness can range from one to multiple
particle diameters. The flow becomes streamlined and the ease of movement of
10
colloidal particles reduces the viscosity of the system. At the onset of shear thickening,
a hydrodynamic instability causes particles to be driven out of these layers. Particles
interact through clustering with lubrication and frictional forces involved. The difficulty
of particles flowing around each other results in a higher rate of energy dissipation and
increase in viscosity. Rheo-optical measurements also confirmed the predictions of the
simulation [14; 15].
5 Applications
5.1 Liquid Armor
During the last 10 years the US Army has shown great interest in shear thickening fluids
(STF). They came to the conclusion that impregnating STF into Kevlar fabrics
improved its performance. It seems that the most suitable STF for such application is
the suspension of silica particles in ethylene glycol. The average particle diameter in
research letters was measured by dynamic light scattering and determined to be 446 nm
[16]. Ethylene glycol is an organic compound widely used as automotive antifreeze. It
was chosen as a solvent due to its volatility and thermal stability. The silica particles
were, for better dispersibility, predispersed in methanol and then blended with ethylene
glycol. Afterwads the dispersion was treated by homogenization for 1h and sonication
for 10h to further improve the dispersibility [17]. At the end density of the silica
particles in solution was estimated to be 1.78 kg/L (
). Experimental research
suggests that the best way to include STF into body armor vest is to impregnate it into
Kevlar fabrics. First of all, the fabrics are immersed in the prepared STF, and are then,
when wet, squeezed by a 2-roll mangle to set the specific wet pick up and to improve
the STF infiltration. At the end, the fabrics are dried in a vacuum oven at 65°C for 20
min. Figure 7 indicates the comparison of six different ways of STF inclusion tested.
Figure 7 Six different setups of Kevlar and STF under study (upper-left). Penetration depth of the
projectile in the clay witness (bottom-left) prefers Kevlar impregnated with STF to other configurations.
In the right graph the energy dissipation percentage is calculated for each of the setups [16].
11
Even though the dissipation energy does not vary much, there is another important
factor for protection, that is penetration depth. Body armor standards require that
penetration depth into a clay witness should not exceed 4.3 cm [16]. To satisfy the
conditions, approximately 20-50 layers of neat Kevlar fabric are required. Such a vest is
heavy and not very flexible. Impregnation of STF into Kevlar is therefore the best
option, because it not only absorbs more energy, but also spreads its impact across a
wider area, resulting in a lower value of penetration depth. For comparison see Table 2.
Table 2 Flexibility and thickness for pure Kevlar and STF impregnated Kevlar [16].
The other great advantage of STF impregnated over neat Kevlar is the protection against
stabbing [17]. It is presumed that the STF, which fills the interstices between Kevlar
filaments, keeps the arrangement of the fibers in the yarn, which induces an increase in
the endurance of the yarn under pulling stress (Figure 8).
a)
b)
Figure 8 (a) Photos of front and rear sides of neat Kevlar and STF-Kevlar after ice pick drop test [18]. (b)
Experimental results of spike test with different loads confirm the improvement of armor with STF
impregnated [17].
5.2 Protective Equipment
British company d3o Lab developed shear thickening material for use in protection
purposes [19]. The structure of this material is unknown to the public. They are
currently trying to negotiate a contract with the Ministry of Defense. The material is
called d3o and is widely spread in the sports equipment industry. Ski helmets, footwear,
gloves, jackets etc. are just a few products that contain d3o as a shock absorber. The
most advertised feature is flexibility in contrast to hardness, for when an accident
occurs.
12
5.3 Other Applications
In the oil industry, many costly and dangerous problems can be encountered when
drilling, such as gas influx, lost circulation or underground blowouts [20]. They often
fail to seal the loss zone with lost circulation materials or cements. A blowout usually
occurs when a drill reaches a gas pocket. When drilling, the shear thickening suspension
is still liquid and can be pumped to the wellbore. STF reacts as an immediate patch due
to the stress caused by a sudden blowout and it solidifies. Another wide-spread use of
STF is traction control, which comes into play in the automotive industry [21]. To
provide the power transfer between the front and the rear wheels, some all wheel drive
systems use a viscous coupling unit. Imagine two coaxial cylinders rotating in the same
directions, one at the speed of the front wheels and the other at the speed of the rear
wheels. The space between them is full of STF. When the grip of all wheels is good, the
relative motion is zero, thus low or no shear is presented. At the moment of the front
wheel slip, STF is affected by a high shear stress, which leads to shear thickening. The
power of the front wheels is transferred to the rear wheels which hopefully push the car
out of the slippery zone. Since they used STF with very limited maximum viscosity,
which limits the amount of torque that can be passed across the coupling, this system is
more appropriate for on-road vehicles.
6 Conclusion
There are many good things about shear thickening and there are also industrial
problems especially for material treatment at high frequencies. In both cases it is very
welcome to know the origin and background of this phenomenon. Even if it is not
completely surveyed the science surrounding shear thickening in colloidal dispersions,
it is a good start for further investigations and applicative usage in everyday life.
13
Literature
[1] A. P. Deshpande, J. M. Krishnan and P. B. S. Kumar, Rheology of Complex
Fluids (Springer, New York, 2010).
[2] http://en.wikipedia.org/wiki/Non-Newtonian_fluid (20.1.2011).
[3] C. Tropea, A. L. Yarin and J. F. Foss, Springer Handbook of Experimental Fluid
Mechanics (Springer, New York, 2007).
[4] S. Mueller, E. W. Llewellin and H. M. Mader, Phil. Trans. R. Soc. A 466, 1201
(2010).
[5] R. G. Larson, The Structure and Rheology of Complex Fluids (Oxford, New
York, 1999).
[6] J. M. Brader et al., J. Phys.: Condens. Matter 22, 363101 (2010).
[7] N. J. Wagner and J. F. Brady, Physics Today 62, 27 (2009).
[8] R. L. Hoffman et al., J.Rheol. 42, 111 (1998).
[9] S. W. Lee, S. H. Ryu and C. Y. Kim, J. Non-Newtonian Fluid Mech. 110, 1
(2003).
[10] A. A. Catherall and J. R. Melrose, J. Rheol. 44, 1 (2000).
[11] A. Borštnik, R. Podgornik and M. Vencelj, Rešene naloge iz mehanike kontinuov
(DMFA, Ljubljana, 2001).
[12] A. A. Potanin and W. B. Russel, Physical Review E 52, 730 (1995).
[13] P. J. Mitchell and D. M. Heyes, Molecular Simulation 15, 361 (1995).
[14] S. J. Lee and N. J. Wagner, Ind. Eng. Chem. Res. 45, 7015 (2006).
[15] F. Ianni, Complex behavior of colloidal suspensions under shear: dynamic
investigation through light scattering techniques (doctoral dissertation, 2006).
[16] Y. S. Lee, E. D. Wetzel and N. J. Wagner, J. Mat. Sci. 38, 2825 (2003).
[17] T. J. Kang, K. H. Hong and M. R. Yoo, Fibers and Polymers 11, 719 (2010).
[18] http://www.ccm.udel.edu/STF/images1.html (27.1.2011).
[19] http://www.d3o.com (27.1.2011).
[20] C. Hamburger et al., J. Petrol. Tech. 37, 499 (1985).
[21] http://en.wikipedia.org/wiki/Dilatant (27.1.2011).
14