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Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
Matthew Pelton,
1,
*
Debadi Chakraborty,
2
Edward Malachosky,
3
Philippe Guyot-Sionnest,
3
and John E. Sader
2,4,
1
Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439, USA
2
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
3
James Franck Institute, University of Chicago, Chicago, Illinois 60637, USA
4
Kavli Nanoscience Institute and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
(Received 2 July 2013; published 13 December 2013)
Newtonian fluid mechanics, in which the shear stress is proportional to the strain rate, is synonymous
with the flow of simple liquids such as water. We report the measurement and theoretical verification of
non-Newtonian, viscoelastic flow phenomena produced by the high-frequency (20 GHz) vibration of gold
nanoparticles immersed in water-glycerol mixtures. The observed viscoelasticity is not due to molecular
confinement, but is a bulk continuum effect arising from the short time scale of vibration. This represents
the first direct mechanical measurement of the intrinsic viscoelastic properties of simple bulk liquids, and
opens a new paradigm for understanding extremely high frequency fluid mechanics, nanoscale sensing
technologies, and biophysical processes.
DOI:
10.1103/PhysRevLett.111.244502
PACS numbers: 47.50.

d, 47.61.

k, 62.25.Fg, 87.15.ht
A fluid is said to be Newtonian if it exhibits a simple
linear relationship between shear stress and strain rate.
This description, which underlies conventional fluid dy-
namics, provides an excellent approximation to the behav-
ior of real fluids, provided the time scale for measurement
is long compared to the time required for stresses to
propagate in the fluid. In simple liquids, including water
and glycerol, these ‘‘relaxation times’’ are on the order of
1 ps–1 ns [
1
3
]. These time scales are short compared to
the time scales associated with the motion of macroscopic
objects in the fluids, which allows interactions between
solid structures and simple fluids to be described by clas-
sical Navier-Stokes treatments [
4
,
5
]. These treatments
hold even for micrometer-scale objects, such as the canti-
levers found in atomic force microscopes and microelec-
tromechanical devices, because they have characteristic
vibrational frequencies in the kHz to MHz range [
6
,
7
].
Non-Newtonian fluid mechanics is therefore convention-
ally associated only with complex fluids that have long
relaxation times, such as polymer solutions and melts,
dense colloidal suspensions such as corn starch in water,
and fluids near their phase transitions [
8
,
9
]. Scaling objects
down to nanometer size scales increases their characteristic
vibrational frequencies up to the GHz or THz range [
10
].
Fluid-structure interactions on these length scales thus
have the potential to show significant deviations from
Newtonian behavior, even for simple liquids.
Departures from Newtonian behavior have been
reported for simple liquids under extreme confinement,
due to structural reorganization and surface effects on the
molecular scale [
11
13
]. For bulk fluids, by contrast, direct
mechanical observation of non-Newtonian behavior has
been limited to solid structures interacting with dilute
gases [
14
]. In this case, the effects can be predicted rigor-
ously by the Boltzmann equation [
15
]. For simple bulk
liquids, however, rigorous theoretical description of, and
experimental access to, the non-Newtonian regime remain
outstanding problems in the physical sciences.
We access this regime directly for the first time by
exciting and probing vibrations in metal nanoparticles
with ultrafast laser pulses [
16
19
]. In these experiments,
absorption of a pump laser pulse leads to rapid heating and
expansion of the nanoparticle, resulting in nearly impulsive
excitation of coherent acoustic oscillations. These oscilla-
tions modulate the transmission of a second probe pulse
due to changes in the resonance frequency of plasmons in
the nanoparticles. Using highly monodisperse samples of
bipyramidal gold nanoparticles [
20
,
21
] ensures narrow
plasmon resonances, resulting in strong transient optical
signals. It also minimizes the effects of inhomogeneous
dephasing of the mechanical vibrations, so that energy
damping times can be determined from the time-dependent
measurements. The low volume fraction (
<
10

6
) of the
samples minimizes correlations between the particles.
Previous measurements showed good agreement between
the measured rate at which the nanoparticle vibrations are
damped by surrounding low-viscosity solvents and models
that treat the solvents as incompressible Newtonian fluids
[
22
24
]. The models, however, could not account for
the observed damping rates in high-viscosity liquids [
24
],
and behavior in these liquids has previously had no
explanation.
Here, we explore this high-viscosity regime by perform-
ing new measurements on bipyramidal gold nanoparticles
in water-glycerol mixtures. Methods for nanoparticle
synthesis, transient-absorption measurements, and data
analysis follow our previous work [
20
,
22
,
23
] (see the
Supplemental Material [
25
]). Representative nanoparticle
images and transient-absorption data are shown in
Figs.
1(a)
and
1(b)
, and the measured results are
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Delay time (ps)
100
200
300
Wavelength (nm)
700
800
A
0.02
0.007
Mass fraction glycerol
Delay time (ps)
100
150
200
250
50
1
1
750
Delay time (ps)
100
150
200
250
50
Mass fraction glycerol
0.4
0.5
0.6
0.7
0.8
0
0.2
0.4
0.6
0.8
Mass fraction glycerol
Quality factor,
Q
Experiment
Analytical (Newtonian)
Finite-element (Newtonian)
(a)
(b)
(c)
(d)
(e)
0.4
0.5
0.6
0.7
0.8
1
10
30
3
FIG. 1 (color online). Acoustic oscillations of bipyramidal gold nanoparticles. (a) Transmission-electron-microscope image of a
representative gold bipyramid. (b) Representative transient-absorption data from an ensemble of gold bipyramids. Change in extinction

A
is plotted as a function of pump-probe delay and probe wavelength. Results are shown for a mixture of 60% glycerol and 40%
water by mass and for a pump-pulse energy of 360 nJ. (c) Measured oscillations of the plasmon resonance frequency for bipyramids in
glycerol-water mixtures. Frequency shifts were determined by fitting to transient-absorption data, the exponentially decaying
background due to heating of the nanoparticles was removed, and the remaining shifts were normalized to have a minimum value
of

1
. Results are shown for a pump-pulse energy of 120 nJ. (d) Calculated oscillations of the plasmon resonance frequency for
bipyramids in glycerol-water mixtures, plotted as in (c). The calculations are performed using an analytical model that treats the
glycerol-water mixtures as Newtonian fluids. The initial phases of the oscillations are chosen to match the experimental results.
(e) Quality factor
Q
of mechanical oscillations of bipyramids in glycerol-water mixtures, as a function of the mass fraction of glycerol
in the mixtures. Circles are determined by fitting experimental data; the dashed line is the result of an analytical model, assuming a
quality factor due to intrinsic damping
Q
int
¼
35
; diamonds are the results of finite-element calculations, assuming
Q
int
¼
50
,
consistent with previous measurements [
24
]. Both models treat the glycerol-water mixtures as Newtonian fluids.
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summarized in Fig.
1(c)
. The data are first compared to an
analytical model based on Newtonian fluid-structure the-
ory [
24
,
25
]. As shown in Fig.
1(d)
, this model predicts
strong damping as the viscosity of the mixtures increases.
This is in striking disagreement with the observations,
which show little change in the damping rates for glycerol
fractions between 40% and 80%.
The discrepancy between experiment and the analytical
Newtonian model can be made more explicit by quantify-
ing the damping rate in terms of the quality factor
Q
¼
!
res
T=
2
, where
!
res
is the angular vibrational reso-
nance frequency and
T
is the time constant for damping of
the oscillations. As shown in Fig.
1(e)
, the model repro-
duces experimental quality factors well for lower-viscosity
mixtures, up to a glycerol mass fraction of approximately
60%. At higher glycerol concentrations, however, the
theoretical modeling completely fails to reproduce the
experimental trend.
Figure
1(e)
also shows the results of rigorous finite-
element simulations that assume Newtonian fluid mechan-
ics [
24
,
25
]. The predictions of these calculations deviate
only slightly from those of the analytical theory, due to
pressure effects at the particle ends. Including the effects of
fluid compressibility gives similar results, with the quality
factor decreasing monotonically as the glycerol concentra-
tion rises (see the Supplemental Material [
25
]).
One might suspect that theory and experiment disagree
because both the analytical model and the finite-element
calculations use the conventional no-slip boundary condi-
tion at the liquid-solid interface. This condition has come
under scrutiny in recent years, with a number of studies
suggesting that it may be violated at nanometer length
40%
50%
60%
65%
70%
75%
80%
Peak shift (arb. units)
Delay time (ps)
0
100
200
50
250
150
(a)
0
0.2
0.4
0.6
0.8
Mass fraction glycerol
Experiment
Analytical (viscoelastic)
FE (viscoelastic)
FE (Newtonian)
Resonant frequency (GHz)
(c)
Quality factor,
Q
0
0.2
0.4
0.6
0.8
Mass fraction glycerol
Experiment
Analytical (viscoelastic)
FE (viscoelastic)
FE (Newtonian)
(b)
1
10
30
3
19
20
21
22
23
FIG. 2 (color online). Nanoparticle vibrations in Newtonian and viscoelastic liquids. (a) Oscillations in the plasmon resonance
frequency for bipyramids in glycerol-water mixtures. Points are experimentally measured values for a pump-pulse energy of 120 nJ
[see Fig.
1(c)
]. Lines are the results of analytical calculations, with initial phases chosen to match the experimental results. Dashed
lines are based on treating the glycerol-water mixtures as Newtonian fluids, and solid lines are based on treating them as viscoelastic
fluids, described by a Maxwell model. Traces are offset vertically for clarity. (b) Quality factor
Q
and (c) frequency of mechanical
vibrations of bipyramids in glycerol-water mixtures. Circles are determined by fitting experimental data; the dashed line is the result of
an analytical model that treats the glycerol-water mixtures as viscoelastic fluids, described by a Maxwell model; triangles are the result
of finite-element (FE) calculations that treat the mixtures as viscoelastic fluids, and diamonds are the result of FE calculations that treat
the mixtures as Newtonian fluids. Analytical calculations assume a quality factor due to intrinsic damping,
Q
int
¼
35
, and FE
calculations assume
Q
int
¼
50
[
24
].
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scales [
12
,
26
]. To assess the impact of such a violation, we
adjust the calculations to allow for an arbitrary amount
of slip. We find that the calculated quality factors still
decrease monotonically as the solution viscosity is
increased, so that this assumption cannot be responsible
for the disagreement between theory and experiment (see
the Supplemental Material [
25
]).
The remaining critical assumption involved in the theo-
retical models is that the fluid is Newtonian. The validity of
this assumption can be quantified in terms of the Deborah
number,
De
¼
!
res

, where

is the shear relaxation time
of the fluid: elastic effects can be ignored only for
De

1
[
8
]. For the bipyramids used here, the longitudinal vibra-
tion period is approximately 50 ps [see Fig.
1(c)
]. This
means that
De

1
for vibrating bipyramids in a water-
glycerol mixture with a glycerol mass fraction of 60% [
1
].
For higher glycerol concentrations, strong viscoelastic ef-
fects and deviations from Newtonian behavior can be
expected.
In the simplest model of a viscoelastic fluid subject to
small strains, known as the linear Maxwell model, viscous
and elastic effects combine in series [
8
],
T
¼
p
I
þ
S
;
S
þ

@
S
@t
¼
2

D
;
where
T
is the stress tensor,
p
is the pressure,
I
is the
identity tensor,
S
is the deviatoric stress tensor as specified
in the second equation,
D
is the rate-of-strain tensor,
t
is
time, and

is the shear viscosity. Conceptually, the behav-
ior of the Maxwell model can be visualized as that of a
linear spring, with spring constant
k
, linked at its end to a
viscous damper, with damping coefficient
b
; the system
recovers from sudden deformation with a characteristic
time

¼
b=k
. The Maxwell constitutive equation can
readily be incorporated into the theoretical models (see
the Supplemental Material [
25
]). As shown in Figs.
2(a)
and
2(b)
, the resulting predictions reproduce the experi-
mentally observed quality factors.
Strikingly, the viscoelastic model predicts a vibration
frequency that increases with high glycerol content. This is
in agreement with the experimental results, as shown in
Fig.
2(c)
. By contrast, the Newtonian model always pre-
dicts a vibration frequency that decreases monotonically
with increasing glycerol content. This qualitative differ-
ence provides clear evidence that the mechanical behavior
of the nanoparticles is strongly affected by non-Newtonian
fluid mechanics.
The nearly constant quality factor and increasing reso-
nance frequency observed for increasing glycerol concen-
tration can be understood as a consequence of the
increasing importance of liquid elasticity. Quantitatively,
Q
¼
2

ð
E
stored
=E
diss
Þ
, where
E
stored
is the maximum en-
ergy stored and
E
diss
is the energy dissipated per cycle,
with both quantities evaluated at resonance. At low glyc-
erol concentrations, where
De

1
, the elastic component
of the liquid response exerts a negligible effect, and energy
is stored only in the solid nanoparticle. In this Newtonian
limit, increasing the glycerol concentration thus only raises
the viscosity and increases
E
diss
, resulting in lower
Q
.For
higher glycerol concentrations, however, where
De

1
,
energy can also be stored elastically in the liquid. As shown
in Fig.
3
, increasing the glycerol concentration in this vis-
coelastic regime leads to more energy being stored in the
liquid. This, in turn, increases the total
E
stored
, compensating
for the increasein
E
diss
astheviscosityincreases. Thequality
factor thus no longer decreases with increasing glycerol
content, and can even increase, as observed in Fig.
2(b)
.
The additional stored energy also increases the total stiffness
of the fluid-structure interaction, and is thus responsible for
the rise in vibrational frequency shown in Fig.
2(c)
.
The good quantitative agreement between theory and
measurement validates the use of a simple Maxwell model
to describe viscoelastic effects in liquid-nanoparticle inter-
actions. It also provides strong support for the theoretical
description of these nanoscale systems using continuum
mechanics. In particular, there is no need to invoke
molecular effects or to depart from no-slip boundary con-
ditions, in clear contrast to previous reports of non-
Newtonian behavior in liquids under extreme confinement
[
11
13
]. Likewise, molecular ordering of the liquid and/or
a change in conformation of the polystyrene sulphonic acid
(PSS) layer surrounding the particle surface cannot
account for our observation (see the Supplemental
Material [
25
]). The small discrepancies between theory
and experiment may be due to limitations of the simple
Maxwell model: instead of a single relaxation time, the
response of the glycerol-water mixtures may better be
described by a spectrum of relaxation times [
1
].
Furthermore, any shortcomings in tabulated relaxation
times [
1
] will clearly affect the comparison. In fact, the
current nanoparticle experiments provide independent
measurements of these relaxation times. Unlike the
40
1
10
-2
10
-4
10
-6
10
-8
10
-10
Normalized Stored Energy Density
30nm
E
f
/
E
TOTAL
= 0.003
0.029
0.17
0% glycerol
40% glycerol
60% glycerol
80% glycerol
0.74
FIG. 3 (color online). Maximum energy density stored in a
vibrating gold nanoparticle and in surrounding water-glycerol
mixtures. The color scale indicates the stored energy density,
normalized by the energy density in the center of the nano-
particle. Results are obtained from finite-element calculations
that treat the particles as linear elastic solids and the liquids as
viscoelastic Maxwell fluids. The fraction of the total energy
stored in the fluid is indicated below each spatial plot.
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ultrasound or inelastic-scattering methods previously used
[
1
3
], nanoparticle vibrations provide a direct, mechanical
probe of non-Newtonian effects in simple liquids. For
smaller objects with higher vibrational frequencies, visco-
elastic effects are expected to become important even for
lower-viscosity liquids with faster shear relaxation times.
Measurements of vibrations in gold nanoparticles with
diameters close to 1 nm, for example, have shown vibra-
tional periods of less than 0.5 ps [
27
,
28
], which means that
they should induce strong viscoelastic effects in pure water.
The experiments reported here represent the first time
that the intrinsic viscoelasticity of simple bulk liquids is
observed to affect the mechanical response of vibrating
solids. While high-viscosity liquids can induce strong
damping in large-scale solids, simple liquids exhibit a
strong elastic response at the intrinsically high frequencies
of nanometer-sized structures, reducing the efficacy of this
dissipation mechanism. Such effects should be general to
liquid-structure interactions on the nanoscale [
29
,
30
]. This
will have important implications for the development of a
new generation of high-speed and high-sensitivity molecu-
lar and biomolecular sensors based on the mechanical
vibrations of nanoscale oscillators operating in liquid
[
31
]. It also means that the wealth of knowledge developed
for macro- and microscale non-Newtonian flows of com-
plex fluids can now be brought to bear on the nanoscale
fluid mechanics of simple liquids.
We thank D. Gosztola for valuable assistance with the
transient-absorption measurements. This work was per-
formed, in part, at the Center for Nanoscale Materials, a
U.S. Department of Energy, Office of Science, Office of
Basic Energy Sciences User Facility under Contract
No. DE-AC02-06CH11357. E. M. was supported by NSF
Grant No. CHE1111799. This research was supported by
the Australian Research Council Grants Scheme and by
Caltech’s Kavli Nanoscience Institute.
*
Present address: Department of Physics, University of
Maryland, Baltimore County, Baltimore, MD 21250,
USA.
To whom all correspondence should be addressed.
jsader@unimelb.edu.au
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