Dynamic properties of shear thickening colloidal suspensions

Rheol Acta (2003) 42: 199–208
DOI 10.1007/s00397-002-0290-7
Young Sil Lee
Norman J. Wagner
Received: 22 June 2002
Accepted: 20 November 2002
Published online: 5 February 2003
Ó Springer-Verlag 2003
Y. S. Lee Æ N. J. Wagner (&)
Department of Chemical Engineering
and Center for Composite Materials,
University of Delaware, Newark,
DE19716, USA
E-mail: wagner@che.udel.edu
Introduction
When subjected to increasing shear stress, concentrated
colloidal suspensions can exhibit a steep rise in viscosity
(Lee and Reder 1972; Hoffman 1974, 1997; Barnes
1989). This shear thickening phenomenon can damage
processing equipment and induce dramatic changes in
suspension microstructure, such as particle aggregation,
which results in poor fluid and coating qualities. On the
other hand, this behavior can be exploited in the design
of damping and control devices, whereby the fluid can
limit the maximum rate of flow through a highly nonlinear
response (Laun et al. 1991; Helber et al. 1990).
It has been demonstrated that reversible shear
thickening in concentrated colloidal suspensions is due
to the formation of jamming clusters bound together by
hydrodynamic lubrication forces, often denoted by the
term ‘‘hydroclusters’’ (Bossis and Brady 1989; Farr et al.
1997; Foss and Brady 2000; Catherall et al. 2000). The
microstructure of shearing suspensions has been studied
ORIGINAL CONTRIBUTION
Dynamic properties of shear thickening
colloidal suspensions
Abstract The transient shear rheology
(i.e., frequency and strain dependence)
is compared to the steady
rheology for a model colloidal dispersion
through the shear thickening
transition. Reversible shear thickening
is observed and the transition
stress compares well to theoretical
predictions. Steady and transient
shear thickening are observed to
occur at the same value of the average
stress. The critical strain for
shear thickening is found to depend
inversely on the frequency at fixed
applied stress for low frequencies
(high strains), but is limited to an
apparent minimum critical strain at
higher frequencies. This minimum
critical strain is shown to be an artifact
of slip. Lissajous plots illustrate
the transition in material
properties through the shear thickening
transition, and the energy
dissipated by a shear thickening
suspension is analyzed as a function
of strain amplitude.
Keywords Shear thickening Æ
Suspension rheology Æ Colloid Æ
Dispersions Æ Dilatancy
by rheo-optical experiments (D’Haene et al. 1993;
Bender and Wagner 1995), neutron scattering (Laun et
al. 1992; Bender and Wagner 1996; Newstein et al. 1999;
Maranzano and Wagner 2001a, 2002) and stress-jump
rheological measurements (Kaffashi et al. 1997). The
onset of shear thickening in steady shear can now be
quantitatively predicted (Maranzano and Wagner
2001a, 2001b) for suspension of hard-spheres and electrostatically
stabilized dispersions.
Although a significant number of experimental and
simulation studies have addressed shear thickening in
steady shear flow, only limited work has addressed the
viscoelastic properties of a shear thickening fluid (Laun
et al. 1991; Boersma et al. 1992; Raghavan and Khan
1997; Mewis and Biebaut 2001). This is partly a consequence
of the difficulty of studying shear thickening,
which is a highly nonlinear response that is often triggered
by relatively high stress levels.
Nonlinear oscillatory shear experiments are useful for
characterizing the onset of shear thickening as well as for

200
determining the time scales required to generate the
shear thickening response. Of interest for technological
applications is the time scale and the minimum strain
required to generate the hydrocluster microstructure
underlying the shear thickening response. For example,
electrostatically stabilized dispersions have been investigated
in both steady and dynamic oscillatory shear
flows for possible damping applications. Laun et al.
(1991) reported the dynamic strain hardening behavior
of a polymer latex dispersion. Their oscillatory test
protocol increased the strain amplitude at a given frequency,
which leads to a shear thickening rheology.
The critical strain for dynamic shear thickening (cc) was observed to decrease with increasing frequency
(x), but eventually plateaued at higher frequencies.
The low frequency behavior was interpreted in terms
of the steady shear behavior, where a critical shear rate
( _c dynamic
c _c steady
c ) must be achieved to thicken. The high
frequency limiting value suggested that a minimum
shear strain ( 50%) is necessary in each half cycle to
cause the dispersion to switch to the high viscosity state.
Similar conclusions for low frequencies were reached by
Boersma et al. (1992), who investigated monodisperse
silica particles suspended in a mixture of glycerol and
water. They reported ‘‘flow blockage’’ in oscillatory
testing, which was also related to steady shear thickening
at low frequencies. Intermediate frequencies yielded a
weaker frequency dependence, but no plateau value.
Finally, at high frequencies the critical deformation for
shear thickening was found to be independent of particle
volume fraction, but again scaled inversely with
frequency. This latter behavior was attributed to the
solid-like response for a sample that is fully in the
hydrocluster state. Note that these experiments were
also performed on a controlled strain rheometer and
the critical strain amplitudes were of order O(10 -2 ) at the
highest frequencies.
Studies of near hard sphere dispersions (Bender
1995) also confirm the agreement between steady shear
thickening and the low frequency dynamic oscillatory
response. Raghavan and Khan (1997) observed similar
congruence at low frequencies, as well as a high frequency
limiting critical strain (cc O(1)) for fumed silica
dispersions in poly(propylene glycol). Recently,
Mewis and Biebaut (2001) also observed dynamic
shear thickening in sterically stabilized colloidal suspensions.
They observed that the peak shear stress at
the onset of shear thickening in oscillatory flow corresponds
to the same steady shear stress measured at
the onset of shear thickening, with no evidence of a
limiting critical strain down to shear strains of order
0.5. Notably, Mewis and Biebaut (2001) also investigated
the shear thickened state by parallel superposition,
observing a viscoelastic liquid response in the
shear thickened state, not the solid response suggested
by Boersma et al. (1992).
In summary, the literature data to date suggests that
the onset of strain hardening at low frequencies for
concentrated suspensions in oscillatory shear flow can
be interpreted in terms of the onset of steady shear
thickening. However, there is contradictory evidence as
to whether a critical strain is required for oscillatory
strain hardening, and as to whether a third, solid-like
regime exists at higher frequencies. This issue is relevant
for the design of devices based on the shear
thickening response (Helber et al. 1990). The goal of
this work is to relate the nonlinear viscoelastic properties
to the steady shear response for a shear thickening
fluid, and to determine if a minimum critical
strain is necessary for shear thickening. This is achieved
by rheological investigation of a model dispersion. Of
particular interest is the critical strain amplitude required
for shear thickening in dynamic shear flow and
its dependence on frequency. Finally, Lissajous plots
are constructed to illustrate the ‘‘switching’’ from liquid
to solid observed during deformation, and to determine
the energy dissipation’s depend on strain amplitude in
a shear thickening fluid.
Experimental
Sample preparation and characterization The colloidal silica investigated
here was obtained from Nissan Chemicals (MP4540), which
is provided as an aqueous suspension (pH=8.5 at 25 °C) with a
particle concentration of about 40 wt%. The particle size distribution
has been characterized with dynamic light scattering and
TEM. Figure 1 shows a transmission electron micrograph of the
suspension; which is observed to contain a minor fraction of
smaller particles. The average particle diameter (z-average) was
determined to be 446±8.4 nm by dynamic light scattering, which
agrees with the TEM measurements of the large particle fractions.
The solution density of the particles has been obtained by measuring
the density of the suspension as a function of weight fraction
of the particles. The weight fraction of silica was determined
gravimetrically after drying the sample at 180 °C for 5 h using a
convection oven. The density of the silica calculated from this
method is 1.78 g/cc. The zeta potential has been determined to be
)32 mV from electrophoresis measurements (Brookhaven Zeta-
PALS) at pH=8.5 and CSALT=0.045 mmol/l. This suspension was
concentrated by tabletop centrifugation. The sediment was resuspended
using a vortex mixer after adding of small amount of the
supernatant liquid. Dilution with the mother liquor provided a
series of aqueous silica suspensions. The suspending fluid was also
replaced with ethylene glycol (EG) by repeated centrifugation and
resuspension with a vortex mixer. This process has been repeated
four times to prepare a second series of dispersions of the same
particles in ethylene glycol.
Rheological measurements The experiments were performed primarily
in a stress-controlled rheometer (SR-500, Rheometrics) at
25 °C with cone-plate geometry having a cone angle of 0.1 radian
and a diameter of 25 mm. Complementary measurements were
performed on a Rheometrics ARES controlled strain rheometer. A
parallel plate geometry was also used with varying gap size to
characterize slip. To prevent adhesive slip between the sample and
the rheometer plates, parallel plates of diameter 25 mm were covered
with emery cloth (NORTON, E-Z FLEX METALITE K224)
using double stick tape. The gaps explored varied between 0.05 mm

and 1.5 mm. A solvent trap prevented evaporation of the solvent.
Three types of dynamic experiments have been performed: stress or
strain sweeps at constant frequency, frequency sweeps with a
constant imposed stress, and time transient measurement of strain
response with fixed stress amplitude and frequency. To remove
loading effects, a preshear of 1 s -1 was applied for 60 s prior to
further measurement. All measurements presented here were
reproducible.
Results and discussion
Result of stress and frequency sweeps
Figure 2 shows the steady shear viscosity and the complex
viscosity as a function of the steady shear stress or
average dynamic shear stress, respectively, for the
aqueous suspension at volume fractions of u=0.55 (a)
and 0.60 (b). Shear thickening is evident in both steady
and dynamic measurements. To confirm the reversibility
of the shear thickening behavior, the complex viscosity
was measured for both ascending and descending stress
sweeps, with good agreement.
Shear thickening is known to be a stress controlled
phenomena; hence one might expect to be able to superimpose
steady shear viscosity and complex dynamic
viscosity when plotted against applied stress. The aver-
Fig. 1 Transmission electron microscopy of colloidal silica obtained
from Nissan Chemicals (MP4540) at a magnification of
40,000
201
age dynamic shear stress applied to the fluid during oscillatory
testing is obtained by integrating the absolute
value of the applied stress over one cycle as
sd ¼ x
Z 2p=x
jsdjdt ¼
2p 0
2s0
ð1Þ
p
where sd=s0cosxt and s0 is the maximum stress. Note
that the root mean square value, i.e.,
Fig. 2a,b Reversible shear thickening behavior of: a 55 vol.%; b
60 vol.% colloidal silica dispersed in water for both steady and
dynamic shearing plotted against the average applied dynamic
shear stress

202
sRMS d ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
R
x 2p=x
2p 0 s2 ddt q
¼ sdffiffi p 1:11sd, could be used
2
with equal superposition to within the accuracy of our
measurements. However, the data do not superimpose
when plotted against the maximum applied stress, or the
shear rate. At the point of shear thickening, the dynamic
viscosity is found to be comparable to the steady shear
viscosity measured at the same average stress. Deviations
between the complex and steady shear viscosities
are evident at both low and high shear stresses, as will be
discussed below.
Using the model developed by Maranzano and
Wagner (2001a, 2001b), the critical shear stress for shear
thickening (sc) can be predicted from independent
measurements of the particle size, concentration, surface
potential, and ionic strength. The equation for the critical
shear stress for electrostatically stabilized dispersions
is
sc ¼ 0:024 kBT ðjaÞW 2
ð2Þ
a 2 lb
where l b is the Bjerrum length defined by l b ” e 2 /
(4p 0k BT), a is the radios of the particle, and Y s=w se/
k BT is the dimensionless surface potential. The theoretical
predictions for s c are found to be in good agreement
with the measured values (Table 1). In this table h m is
the characteristic separation distance in the incipient
hydrocluster state, which is determined directly from
j )1 , the Debye length by h m=1.453/j.
The extended Cox-Merz rule equates the steady and
dynamic viscosities at equivalent shear rate and frequency.
As our data is highly nonlinear, this relationship
is observed not to hold. For materials with slow relaxing
microstructure, a modified Cox-Merx rule has been
proposed by Doraiswamy et al. (1991) known also as the
‘‘Delaware-Rutgers’’ rule. The basis of this approach is
that slowly relaxing materials respond to the highest
applied strain rate applied during the dynamic measurement.
Consequently dynamic and steady properties
overlay when the highest shear rate experienced during
the oscillation is taken as the effective steady shear rate
ðc 0x _c). Raghavan and Khan (1997) use this rule to
correlate the steady shear thickening to strain thickening
for their fumed silica suspension. In Fig. 3 we compare
data obtained from a frequency sweep experiment with
imposed maximum stress of 10 Pa to steady shear data
in the sprit of the ‘‘Delaware-Rutgers’’ rule. It is
apparent that the ‘‘Delaware-Rutgers’’ rule applies in
Table 1 Comparison of experimental and theoretical critical shear
stress foraqueous silica particle suspensions
Volume
fraction /
ja hm
(nm)
Theoretical critical
shear stress sc (Pa)
0.55 8.4 39 37 20±0.5
0.60 9.4 35 41 46±0.5
Experimental critical
shear stress sc (Pa)
the region of shear thinning for the suspension of 60%
silica particle, but that the shear thickening transitions
do not superimpose. Closer inspection of the data of
Raghavan and Khan (Fig. 10 of their work) shows a
similar level of disagreement upon shear thickening.
Consequently neither extended Cox-Merz nor the Delaware-Rutgers
rule correlates dynamic and steady viscosities
in the shear thickening regime. This is further
confirmed by Mewis and Biebaut (2001), who report
qualitative differences in the parallel superposition viscoelastic
measurements between shear thinning and
shear thickening parts of the flow curve. Thus, although
the ‘‘Delaware-Rutgers’’ rule applies to the shear and
frequency thinning regions of the flow curve, it fails to
correlate the viscosity in the shear thickened state or the
onset of the shear thickening transition.
This result is consistent with the concept that the
shear thickening transition is stress controlled, as demonstrated
in Fig. 2 above. As the shear-thickening
transition is relatively fast for our materials, and the
response is in the nonlinear regime, neither the extended
Cox-Merz rule nor the Delaware-Rutgers approach
would be expected to hold in the shear thickening
regime, as demonstrated.
Shown in Fig. 4 is the shear thickening behavior as a
function of strain amplitude for the aqueous suspension
at 60 vol.% silica. The strain reported here is the measured,
peak strain amplitude (this experiment is
Fig. 3 Application of Cox-Merz and Delaware-Rutgers rules for
60 vol.% aqueous silica dispersion

Fig. 4 Strain thickening behavior of 60 vol.% aqueous silica
dispersion as a function of dynamic frequency
performed on a stress controlled instrument). From this
data, the critical strain for strain thickening was found
to depend inversely on the dynamic frequency for low
frequencies, as shown in Fig. 5. The frequency dependence
of the critical amplitude has been reported for
various systems. Laun et al. (1991) and Raghavan and
Khan (1997) observed transitional behavior, namely that
at low frequencies the critical amplitude decreases inversely
with the frequency, while at very high frequencies
(x>100 rad/s) the critical amplitude approaches a
constant value as shown in Fig. 5. In the present experiments
the frequency for the onset of shear thickening
is around 30 rad/s and the minimum critical strain was
found to be 2. Figure 5 also shows the critical strain
for the ethylene glycol suspension of 62 vol.% silica. The
overall behavior is similar to that of the aqueous silica
suspension, but the minimum critical strain of the ethylene
glycol based suspension is lower ( 0.7).
The low frequencies data reported in Fig. 5 can be
understood within the context of the model presented by
Maranzano and Wagner (2001a, 2001b). As shown, the
dynamic oscillatory measurement show shear thickening
at the same value of the average applied stress as the
steady shear measurements. During the oscillation, shear
thickening is observed to occur when s>sc. One can
thus estimate the strain required by
sc
cc ; ð3Þ
gcx where g c is the shear viscosity at the critical point
for shear thickening and s c is the critical stress, both
determined from the steady shear data. The lines in
Fig. 5 are calculated from the measured steady shear
data. The predictions work well for our data (Fig. 5a)
and that of Raghavan and Khan (1997) and Mewis and
Biebaut (2001) (Fig. 5d) at low frequencies. However,
deviations from this prediction become evident at higher
frequencies. The data of Boersma et al. (1992) (Fig. 5b)
may be limiting toward the predictions, whereas the data
of Laun et al. (1991) does not have the corresponding
steady shear data for making the prediction. However,
note that Laun’s data qualitatively displays the predicted,
low frequency behavior.
Wall slip has been postulated to influence the high
frequency results (Boersma et al. 1992). Figure 6 is a
schematic drawing of the postulated deformation with
slip between parallel plates, where the slip distance at the
wall has been defined as D slip and h is the gap size. Thus,
the measured or apparent strain is the sum of the true or
real strain in the sample and the strain due to slip as
cc;app ¼ cc;real þ 2Dslip
: ð4Þ
h
Thus, at the point of shear thickening cc,app(=Dapp/h)
is the apparent critical strain for shear thickening and
cc,real (=Dreal/h) is the real critical strain for shear
thickening. The apparent critical strain has been measured
using 25-mm parallel plates where the gap size is
varied from 1.5 mm to 0.05 mm. Figure 7 shows the
measured apparent critical strain for shear thickening
for silica dispersed in both water and ethylene glycol at
25 °C. The lines correspond to a fit of Eq. (4) to determine
the slip distance. The fits yield slip distances of
0.44 mm for the ethylene glycol based shear thickening
fluid and 1.45 mm for the water based shear thickening
fluid. However, with the use of roughened plates the
critical strain for shear thickening obtained during strain
sweep experiments were found to be nearly independent
of gap size, as shown in Fig. 7, demonstrating that the
slip was substantially reduced.
This analysis can explain why the measured critical
strain for shear thickening limits at high frequencies. We
can model the critical strain shown in Fig. 5 by accounting
for a slip distance at the fixture walls as
c c;app ¼ sc
g cx
203
2Dslip
þ : ð5Þ
h
In the comparison, we take h 1.25 mm, the edge
gap, to be a characteristic distance for the cone-plate
geometry used for these measurements. In Fig. 8 the
lines representing the predictions of Eq. (5) are in
good agreement with the observed critical strain as a
function of frequency. For comparison, we used our
slip measurements and the reported steady shear data
to predict the critical strains reported by Raghavan
and Khan (1997) for fumed silica in polypropylene
glycol. Their data fits well with their steady shear data

204
Fig. 5a–d Critical strain for shear thickening plotted as a function
of frequency for: a 60 vol.% aqueous and 62 vol.% ethylene glycol
based silica dispersions; b silica in glycerol/water by Boersma et al.
(1992); c latex in ethylene glycol by Laun et al. (1991); d fumed
silica in PPG by Raghavan and Khan (1997) and silica in octanol
by Mewis and Biebaut (2001). The lines are predictions of Eq. (3)
using the measured steady shear rheology when available
and our slip distance characterized for a silica suspension
in ethylene glycol. According to our experiments
with roughened plates, the slip can be greatly
reduced (i.e., Fig. 7), such that the critical strain for
shear thickening no longer exhibits a plateau value for
high frequencies (Fig. 8). This shows that the high
frequency limiting value of the critical strain observed
in previous experiments with smooth tooling is an
artifact of adhesive failure and slip.
The experimental analysis of slip demonstrates that
the high frequency plateau in the critical strain can be
explained as wall slip. The slip distance is expected to
depend on the properties of solvent, which to first
order might be expected to correlate with the wetting
of the tool by the solvent. Indeed, we observe more
slip for water than ethylene glycol, which qualitatively
follows the expected wetting behavior (Vidal 2002)
(contact angles on alumina are reported to be 75.9°
for water and 50.4° for EG). Interestingly, this analysis
confirms the postulate of Boersma et al. (1992),
who surmised that their high frequency behavior, for

Fig. 6 Schematic illustration of the displacement of a suspension
between parallel plates with wall slip
Fig. 7 Apparent critical strain for shear thickening as a function of
gap size between parallel plates. Silica suspension in water (filled
squares), ethylene glycol (open circles) with cone and plate and
ethylene glycol (filled circles) with emery cloth covered parallel
plates
which the critical strain for shear thickening does not
plateau (Fig. 5b), was due to a solid-like response
without slip.
Lissajous plots
205
Fig. 8 Measured and predicted critical strain as a function of
angular frequency for silica suspension in water (open squares),
ethylene glycol (open circles) measured with cone and plate,
ethylene glycol (filled circles) measured with emery cloth covered
parallel plates and fumed silica suspension in polypropylene glycol
(open triangles) from Raghavan and Khan (1997)
To understand further the dynamical nature of the shear
thickened state, the dynamic oscillatory measurements
were analyzed as flows. The energy dissipated by the
shear thickening fluid can be obtained by integrating the
area contained in a plot of stress vs strain for a dynamic
oscillatory test. The flow mechanism can be understood
from the shape of the resulting stress-strain curve (Lissajous
plot). At 5 Pa of maximum imposed stress and
0.1 rad/s of the frequency, an elliptical hysteresis loop is
recorded as shown in Fig. 9. Figures 9, 10, and 11 show
the data taken at stress amplitudes of 5 (linear viscoelastic
region), 50 Pa (strain thinning nonlinear region),
and 500 Pa (strain thickening nonlinear region). For this
sample the critical maximum stress (s 0) for stress thickening
in a dynamic test at x=0.1 rad/s was around
s 0=80 Pa and the critical stress for shear thickening in
steady shear was 46 Pa, which approximately corre-
sponds to sd ¼ 2s0
p .
The area enclosed by the ‘‘loop’’ can be interpreted as
viscous damping. The angle between the primary axis of
the ellipse and the horizontal axis indicates the elastic
modulus; a zero degree orientation is Newtonian. As seen,
the colloidal dispersion at s\ p
2 sd yields a cycle that has
both elastic and viscous character, indicating viscoelasticity.
In order to explore the difference in mechanical

206
Fig. 9 Lissajous plot for aqueous silica dispersion at 60 vol.% and
a frequency of 0.1 rad/s and the applied stress amplitude of 5 Pa
which is relatively low
Fig. 10 Lissajous plot for aqueous silica dispersion at 60 vol.%
and a frequency of 0.1 rad/s and the applied stress amplitude of
50 Pa which is close to the point of shear thickening
properties between shear thinning and thickening
region, the angular frequency was held fixed at 0.1 rad/s
while the maximum stress was increased stepwise. As
seen, the area enclosed increases with increasing stress
amplitude, indicating an increase in viscous dissipation.
Above maximum imposed stresses of 50 Pa the loops
Fig. 11 Lissajous plot for aqueous silica dispersion at 60 vol.%
and a frequency of 0.1 rad/s and the applied stress amplitude of
500 Pa, which is well above the critical stress for shear thickening
deviate strikingly from an elliptical shape. Further
increases in stress amplitude above the critical stress
converge to a highly non-elliptical shape. This can be
interpreted as the superposition of a primarily fluid
response for low stresses in the cycle with a primarily
elastic response for stresses exceeding the critical stress
for shear thickening. Although qualitative, the shape
analysis immediately distinguishes this fluid from other,
complex behaviors (such as a yielding fluid), and signals
the onset of shear thickening in the dispersion. Notice
that, on the time scale of the oscillation, the fluid is
thickening and ‘‘melting’’, such that the material
response time for shear thickening is substantially faster
than the experiment’s frequency.
The normalized strain (c/cmax) is plotted as a function
of the normalized applied stress (s/s0) (sinusoidal frequency
x=0.1 rad/s) on a period (Fig. 12). Table 2
gives the normalizing factors for Fig. 12. Below the
transition stress for shear thickening (46 Pa) the distortion
of stress-strain curve is increasing with stress amplitude.
However, above the transition stress for shear
thickening, the Lissajous diagrams show the same pattern
with increasing imposed maximum stress. Note that
the maximum strain limits at high stress (Table 2), which
is in agreement with the slip analysis if the sample itself
exhibits primarily a solid response. This result is also in
good agreement with the observation of Mewis and
Biebaut (2001), who observed a unique, viscoelastic
master curve for their shear thickening dispersions using
parallel superposition. Note, however, that this pattern
and the parallel superposition spectrum observed by

Fig. 12 Scaled Lissajous plot obtained for 60 vol.% aqueous silica
dispersion at x=0.1 rad/s and stress amplitude of 1, 5, 10, 50, 100,
500, and 1000 Pa
Table 2 Scaling values for Fig. 12
Angular frequency
x (rad/s)
Maximum
stress s0 (Pa)
Maximum
strain c0
0.1 1 2.8·10 )4
0.1 5 4.8·10 )3
0.1 10 1.3·10 )2
0.1 50 16
0.1 100 16
0.1 500 17
0.1 1000 18
Mewis and Biebaut (2001) must also reflect the large
amount of slip observed in the shear thickening state.
Analysis of the shear thickening fluid subjected to a
dynamic frequency of 0.1 rad/s with stress amplitudes
varying from 1 to 1000 Pa shows (Fig. 13) that the
energy dissipated is an increasing function of strain
amplitude (peak value of measured strain). The energy
dissipated during a cycle (U d) is given by the area
enclosed by the Lissajous plot (Yziquel et al. 1999) (i.e.,
Ud ¼ H sdc). The following phenomenological relation is
often used to relate dissipation energy to strain amplitude
(Citerne et al. 2001):
Ud ¼ Jc n 0 : ð6Þ
where J and n are material constants. The constant J is
referred to as the damping constant and n is called the
damping exponent. The energy dissipated per volume
per cycle for our silica suspension is proportional to the
strain raised to the second power when the fluid is deformed
in linear viscoelastic region, as expected. However,
above the linear viscoelastic region, the damping
exponent is very large of c0 (n 15). This abrupt change
in behavior signals a change in microstructure and is
taken to be a signature of the hydroclustered state.
Conclusions
207
Fig. 13 Energy dissipated per cycle per volume as a function of
measured strain at x=0.1 rad/s for 60 vol.% aqueous silica
dispersion obtained from integrating the area enclosed in the
Lissajous plots
Reversible shear thickening is measured in both oscillatory
and steady shear flow for a charge stabilized
colloidal suspension in two solvents. The Delaware-
Rutgers rule can correlate suspension rheology prior to
shear thickening, but is violated in the shear thickened
state. Instead, a superposition of steady and the dynamic
viscosities at the point of shear thickening regime is
achieved if plotted vs the average stress magnitude. The
critical strain required for dynamic shear thickening
scales inversely with the frequency for small frequency,
while for high frequency, slip leads to an apparent plateau.
Consequently, measurements of steady shear
thickening and the slip enable predicting both the onset
of shear thickening in oscillatory flow, as well as the
behavior of the strain amplitude with frequency. The

208
damping characteristics of a shear thickening suspension
as calculated from the Lissajous plots show a dramatic
increase in viscous dissipation upon shear thickening,
which is due to the jamming inherent in the hydroclustered
state.
References
Barnes HA (1989) Shear-thickening (‘‘dilatancy’’)
in suspensions of nonaggregating
solid particles dispersed in
Newtonian liquids. J Rheol 33:329–366
Bender JW (1995) Rheo-optical investigations
of the microstructure of model
colloidal suspensions. Thesis, Newark
Bender JW, Wagner NJ (1995) Optical
measurement of the contributions of
colloidal forces to the rheology of
concentrated suspension. J Colloid
Interface Sci 172:171–184
Bender JW, Wagner NJ (1996) Reversible
shear thickening in monodisperse and bi
disperse colloidal dispersion. J Rheol
40:899–916
Boersma WH, Laven J, Stein HN (1992)
Viscoelastic properties of concentrated
shear-thickening dispersion. J Colloid
Interface Sci 149:10–22
Bossis G, Brady JF (1989) The rheology of
Brownian suspensions. J Chem Phys
91:1866–1874
Catherall AA, Melrose JR, Ball RC (2000)
Shear thickening and order-disorder
effects in concentrated colloids at high
shear rates. J Rheol 44:1–25
Citerne GP, Carreau PJ, Moan M (2001)
Rheological properties of peanut butter.
Rheol Acta 40:86–96
D’Haene PD, Mewis J, Fuller GG (1993)
Scattering dichroism measurements of
flow-induced structure of a shear thickening
suspension. J Colloid Interface Sci
156:350–358
Doraiswamy D, Mujumdar AN, Tsao I,
Beris AN, Danforth SC, Metzner AB
(1991) The Cox-Merz rule extended: a
rheological model for concentrated
suspensions and other materials with a
yield stress. J Rheol 35:647–685
Farr RS, Melrose JR, Ball RC (1997)
Kinetic theory of jamming in hardsphere
startup flows. Phys Rev E
55:7203–7211
Foss DR, Brady JF (2000) Structure, diffusion
and rheology of Brownian suspensions
by Stokesian dynamics
simulation. J Fluid Mech 407:167–200
Helber R, Doncker F, Bung R (1990)
Vibration attenuation by passive stiffness
switching mounts. J Sound Vib
138:47–57
Hoffman RL (1974) Discontinuous and
dilatant viscosity behavior in concentrated
suspensions. II. Theory and
experimental tests. J Colloid Interface
Sci 46:491–506
Hoffman RL (1997) Explanations for the
cause of shear thickening in concentrated
colloidal suspensions. J Rheol
42:111–123
Kaffashi B, O’Brien VT, Mackay ME,
Underwood SM (1997) Elastic-like and
viscous-like components of the shear
viscosity for nearly hard sphere,
Brownian suspensions. J Colloid Interface
Sci 187:22–28
Laun HM, Bung R, Schmidt F (1991)
Rheology of extremely shear thickening
polymer dispersions (passively viscosity
switching fluids). J Rheol 35:999–1034
Laun HM, Bung R, Hess S, Loose W, Hess
O, Hahn K, Ha¨ dicke E, Hingmann R,
Schmidt F, Lindner P (1992) Rheological
and small angle neutron scattering
investigation of shear-induced particle
structures of concentrated polymer dispersions
submitted to plane Poiseuille
and Couette flow. J Rheol 36:743–787
Lee DI, Reder AS (1972) The rheological
properties of clay suspensions, latexes
and clay-latex systems. TAPPI Coating
Conference Proceedings, pp 201–231
Acknowledgments Dr. Eric Wetzel of Army Research Laboratory
(MD) is acknowledged for helpful discussions. This work has been
supported through the Army Research Laboratory CMR program
(Grant No. 33-21-3144-66) through the Center for Composite
Materials of the University of Delaware. Funding for equipment
from the NSF-MRI (CTS-9977451) is gratefully acknowledged.
Maranzano BJ, Wagner NJ (2001a) The
effect of interparticle interactions and
particle size on reverse shear thickening:
hard-sphere colloidal dispersions.
J Rheol 45:1205–1222
Maranzano BJ, Wagner NJ (2001b) The
effect of particle size on reverse shear
thickening of concentrated colloidal
dispersion. J Chem Phys 114:
10,514–10,527
Maranzano BJ, Wagner NJ (2002) Flowsmall
angle neutron scattering
measurements of colloidal dispersion
microstructure evolution through the
shear thickening transition. J Chem
Phys 117:10291–10302
Mewis J, Biebaut G (2001) Shear thickening
in steady and superposition flows effect
of particle interaction forces. J Rheol
45:799–813
Newstein MC, Wang H, Balsara NP,
Lefebvre AA, Shnidman Y, Watanabe
H, Osaki K, Shikata T, Niwa H,
Morishima Y (1999) Microstructural
changes in a colloidal liquid in the shear
thinning and shear thickening regimes.
J Chem Phys 111:4827–4838
Raghavan SR, Khan SA (1997) Shearthickening
response of fumed silica
suspension under steady and oscillatory
shear. J Colloid Interface Sci 185:57–67
Vidal M (2002) Surface energy study.
http://www.mse.gatech.edu/research/
surf/surt99/vidal.htm
Yziquel F, Carreau PJ, Tanguy PA (1999)
Non-linear viscoelastic behavior of
fumed silica suspensions. Rheol Acta
38:14–25