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Measuring nanoparticle flow with the image structure
function
Cite this: Lab Chip, 2013, 13, 2359
Maria Dienerowitz,* Michael Lee, Graham Gibson and Miles Padgett
Received 5th January 2013,
Accepted 28th March 2013
We present a technique to measure the velocity and flow profiles of a nanofluid in a microfluidic channel.
Importantly, we extract the flow velocity from a series of standard brightfield images without employing
particle tracking or laser-enhanced methods. Our analysis retrieves the flow information from the image
DOI: 10.1039/c3lc00028a
structure function of sub-diffraction limited nanoparticles in suspension. We are able to spatially resolve
www.rsc.org/loc
the flow velocity and map out the parabolic flow profile across the width of a microfluidic channel.
1 Introduction
Lab-on-a-chip and microfluidic devices enable novel applications and research in a variety of fields, such as biology,
physics and chemistry.1–4 An important aspect of microfluidic
technology is the ability to measure flow velocities and resolve
flow profiles spatially on the micron scale. Various microscopic techniques, for example laser Doppler anemometry or
micro-particle image velocimetry (mPIV), are at hand to
perform those tasks for fluids containing micron-sized
particles or cells.5–7
Nanoparticles suspended in fluids, often referred to as
nanofluids, play a significant role in future nanotechnology
research. A promising feature of nanofluids are their specific
transport properties enhancing the capabilities of the base
fluid, for example an increased thermal conductivity in heat
transfer fluids8 or targeted in vivo drug delivery by intravenously manipulating ferromagnetic nanofluids with a magnetic flux.9 However, measuring the flow profile and velocity of
a nanofluid poses a significant challenge as the suspended
nanoparticles are below the diffraction limit and thus escape
standard microscopic image detection techniques.10 Although
it is possible to introduce micron-sized tracer particles and
determine the flow profile from their motion, these are likely
to disturb the original sample and struggle to portray the
specific properties of the nanoparticles correctly. Thus, there is
a need for a non-invasive and straightforward measurement
technique to observe and quantify nanoparticle flow phenomena.
Recently, Cerbino et al. introduced Differential Dynamic
Microscopy (DDM), a technique capable of retrieving particle
dynamics from a microscopic real space image.11 Instead of
video tracking individual entities, DDM obtains the structure
SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ,
UK. E-mail: [email protected]; Fax: +44 (0)141 330 2893;
Tel: +44 (0)141 330 6403
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and dynamics of the entire sample,12 similar to conventional
light scattering experiments, such as multi-angle Dynamic
Light Scattering (DLS). With DDM it is possible to quantify the
Brownian motion of the particles in the sample, even if their
size is below the resolution limit of the microscope. Although
it is impossible to resolve individual nanoparticles with
brightfield imaging, their motion introduces refractive index
changes for the passing illumination resulting in intensity
fluctuations in the near field. Imaging the sample onto a
camera records these intensity fluctuations. By processing a
sequence of the images in the reciprocal Fourier space, the
dynamics of the particles in the sample are retrieved. This
method allows monitoring of the aggregation of nanoparticles,13 but has also been applied to larger objects to study
bacteria motility.14 DDM does not require laser illumination; it
works with a low coherence source and standard brightfield
high-speed video images from any microscope.
We have developed a technique analysing the image
structure function introduced by DDM to determine nanoparticle flow from standard brightfield microscope images. We
measure flow speeds of a few mm s21 of suspensions
containing gold nanoparticles as small as 40 nm.
Furthermore we are able to obtain the spatial flow information
of a nanofluid containing 100 nm gold particles in a
microfluidic channel, mapping out a parabolic flow profile
across the width of the channel with micrometer resolution.
Our measurements retrieve the flow data from brightfield
video images captured with an Ethernet-interfaced CMOS
camera.
2 Materials and methods
2.1 Experimental setup
We conducted our experiments on a purpose-built microscope,
including standard brightfield Koehler illumination. A bright
halogen fibre lamp is our illumination source and is focussed
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Lab on a Chip
through a 106 objective (Olympus, Plan N, NA 0.25) onto the
sample. For the first set of experiments, which measure
constant flow, we view the sample with a 1006 microscope
objective (Nikon Plan Apo a, 1006, oil immersion, NA 1.45),
and the sample cell confines a drop of gold nanoparticle
solution (BBInternational, various sizes: 40 nm, 100 nm) or
polystyrene bead solution (2 mm, Bangs Laboratories, Inc.)
between two coverslips. In the second set of experiments we
changed to a 606 microscope objective (Nikon Plan Apo a,
606, oil immersion, NA 1.4) simply to acquire a larger field of
view. A square glass capillary tube (Vitrocom, inner diameter
200 mm) connected to two syringes formed our microfluidic
channel. We initiated flow by adjusting the height difference
of the syringes. Again, we used a 100 nm gold nanoparticle
solution (BBInternational). An x–y stage (ProScan Prior x–y
stage, H101) holds the sample cell and an additional z stage
(Newport motorized linear stage, M-MFN25CC) moves the
microscope objective for focussing. We record brightfield
images of the sample at 90 Hz with a high-speed Ethernetinterfaced camera (Dalsa GenieTM, CR-GM00-H6400) for 10 s.
The images are then further processed in our own LabVIEW
analysis programme.
2.2 Image structure function and measuring flow velocity
Differential Dynamic Microscopy derives information about
the Brownian motion of particles in the sample from the
image structure function12
D(q,Dt)~S jD^I(q; Dt)j2 T,
(1)
with the fast Fourier transform of the difference signal DI(x;
Dt) = I(x, Dt) 2 I(x; 0) of the image intensity I for a specific time
delay Dt between two images. Its form is determined by the
geometry of the particles in the sample and the transfer
function of the microscope. The structure function is
mathematically related to the correlation function, but more
robust to drift and more accurate for smaller datasets.15 For
each flow measurement we record 1000 frames, calculate the
structure functions D(q, Dt) for a set time delay (Dt = 1frame,
2frames, 5frames, ....) and average over all structure functions
of the same time delay Dt. In its normal application, DDM
retrieves the mass diffusion coefficient Dm of the particles by
fitting an exponential to the structure function.11 This analysis
is similar to the exponential fitting applied to the intermediate
scattering function in Dynamic Light Scattering.16 Contrary to
these standard approaches we extract the flow information
from the structure function without prior knowledge of the
temperature or viscosity of the sample. To determine the flow
velocity, we focus on investigating a specific feature of the
structure function instead.
The structure function of a sample performing Brownian
motion is rotationally symmetric since Brownian motion is
homogenous (see Fig. 1a). However, in the presence of a fluid
flow this symmetry is broken and the structure function
displays additional features. As we show in Fig. 1b, a constant
fluid flow introduces a fringe pattern perpendicular to the flow
velocity. For increasing time delays Dt, the number of fringes
increase and move closer together. Determining the periodi-
2360 | Lab Chip, 2013, 13, 2359–2363
Fig. 1 (a) The image structure function D(q, Dt) is rotationally symmetric for
particles subjected to homogeneous Brownian motion. (b) The image structure
function for a sample with directional flow shows an additional fringe pattern
originating from the now asymmetric motion. We chose these sample images
from 2 mm polystyrene beads for clarity, since the fringe pattern is hard to see
for smaller particles.
city of the fringes at a certain time delay Dt gives us the average
distance travelled by the nanoparticles during the time Dt,
which yields the flow velocity in the sample. To explain the
fringe pattern, it is more apparent to look at the structure
function of a large bead. If one thinks of the bead as a pinhole
that moves along x, subtracting two images of this moving
bead at two different times results in a difference image with
two pinholes. The Fourier transform (or diffraction pattern) of
these two pinholes is a periodic interference pattern. This
interference pattern is multiplied with the diffraction pattern
of a single pinhole and results in fringes in the otherwise
homogeneous structure function of a sample with no flow. In a
sample with a large number of particles, a directional flow
creates a multitude of double slit pairs in the difference image
and results in a fringe patterned structure function as in the
single particle case.
3 Results and discussion
We divided our work into two parts. In the first set of
experiments we applied our approach to measure constant
flow velocities. We focussed on determining the slowest
possible flow velocities our technique is able to resolve for a
specific nanoparticle size. The detectable flow speed decreases
for larger particles, as one would expect from an increase in
the Péclet number (Pe = Lv/Dm with characteristic length L =
channel width, flow velocity v and mass diffusion coefficient
Dm).17 The Péclet number is the ratio of convection to diffusion
and increases as the flow dominates over the Brownian
diffusion. We discuss the interplay of Brownian diffusion
and fluid flow in more detail in Section 3.1. In the second set
of experiments we mapped out a flow profile across a
microfluidic channel where the flow velocity changes with
distance to the channel walls. For all measurements we
recorded a set of 1000 frames (640 6 480 pixels) at a rate of 90
Hz.
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Fig. 2 The images on the left show the averaged structure functions D(q, Dt) of
a 100 nm gold nanoparticle suspension at different time delays Dt. We extract
the flow velocity v = Ds/Dt by determining the spacing of the fringes Ds in the
structure function. As mentioned in the text, the fringe structure is less obvious
for nanoparticles, but is retrieved by applying a second FFT to the structure
function D(q, Dt), pictured on the right.
3.1 Constant flow measurements
For the constant flow measurements we confine a drop of
nanoparticle solution between two coverslips. We image the
sample in the middle of the cell and simulate fluid flow by
setting the microscope stage to move at a certain speed. This
controlled movement of the stage provides a measure to
compare the flow velocity determined by analysing the
structure function. Depending on the particle size and flow
velocity, the structure function starts to show a fringe pattern
at a time delay Dtmin. The spacing of the fringes Ds is related to
the average distance the nanoparticles moved during Dt.
Contrary to the clearly visible fringe pattern in the structure
Paper
function for micron-sized particle flows (as shown in Fig. 1b),
the fringe pattern for nanoparticle flows is less obvious. To
correctly extract the spacing of the fringes Ds we perform
another fast Fourier transform (FFT) on the central region
(width = 10 pixels) of the structure function, as illustrated in
Fig. 2. This is equivalent to calculating the spacing of the
diffraction orders created by a diffraction pattern with specific
periodicity. We obtain a velocity v = Ds/Dt for each delay time
Dt, as we present in Fig. 3.
Our results show that the minimum delay time Dtmin to
measure a distinct directional flow velocity is smaller for faster
flow velocities. It is determined by the Brownian motion of the
nanoparticles and the resolution limit of our system. In order
to measure the flow velocity, the directional displacement of
the particle due to the flow (x = v?Dt) has to exceed the root
mean square displacement
the particle performing
pffiffiffiffiffiffiffiffiffiffi pof
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Brownian motion ( Sx2 T~ 2Dm Dt). The minimum time
delay before the flow starts to dominate the nanoparticles’
2Dm
displacement is therefore Dtmin w 2 and decreases for larger
v
flow velocities v. As the mean square displacement is larger for
smaller particles, Dtmin for a specific flow velocity is longer the
smaller the observed nanoparticles’ are.
Additionally, the structure function has a specific size
according to the scattering properties of the particles and
the resolution of our imaging system. The largest separation of
two fringes is limited by the diameter of the structure
function. This separation corresponds to a minimum distance
the particle must have travelled in real space for its motion to
be resolvable via the fringe pattern on the structure function.
For example, the radius of the structure function of 100 nm
gold particles is 25 pixels, which corresponds to 1 mm
displacement in real space. We confirmed this value by
measuring the velocity of immobile 100 nm gold particles
fixed to a glass slide (not shown).
There is a maximum time delay Dtmax to determine flow
velocities, which is limited by the smallest spatial frequency
we are able to resolve with our system. The time it takes for a
particle being displaced across half the field of view is the
longest measurement time for each velocity. For example, at v
Fig. 3 These graphs present the constant flow velocity results for fluids containing 40 nm gold particles, 100 nm gold particles and 2 mm polystyrene beads. We
obtain a velocity measurement for each time delay Dt and we subsequently determine the average value of the fluid velocity from all measurements at different Dt.
For each particle size and flow velocity we acquired 1000 frames at approximately 90 Hz.
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= 10 mm s21, the limit is reached after tmax = 2.4 s. This agrees
with our data for 100 nm and 2 mm particles. For 40 nm
particles tmax is even shorter since a further restriction applies,
e.g., how long the particle remains in the detectable region
along the optical axis. The particle does not have to stay in
focus, but its diffraction pattern needs to be resolvable. When
the root mean square displacement exceeds this distance, the
particle will no longer contribute to the structure function. The
maximum time delay equates to Dtmax = ,x2./(2Dm) and is
tmax = 4.3 s for 100 nm and tmax = 1.5 s for 40 nm particles. This
is the time it takes for a particle to be in focus and leave the
focal region, and is slightly longer if the particle diffuses from
out of focus into focus and out again.
These limitations result in a minimum measurable flow
speed for a certain particle size. If tmin ¢ tmax the diffusion
dominates the motion of the particles. As we present in Fig. 3,
we are able to detect flow velocities down to 4 mm s21 for 40
nm gold nanoparticles. For 100 nm gold nanoparticles the
slowest flow velocity we are able to detect is 3 mm s21. This
limit decreases even further for 2 mm polystyrene beads,
however the slowest velocity we are able to set with our
microscope stage is 0.5 mm s21. One set of recorded images
(1000 frames) results in a number of velocity measurements
v(Dt), one for each Dt. An average over all v(Dt) determines the
standard deviation as indicated in Fig. 3. The small standard
deviation demonstrates the precision of our method.
3.2 Shear flow measurements
In the next step we measure the flow velocity across the width
of a microfluidic channel. The geometry of the square
microfluidic channel is outlined in Section 2.1. We imaged
the nanofluid flow 30 mm deep in the channel to ensure a
constant fluid velocity within the focal depth along the optical
axis (z-axis). Our field of view is too small to capture the entire
200 mm wide microfluidic channel at once on the camera chip.
We thus divide the channel into four sections and record 1000
frames of each section. Since the flow velocity is now changing
across the field of view, performing a FFT on the entire image
does not reveal the flow velocity at a specific position in the
channel. Therefore we perform a one-dimensional FFT on
every pixel column of the image (see Fig. 4a). The columns are
aligned in the direction of the nanofluid flow along the
microfluidic channel.
Instead of a circular structure function, we obtain a 1D
structure function with superimposed fringes for each
column. Clearly, a nanoparticle has to pass the corresponding
space along such a column in the image during the
measurement in order to generate a structure function. The
position of the column remains in real space, only the signal
along that column is in reciprocal space. After applying an FFT
(as described in the previous section) to the one-dimensional
structure function, we retrieve a velocity for potentially any
pixel column, and thus its position within the microfluidic
channel. Since the Reynolds number is very low in our
microfluidic channel (Re % 1), the fluid flow is laminar and we
obtain a parabolic flow profile, as pictured in Fig. 4c. Hagen–
Poiseuille18,19 describes the velocity profile in a Newtonian
fluid flow as a function of the distance from the centre of a
square microfluidic channel by
2362 | Lab Chip, 2013, 13, 2359–2363
Lab on a Chip
Fig. 4 (a) We calculate the 1D structure function for each pixel column across
the image of the microfluidic channel. (b) An additional FFT retrieves the
spacing of the fringes Ds in the 1D structure function (Dt = 0.18 s). (c) We
present the measured flow velocities across the width of the microfluidic
channel for two different maximum velocities at its centre. The error bars
indicate the standard deviation of each individual flow measurement averaged
over all Dt. The parabolic flow profile fit according to the Hagen–Poiseuille
equation (see text) and agrees very well with our data.
x
y
v(x,y)~2:12 U½1{( )2:2 ½1{( )2:2 a
a
U~
DP a2
Dx 28:3g
(2)
(3)
with the velocity v(x, y), the average velocity U, the dynamic
viscosity g, the pressure drop along the channel DP over a
distance Dx in the direction of the flow, the width of the
channel 2a and the distance from the channel centre x and y =
70 mm. The theoretical prediction agrees very well with our
DP
results in Fig. 4c. We use
as fitting parameter and choose
Dx
half the width of the channel as a = 108 mm. This is slightly larger
than the nominal channel diameter of the capillary tube and
indicates either a non-perfect or a no-slip boundary condition at
the channel walls,20 a wider channel due to manufacturing
tolerances or both.
4 Conclusions
This paper presents a technique to measure nanofluid flows in
constant or shear flow conditions based on analysing the
image structure function. We extract the flow velocities by
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computationally accessing the scattering information from the
microscopic images of our samples. Our approach is applicable to any standard microscope without the need for
specialist laser equipment, microscopic tracer particles nor
prior knowledge of the temperature and viscosity of the fluid.
This makes it easy to operate for a non-specialist, and thus
very appealing for interdisciplinary work. We demonstrate that
we are capable of measuring very slow nanofluid flows
containing particles down to a size of 40 nm. To illustrate
the effectiveness of our technique we spatially resolve the
parabolic flow profile with mm resolution in a microfluidic
channel. Expanding this method to dielectric nanoparticles
promises a wide range of applications in microrheology and
lab-on-a-chip in general.
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