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SUPPLEMENTAL MATERIAL
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, IL, 60439, U.S.A.
2
Department of Mathematics and Statistics, The University of Melbourne, Victoria 3010, Australia
3
James Franck Institute, University of Chicago, Chicago, IL, 60637, U.S.A.
4
Kavli Nanoscience Institute and
Department of Physics
, California Institute of Technology, Pasadena,
CA
91125, U.S.A.
I.
MATERIALS AND METHODS
A.
Sample Preparation
Bipyramidal gold nanoparticles are synthesized in aqueous solution using a seed
-
mediated growth
method. The synthesis follows the procedure described in Ref.
1
.
As
well as bipyramidal gold
nanoparticles, this synthesis also results in an irregular, spheroidal byproduct. The byproduct has a
plasmon resonance around a wavelength of 550 nm, far away from the longitudinal plasmon resonance
in the bipyramidal particles,
which is centered around a wavelength of 760 nm. Optical measurements
at the longitudinal
-
plasmon resonance frequency therefore probe only the bipyramids, and not the
byproduct.
As synthesized, the particles are stabilized in aqueous solution with a larg
e excess of
cetyltrimethylammonium bromide (CTAB
). This excess surfactant complicates transfer of the
nanoparticles into different solvent mixtures, and can modify the viscosity and other physical properties
of the solvent mixture. We therefore replace the CTAB with
polystyrene sulphonic acid (PSS
), a
negatively charged polymer that coats the nanoparticles and stabilizes them without requiring excess
polymer in solution
.
1
In order to do this, the aqueous nanopa
rticle solution is first diluted to one fifth of
its original volume and is then centrifuged at 5000
g
for five minutes. The supernatant is then decanted,
and the remaining pellet is suspended to the diluted volume in a 2:1 mixture of water and 0.1% PSS
s
olution (by weight). The solution is left to stand for two hours, and is then again centrifuged at 5000
g
for five minutes. The supernatant is decanted, and the pellet is resuspended in water to its original
volume.
Small volumes of the resulting solutio
ns of PSS
-
functionalized nanoparticles in water are added to
larger volumes of water and glycerol to form the solutions that are measured optically. The quantities
of water and glycerol to be mixed are measured by mass, with the aqueous nanoparticle solut
ion
SUPPLEMENTAL MATERIAL
Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
2
included in the water portion of the mixture. The solutions are stirred thoroughly to ensure that the
mixtures are homogeneous. They are then transferred to optical cuvettes with a path length of 2 mm,
taking care that no bubbles are introduced into
the high
-
viscosity solutions in the process. The final
concentration of the bipyramids in the solutions is such that the optical density in the 2
-
mm cuvettes at
the longitudinal
-
plasmon resonance wavelength is between 0.1 and 0.2.
B.
Transient
-
Absorption
Measurements
Optical measurements are performed using a transient
-
absorption
-
spectroscopy system (Ultrafast
Systems HELIOS). Measurement procedures follow those described in our previous publications
.
2
,
3
Thermal artifacts are avoided by stirring the sample with a magnetic stir bar. Our previous
measurements on high
-
viscosity samples involved mechanically translating the cuvette during
measurements; we found, however, that this meant that the amount of pu
mp light scattered into the
detector varied over time, as bubbles in the solution and contaminants on the surfaces of the cuvette
were translated through the pump beam. When the sample is stirred, by contrast, this scattering
background is less variable o
ver time, and can thus largely be subtracted from the measured signal by
averaging eight spectra at negative time delays. We therefore limited our measurements to samples
that could be stirred during the measurements, which meant that we could measure sol
utions up to a
maximum mass fraction of 80% glycerol.
The temperature of the sample was monitored during measurements using a thermocouple
immersed into the solution. The cuvette was sealed with the thermocouple in place, in order to prevent
absorption of
water from the environment into the water
-
glycerol mixtures.
C.
Experimental Data Analysis
The background in the measured transient spectra due to scattered pump light is subtracted from all
spectra, the measured signal is corrected for the fact that dif
ferent probe wavelengths arrive at the focal
spot at different times, and the time axis is adjusted so that the pump and probe overlap at time zero.
The resulting spectrum at any given pump
-
probe time delay is then taken to be due to a shift in the
longit
udinal plasmon resonance frequency of the bipyramids and a broadening of the plasmon
resonance line. Assuming that the extinction spectra with and without the pump laser are both
Lorentzian functions, each transient spectrum thus has the following form
:
2
SUPPLEMENTAL MATERIAL
Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
3
Δ
1
Δ
ω
!
+
1
4
ΔΛ
Δω
!
+
1
4
!
.
(
S1
)
This expression involves the following normalized frequencies:
Δ
!
Λ
!
/
!
,
Δ
Λ
(
)
Λ
(
)
Λ
!
/
!
, and
(
)
(
)
/
!
, where
!
is the probe frequency;
Λ
(
)
and
(
)
are the resonance
frequency and linewidth, respectively, for a pump
-
probe delay
; and
Λ
!
and
Z
!
are the plasmon
resonance frequency and linewidth in the abs
ence of the pump pulse.
These last two values are
determined
from the measured linear extinction spectrum of each sample. The remaining values are the
fractional frequency shift and line broadening; these are determined for each time delay by least
-
squar
es fitting of the transient spectra for wavelengths between 650 and 830 nm. For measured spectra
where there is a strong background of scattered pump light, wavelengths close to the pump wavelength
are excluded from the fit. For spectra at certain time d
elays, the signal
-
to
-
noise ratio is not sufficient to
allow for a good fit; these delays are omitted from the resulting time traces.
The time dependence of the peak shift, for pump
-
probe delays greater than 30 ps, consists of a
damped oscillation on a deca
ying background. The oscillations are due to longitudinal acoustic
vibrations of the bipyramids, and can be approximated as an exponentially damped sinusoid with
frequency
and decay time
!"!
.
3
,
4
,
5
,
6
The background is due to the increased lattice temperature of the
nanoparticles, and can be described over the measured time range as a decaying
exponential with time
constant
cool
.
We therefore fit the time
-
dependent peak shift using a least
-
squares method to the
following function:
ΔΩ
=
!
exp
!
!
tot
sin
휔푡
+
+
!
exp
!
!
cool
,
(
S2
)
where
!
,
!
, and
are fitting parameters
.
2
,
3
After the fit has been performed, the fitted second term
can be s
ubtracted from the data to obtain time traces that isolate the effects of acoustic oscillations, as
in Figs. 1C and 2A.
For each water
-
glycerol mixture, measurements are made with pump
-
pulse energies of 120 nJ, 240
nJ, and 360 nJ, and values of
!"!
and
are determined for each of these pump energies. A linear
least
-
squares fit is then performed for the dependence of
!"!
and
on pump power. The result of this
fit is used to obtain values at zero power, which are the final values reported. The fitt
ed decay time,
!"!
, includes effects of the energy decay time,
, and the inhomogeneous dephasing time,
!"#
. The
inhomogeneous decay time depends on the distribution of nanoparticle dimensions. From analysis of
SUPPLEMENTAL MATERIAL
Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
4
transmission
-
electron
-
microscope (TEM)
images, we obtain the mean length,
!
, and standard
deviation in length,
!
, of the bipyramidal nanoparticles. Assuming that the lengths follow a normal
distribution, the inhomogeneous damping time can be approximated as
3
inh
2
!
!
.
(
S3
)
For sufficiently weak inhomogeneous damping, the energy decay time
can then be
approximated
according to
3
1
1
tot
1
inh
.
(
S4
)
A single adjustable parameter,
in
t
, is also included when comparing measurements to theory. This
intrinsic quality factor accounts for damping within the nanoparticles themselves, and has been
determined from previous measurements of nanoparticles in low
-
viscosity fluids
.
7
The total theoretical
damping rate is then given by
1
=
1
fluid
+
1
int
.
The vibrational resonance frequency is
!"#
=
/
1
1
/
(
4
!
)
.
SUPPLEMENTAL MATERIAL
Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
5
II.
ANALYTICAL MODELS
A.
Newtonian Fluid
We assume that
the oscillation amplitude of the particle is
much
smaller than any other geometric
length scale, so that all nonlinear effects can be ignored
.
8
T
his
is the practical case and
enabl
es
linearization of the equations of motion
.
7
,
8
The required governing equation for the fluid motion is th
en
the
incompressible unsteady
Stokes equation:
=
0
,
휕푡
+
훁퐯
=
,
(
S5
)
where
v
is the fluid velocity field
and
ρ
is
the fluid density.
The Cauchy stress tensor is
=
+
,
where
p
is the fluid pressure,
is
the identity tensor, and
S
is
the
deviatoric stress tensor. For an
incompressible Newtonian fluid,
=
2
, where
μ
is
the fluid shear viscosity and
=
훁퐯
+
훁퐯
!
/
2
is the rate
-
of
-
strain tensor.
The corresponding equation for the solid
nanoparticle
is Navier
’s
equation:
!
!
!
=
2
1
+
+
1
1
2
,
(
S6
)
where
u
is the solid displacement field,
!
is
the solid density,
E
the Young’s modulus of the solid, and
σ
is the Poisson ratio of the solid.
Since the motion is oscillatory, all
time
-
dependent variables
,
such as the solid displacement
,
fluid
velocity
a
nd
fluid pressure
,
are expressed in terms of the explicit time dependence
!
!
!
!
:
,
,
=
,
Ω
)
!
!
!
!
,
(
S7
)
where
i
is the usual imaginary unit and
X
denotes any time
-
dependent quantity
.
For simplicity
,
we
henceforth omit the superfluous ‘
’ notation, noting that the above relation holds universally
for
harmonic oscillation
.
At the interface between the fluid
domain
and the solid particle, the con
ditions of continuity of
stress, velocity, and
displacemen
t are imposed
. The no
-
slip boundary condition is enforced at the solid
interface
.
This provides direct coupling between the Navier
-
Stokes equation for the fluid
, equation
(S5),
and Navier’s equation for the solid
, equation (S6)
.
SUPPLEMENTAL MATERIAL
Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
6
We simplify the calculat
ions by approximating the nanoparticles as axisymmetric. A bipyramid is
thus approximated as a pair of truncated cones, as illustrated in
Fig. S1
. Each half of the particle along
its symmetry
axis
is a cone with a base diameter of
!"#
and a height of
!
/
2
, truncated to a height of
/
2
, so that the total length of the particle is
.
Figure S1
:
Schematic of the
bipyramid
particle geometry used for calculations.
The origin is at
the center of the particle.
An analytical solution to this problem
can be obtained by assuming that the length of the particle
greatly exceeds both the viscous penetration depth and the particle radius
.
7
This allows for
approximation of the hydrodynamic load at any axial position along the particle by that due to a local
shear flow; pressure does not affect the flow in this limit. We then obtain the following normalized,
complex eigenfrequency for the vibration
:
7
Ω
!
!
!
Ω
!
!
!
푑푧
!
!
!
!
+
1
+
!
!"#
1
Ω
!
푑푧
!
!
!
,
(
S8
)
where
Ω
is the complex eigenfrequency,
(
)
is the position
-
dependent radius of the particle,
is the
distance
along
the
particle,
normalized
by
,
and
=
!
2
푖훽
Ω
(
)
/
!"#
/
!
2
푖훽
Ω
(
)
/
!"#
.
In this expression, the Reynolds
number is
Ω
=
Ω
!"#
!
2
.
(
S9
)
The integrals in equation (S8) must be evaluated numerically. The equation then becomes a
transcendental equation for the complex eigenfrequency,
Ω
. The resonant frequency,
res
, and quality
factor,
!"#$%
, of the nanoparticle vibrations can be determined from this eigenfrequency:
SUPPLEMENTAL MATERIAL
Viscoelastic Flows in Simple Liquids Generated by Vibrating Nanostructures
7
res
=
Ω
!
!
+
Ω
!
!
,
(
S10
)
and
fluid
=
res
2
Ω
i
,
(
S11
)
where
Ω
r
and
Ω
i
are the real and imaginary components of
Ω
, respectively.
B.
Linear Maxwell Fluid
The model developed above can readily be generalized to the case of a linear Maxwell fluid. In this
case, the deviatoric stress tensor is given by
9
+
t
=
2
,
(
S12
)
where
is the shear relaxation time of the fluid
. Applying the harmonic time dependence of equation
(S7), equation (S11) becomes
S
Ω
S
=
2
, which has the explicit solution
=
2
1
Ω
.
(
S13
)
The solution for a Maxwell fluid can thus be rigorously obtained from that of a Newtonian fluid, under
the substitution
1
Ω
.
(
S14
)
In particular, making this substitution in equation (S9) enables calculation of the dynamic response of a
nanoparticle immersed in a linear Maxwell fluid, from equation (S8).