SHORT COMMUNICATION

Parametric study of acoustic focusing of particlesin a micro-channel in the perspective to improvemicro-PIV measurements

Olivier Dron Æ Claire Ratier Æ Mauricio Hoyos ÆJean-Luc Aider

Received: 27 March 2009 / Accepted: 27 June 2009 / Published online: 30 July 2009

� Springer-Verlag 2009

Abstract In this study, we propose a new technique to

bring most of the particles in a horizontal plane at mid-

height of a micro-channel in order to improve the quality of

the micro-PIV measurements. The basic principle is to

create a stationary acoustic wave along the channel height

so that the resulting acoustic force moves the particles

toward the pressure node. A parametric study has been

carried out without mean flow to characterize the motion

of the particles toward the nodal-plane. We found that

focusing speed grows with the acoustic pressure amplitude,

with the concentration of particles in the suspension and

with the particles diameter. We also led a preliminary

investigation of acoustic focusing together with a mean

stationary flow. We still observed an important focusing of

particles for low freestream velocities. Nevertheless,

acoustic focusing is inefficient beyond a given critical

freestream velocity U0 for a given set of acoustic param-

eters and a given type of particles. It was also shown that

other phenomena, like clumps formation, can be observed

without mean flow if the acoustic focusing lasts long

enough.

Keywords Flow field velocimetry � Acoustic streaming �Standing wave

1 Introduction

While the last 10 years have seen enormous progress in

the general areas of Micro Electro Mechanical Systems

(MEMS) design and fabrication (Gad-el-hak 1999), the

understanding of fundamental physical processes on these

small scales is hampered by the lack of suitable quantita-

tive measurement and simulation tools.

Velocity measurements at small scale is a challenge and

requires dedicated techniques. Several common macro-

scopic measurement techniques have been extended to

microscopic length scales, like micro-LDV (Laser Doppler

Velocimetry) (Thieu et al. 1995), micro-MTV (Molecular

Tagging Velocimetry) (Garbe et al. 2008), and micro-

PIV (Particle Image Velocimetry) (Santiago et al. 1998;

Meinhart and Santiago 1999). Micro-LDV will give accu-

rate time dependant velocity measurements in a given point

of the volume, while the micro-MTV technique allows

mainly for velocity measurements along a line.

For in-plane velocity measurements, one has to use

micro-PIV, which is actually similar to the macroscopic

PIV method, which means bulk seeding and auto-correla-

tion calculations to get velocity fields. However, the laser

sheet illumination is irrelevant for micro-PIV since the laser

sheet thickness (the laser Waist) and the micro-channel

height are roughly of the same value. Instead, the whole

volume of the flow is illuminated at the same time, which

means all the tracer particles are illuminated, even the ones

that are not in the measurement plane. Those particles

generate background noise on the images, and lower the

resolution. Several techniques to improve micro-PIV have

O. Dron (&) � J.-L. Aider

Team ‘‘Instability, Control and Turbulence’’ ESPCI,

Laboratoire de Physique et Mecanique des Milieux Heterogenes,

10 rue Vauquelin, Paris Cedex 05 75231, France

e-mail: dron@pmmh.espci.fr

J.-L. Aider

e-mail: aider@pmmh.espci.fr

C. Ratier � M. Hoyos

Team ‘‘Separation’’ ESPCI, Laboratoire de Physique et

Mecanique des Milieux Heterogenes, 10 rue Vauquelin,

Paris Cedex 05 75231, France

123

Microfluid Nanofluid (2009) 7:857–867

DOI 10.1007/s10404-009-0477-7

already been established, based on image processing after

acquisition (Bourdon et al. 2004) or hydrodynamic focus-

ing (Mielnik and Saetran 2006). In this article, we describe

an alternative method to create thin sheet of particles in the

micro-channel instead of using a laser sheet to illuminate

particles evenly distributed throughout the whole channel.

The method consists in focusing the particles using acoustic

force created by an acoustic standing wave along the

height of the channel. Before using acousting focusing to

improve micro-PIV measurements, it is necessary to study

the dynamics of focalization without mean-flow and its

dependance on different physical parameters. This is the

objective of the present article which is organized as

follows.

First, the experimental setup and experimental tech-

niques are presented. Then, the principle of acoustic

focusing of particles is explained and some characteristic

times are derived from the equation governing the particles

motion. A detailed parametric study of acoustic focusing

without mean flow is then described before showing an

example of acoustic focusing together with a mean sta-

tionary flow. Finally, some other phenomena observed after

long focusing times are shortly described in the discussion

before turning to the conclusion.

2 Experimental setup and methods

2.1 Flow facility

The experiments were carried out using a 0.36 mm (height

h) 9 3 mm (width) 9 40 mm (length) micro-channel. The

channel is composed of one mylar sheet stuck between two

1 mm plexiglas plates. The drawing of the micro-channel is

cut into the mylar sheet so that the channel height is

imposed by the mylar sheet thickness (0.36 mm in this

case). The micro-channel is designed for low velocity

flows, but the velocities are large enough to study acoustic

focusing and its limits. The height of the channel is chosen

to fit an acoustic purpose that will be described later. The

fluid used for these experiments was distilled water.

2.2 Micro-PIV setup

As shown on Fig. 1, a 15 Hz pulsed Nd:YAG laser illu-

minates the fluorescent flow-tracing particles through an

epi-fluorescent inverted microscope. Fluorescent particles

absorb the green light (k & 532 nm) and reemit the light

to a slightly higher wavelength (k & 560 nm). To reduce

the noise induced by the incoming wavelength, a filter cube

is used with a narrow filter band around k & 560 nm so

that only the fluorescent light is recorded by the double-

frame CCD camera. The micro-PIV set-up is the same as

the one proposed by Meinhart and Santiago (1999).

The image resolution is 1200 9 1600 pixels with a 12

bits dynamic range and a sampling frequency of f = 15 Hz.

The whole PIV process from acquisition to vector post-

processing is carried out with Davis 7.1 software from

LaVision�.

2.3 Seeding particles

The tracer particles for the l-PIV are red fluorescing

polymer microspheres 1% solids, with a 2–7 lm diameter

ðUÞ: It is small enough for the particles (with a specific

density of qp = 1.05 g cm-3) to follow the flow faithfully

without perturbing it, and large enough for them to scatter

sufficient light without being affected by Brownian motion.

We indeed observed that the Brownian motion of those

particles does not affect velocity measurements. These

particles are doped with a fluorescent dye with a peak

excitation wavelength of kabs & 532 nm and a peak

emission wavelength of kemit & 560 nm. The choice of the

initial concentration of flow-tracing particles comes from

auto-correlation techniques criteria and is also one of the

parameters that will be varied in the following.

The flow is produced by a syringe pump which allows

flow rates from 1–99 times a reference flow rate Q01

= 0.7 ml h-1. It corresponds to velocities ranging from

U0 = 0.18 mm s-1 to 18 mm s-1, leading to a Reynolds

number based on the micro-channel height ranging from

Reh ¼ hUm � 0:06 to 6, where m = 10-6 m2 s-1 is the

kinematic viscosity of water. As expected the flow is in the

laminar regime.

Fig. 1 Description of the micro-PIV setup. Flow tracing particles are

illuminated by two pulsed Nd:YAG lasers through an epi-fluorescent

inverted microscope. The particle images are recorded by a double-

frame CCD camera (LaVision� Adapted from Meinhart and Santiago

(1999)

858 Microfluid Nanofluid (2009) 7:857–867

123

2.4 Acoustic setup

An 8 mm diameter transducer, whose functional frequency

is roughly fac = 2 MHz, is used to generate the acoustic

wave. The voltage applied to the transducer is the param-

eter used to control the acoustic amplitude. As shown on

Fig. 2a, the transducer is set on one side of the micro-

channel, while the inverted microscope is on the other side.

Thanks to this setup, one can see the motion of the particles

right under the acoustic transducer which is necessary to

characterize the dynamics of the focusing of the particles

with or without mean flow.

3 Acoustic focusing

3.1 Theory

3.1.1 Principle

The main objective of this study is to demonstrate and

characterize the acoustic focusing of particles in a micro-

channel which could be a very useful phenomenon to

improve micro-PIV visualizations. Of course, acoustic

focusing can also be helpful in any situations where one

wants to move particles in a given direction without

changing the mean flow. The acoustic radiation force,

which is the origin of the acoustic focusing, has already

been used to manipulate particles, such as the formation of

particle lines (Nield et al. 2007; Oberti et al. 2009) or the

separation of particles as a function of their physical

properties (Hoyos et al. 2005; Petersson et al. 2005).

The method is based on the creation of a standing

acoustic wave along the height of the micro-channel to

move the tracer particles toward the acoustic pressure

nodal-plane (Fig. 2b). This will be possible thanks to the

non-linearity of the acoustic force. Indeed, considering

only the linear term of the acoustic force, leads to a time-

average of the acoustic force on the particles equal to zero,

unlike non-linear terms. In order to identify and quantify

the relevant parameters of the phenomenon, one has to

study the acoustic focusing dynamics without mean flow.

In the following, we describe the dynamics of a single

particle in still water and in an acoustic field.

Theoretically, one has to consider a viscous fluid. It

is appropriate to neglect viscous effects as long as the

particle diameter is smaller than the acoustic wavelength

U� kð Þ but larger than the acoustic boundary layer

U� d; with d ¼ mpfac

� �0:5� �

(Doinikov 1996). In our

experiments, d = 0.4 lm, k = 720 lm, and 2 lm�U�7 lm so that the viscous effects are negligible.

The acoustic wave will also have another consequence on

the particles: it will create acoustic streamings that will drag

the particles thanks to the Stokes force (see Sect. 6.2). It is

possible (Sundin et al. 2007), setting the system parameters,

especially the particle size, to make one of the two

effects dominant over the other one. In our experiments, the

acoustic force is dominant over acoustic streaming.

3.1.2 Forces

Particles in motion in still water and in an acoustic field are

subjected to three forces (Fig. 2b):

– The acoustic force FA as defined by Gor’kov (1962),

– The buoyancy FB, and

– The Stokes force FS.

The analytic expressions of those forces are:

FA�! ¼ Vpk EA x; yð Þh i A1 þ

3

2A2

� �sin 2kzð Þez

! ð1Þ

FB�! ¼ �Vp qp � qf

� �gez! ð2Þ

Fig. 2 a A syringe pump

injects distilled water seeded

with particles into the 360 lm

micro-channel. The particles are

exposed to a standing wave

induced by the acoustic

transducer. In the same time,

they are also illuminated and

observed by the micro-PIV

system. b Description of the

forces applied on a particle in

still water and in standing

acoustic wave. The acoustic

force pulls the particle toward

the acoustic pressure node

against the Stokes force and

buoyancy

Microfluid Nanofluid (2009) 7:857–867 859

123

FS�! ¼ 6plR U

!x; zð Þ � u! x; zð Þ

� �ð3Þ

where R is the particle radius, Vp ¼ 43pR3 is the particle

volume, k is the acoustic wave number (andk the wavelength),

EA is the acoustic wave energy (h i stands for the time-

average), qp is the particle density and qf the fluid density, l is

the dynamic viscosity, and U!

and u! are, respectively, the

flow velocity and the particle velocity in the channel frame.

The two coefficients A1 and A2 depends on the density and

sound velocity in both the fluid and the particles:

A1 ¼ 1� c2f qf

c2pqp

; A2 ¼2 qp � qf

� �

2qp þ qf

ð4Þ

In the following experiments, the fluid is water (qf =

1000 kg m-3) and the particles are made of polymer

(qp = 1050 kg m-3). The sound velocity in the fluid and in

the particles are cf = 1435 m s-1 and cp = 2407 m s-1,

respectively. Considering the given physical parameters,

one gets A1 þ 32A2

� �� 0:71 [ 0:

If one considers z = 0 at the bottom of the channel, and

positive upward, then the acoustic force is directed toward

the nodal-plane. When A1 þ 32A2

� �is negative, then the

acoustic force is directed toward the pressure anti-nodal-

plane (Spengler et al. 2003). In the following q will be

used instead of qp, and Dq ¼ qp � qf :

Here, one should emphasize that the acoustic force

defined in Eq. 1 has been derived for an ideal fluid

(Gor’kov 1962). For actual fluids, one should take into

account thermal and viscous dissipation in the derivation

of the acoustic force as proposed by Doinikov (1996). If

we compute with our physical parameters the acoustic

(ka = 720 lm), thermal (kt = 0.15 lm), and viscous

(kv = 0.4 lm) wavelengths as defined by Doinikov, we see

that ðka � kt; kv;/Þ showing that our experimental con-

ditions correspond to the long sound wavelength limit and

weak dissipative effects (kv, kt \ /).

3.1.3 Sedimentation and focusing times

In order to determine characteristic sedimentation and

focusing times, one can integrate the law of resultant force

projected along the z-axis, without the acoustic force for

the sedimentation time, and with the acoustic force for the

focusing time. It leads to the following equation:

1

3U2q€z ¼ �6l _z� 1

3U2Dq g

þ 1

3U2k EA x; yð Þh i A1 þ

3

2A2

� �sin 2kzð Þ ð5Þ

where U is the particle diameter.

Initial conditions are _zðt ¼ 0Þ ¼ 0 (no initial speed)

for the velocity, kzðt ¼ 0Þ ¼ p2

for the sedimentation (the

particle is located at the mid-height of the channel)

and kzðt ¼ 0Þ ¼ p for the focusing (the particle is

located at the top of the channel). One can define the

sedimentation (TS) and focusing (TF) times such as the

time needed for a particle to travel half the height

channel, h:

kzðt ¼ TSÞ ¼h

2and kzðt ¼ TFÞ ¼

h

2:

It leads to the following expressions and numerical values

for our experiments:

TS ¼9lh

U2Dqgð6Þ

TF ¼9

2

lk

U2k EA x; yð Þh i A1 þ 32A2

� � ð7Þ

Using the physical parameters of the present experiment,

we obtain TS & 1340 s and TF & 2.6 s, with a 10% error

for U ¼ 2:2 lm ð�5%Þ; TS � 674 s and TF & 1.3 s, with a

10% error for U ¼ 3:1 lm (±5%) and TS & 132 s and

TF & 0.25 s, with a 36% error for U ¼ 7 lm (±18%). The

way EAðx; yÞh i is estimated experimentally is detailed in

Sect. 6.2. One can conclude from those numerical values

that sedimentation is negligible compared to acoustic

focusing for short times.

The mean focusing times correspond to a mean focusing

velocity uF ranging between 60 and 792 lm s-1. The

Reynolds numbers associated to the motion of the particles

induced by acoustic force ReU ¼ UUm ranges between 10-5

and 4.6 9 10-4 for the three diameters so that there should

not be any wake or added mass effect.

3.2 Acoustic frequency

In the following, the acoustic frequency is tuned to induce

a standing wave with only one pressure nodal-plane along

the height of the micro-channel so that only one focusing

plane will be considered. The micro-channel height cor-

responds to half of the wavelength in that case.

4 Acoustic focusing without mean flow

The main goal of this study is to confirm and to quantify

the acoustic focusing of particles in a micro-channel

without mean flow as a first step before application to

micro-PIV measurements. The second step consists in

studying the acoustic focusing of particles in a stationary

mean flow. It is illustrated by very preliminary results in

the second part of the article. The influence of the fol-

lowing parameters has been studied without mean flow:

860 Microfluid Nanofluid (2009) 7:857–867

123

the acoustic amplitude (through the voltage applied on the

acoustic transducer), the particles concentration, and the

diameter of the particles.

4.1 Observation of acoustic focusing

The focusing dynamics is illustrated in Fig. 3 through

different snapshots of the particles ðU ¼ 2 lmÞ in the

nodal-plane, which is merely the camera focal plane, for

different successive time steps. At time t = 0, the trans-

ducer is turned on. Particles that are located in the focal

plane are seen as white dots. Out-of-plane particles appear

as diffuse spots. The more distant from the focal plane the

particles are, the larger the spots. One can clearly see on

Fig. 3b–d that the number of particles in the nodal-plane is

increasing as a function of time. Actually, one could even

see the diffuse spots getting smaller while the particles

move toward the nodal-plane. After some time, the

focusing slows down, mainly because most of the particles

are already in the nodal-plane (Fig. 3d).

We can conclude from Fig. 3 that the order of magnitude of

the experimental focusing time, with those physical parame-

ters, is around a hundred of seconds. This is a lot more than the

numerical value TF & 2.6 s found in Sect. 3.1, whereas the

parameter values are almost the same. First of all, one has to

keep in mind that the two values merely mean something

different. TF is the time needed for a single particle to move

from the top of the channel to its middle plane. The interac-

tions between particles is neglected, as well as any viscous or

screening effect which could lower the acoustic focusing. If

one can expect some dissipation of acoustic energy, the

acoustic wave might also not be purely a standing wave, but

could be for a small part a traveling wave and a larger part a

standing wave. All those phenomena are difficult to quantify

but all could contribute to slow down the dynamics of acoustic

focusing.

Fig. 3 Time-evolution of the particles ðU ¼ 2 lmÞ concentration in

the nodal-plane. a The focusing has not started yet. The particles

distribution along z is homogeneous. We can see more out-of-plane

particles than in-plane particles. At time t = 0, a 2 MHz acoustic wave

(P = 5 V) is generated. b t = 30 s, c t = 65 s, and d t = 199 s—the

number of focused particles increases with increasing time

Microfluid Nanofluid (2009) 7:857–867 861

123

4.2 Parametric study of the acoustic focusing

A way to quantify the dynamics of the focusing is to

evaluate the number of particles in the nodal-plane as a

function of time. The method used to calculate the number

of particles is the following. Using the software Image J�

we first smoothen the picture. Then we use a filter to

homogenize the number of pixels defining each particle.

Finally, we divide the number of pixels that correspond to

particles by the average number of pixels for one particle.

4.2.1 Influence of the concentration of particles

The particles concentration is an important parameter for

the micro-PIV measurements as it is related to the number

of out-of-plane particles. Note that its influence does not

appear in the analytical expressions (Eqs. 6, 7) merely

because we considered dynamics for a single isolated

particle in the micro-channel, i.e., no collective effects

were taken into account in the force balance. An additional

term would be necessary to take into account the effect of

the concentration. In the following, we set C = 1 as the

concentration resulting of diluting one drop of the particles

with 10 ml of distilled water, rather than giving the exact

value of the concentration. C = 1 is then equivalent to

C = 5 9 106 particles in 1 ml of the resulting solution.

Since we are dealing with number of particles, we have

to normalize the data to make comparison easier. In the

following, the percentage of focused particles will be

considered rather than the number of focused particles.

To study the influence of the concentration, time-evo-

lution of the percentage of particles in the nodal-plane are

plotted for two different initial concentration values, C = 1

and C = 2, respectively (Fig. 4). A concentration too high

would make it impossible to count the particles, while a

low concentration would make the auto-correlation calcu-

lation difficult. This is the reason why we could not study a

wider range of concentration values. As expected, the ini-

tial concentration of particles play an important role in the

focusing dynamics. Except for the first seconds, the per-

centage of particles in the nodal-plane grows linearly with

time. One can clearly see in Fig. 4, and from the growth

rates obtained from the linear fits, that the focusing is faster

for C = 2: it is almost twice as fast as for C = 1. This

result is rather surprising, as one could have expected a

decrease of acoustic focusing when increasing the con-

centration. Unfortunately, it cannot be interpreted in the

framework of Sect. 3.1 where only a single, isolated par-

ticle is considered. It is clearly necessary to include col-

lective effects in the modeling. One possibility would be to

include a new term in Eq. 5, taking into account interaction

forces between particles. Another possibility could be to

describe the dynamics of the distribution of particles along

the height of the channel instead of following a single

particle. In this hypothesis, it should be possible to propose

a model for the dynamics of the concentration profile as a

function of time n(t, z), taking into account the transition

from a nearly homogenous vertical distribution profile to a

nearly gaussian profile as shown later in Sect. 5.1.

4.2.2 Influence of the acoustic pressure amplitude

The real pressure amplitude value is not known, but we do

control the voltage applied on the transducer. If the trans-

ducer and the whole acoustic system are linear, then the

relationship between the pressure amplitude and the volt-

age supply is also linear. In the following, the parameter

called P is the voltage value in Volt, and is proportional to

the acoustic pressure. Three different values of the acoustic

pressure will be considered: P = 3 V, P = 5 V, and

P = 10 V. P = 10 V is actually the maximum of the

voltage applied on the transducer, while for P \ 3 V

acoustic effects are barely observable.

EA, the acoustic energy density, is a quadratic function

of P, thus according to Eq. 7 one can expect a focusing

velocity proportional to P2. Results are shown in Fig. 5.

The influence of the acoustic pressure on the focusing

velocity seems to be even stronger than a quadratic one and

has to be maximized in order to get the most powerful

acoustic focusing. One could use some transducers that

support higher voltage values, but as one will see in Sect. 6,

some limitations appear when a high acoustic pressure is

used (Fig. 6).

4.2.3 Influence of the radius of the particles

The acoustic force (Eq. 1) is a cubic function of the particle

radius. Therefore, there is no doubt that this parameter has

a strong influence on the acoustic focusing. Unfortunately,

Fig. 4 Influence of the particle concentration. On the graphics is

represented, for different values of particle concentration, the

percentage of in-plane particles as a function of time. P = 5 V and

U ¼ 7 lm for every experiment

862 Microfluid Nanofluid (2009) 7:857–867

123

this parameter is not as easy to change as the other

parameters. Only a limited range of radius is manufactured.

In particular no particles are manufactured in the range

between 3 and 7 lm. This is the reason why there is only

three different particle sizes in this study.

The focusing is way faster with U ¼ 7 lm particles, but

there is nearly no difference between U ¼ 2 lm and U ¼3 lm: The result is quite reproducible. It appears that the

quadratic dependency on the radius (Eq. 7) is not properly

recovered in the experiment. It might be the consequence

of a transition between two different regimes, like what Li

et al. (2008) demonstrates for a droplet. Also, Bjerknes

forces (see end of Sect. 6.2), which is not a quadratic

function of the particle size, could interfere with the

acoustic radiation force. In the end, three values of the

parameter U are not enough to conclude. Further investi-

gation with other radius are needed to identify the real

dependency of the particle diameter on the focusing, and

possibly different regimes.

5 Acoustic focusing with a mean stationary flow

5.1 Acoustic focusing velocity limitation

One of the motivations in studying acoustic focusing is to

use it to improve the quality of the PIV velocity fields. The

next step is then naturally to check the efficiency of

acoustic focusing of particles in a stationary flow. To do so,

the parameters are fixed to P = 10 V and C = 1 in order to

optimize the acoustic focusing, according to the previous

results without mean flow. Figure 7 shows for different

flow rates the concentration profiles. Even if there is not yet

precise criterion to determine if the focusing is efficient,

one can guess that the focusing will lead to a significant

improvement of the velocity field measurements only for

Q = 0.7 and Q = 1.4 ml h-1 which lead to peaked particle

concentration profiles in the nodal-plane region. It leads us

to a critical freestream velocity U0 & 0.4 mm s-1 for these

experimental conditions ðP ¼ 10 V;C ¼ 1; and U ¼ 2 lmÞ:The efficiency of the method is actually the result from

the competition between the focusing time (time needed for

a particle to arrive in the nodal-plane) and the time of flight

(time needed for a particle to cross the region influenced by

the acoustic field). If the particles velocity is too large, then

the particles will not have the time to reach the nodal-

plane.

5.2 Preliminary investigation of the influence

of acoustic focusing of particles on PIV fields

In Fig. 8, two velocity fields measured in the same

experimental conditions and post-processed exactly the

same way, are compared. Figure 8b is obtained with a

given acoustic wave (P = 10 V, 2 MHz) while Fig. 8a is

obtained without any acoustic focusing. Both PIV fields are

actually an average of 345 instantaneous velocity fields.

Fig. 5 Influence of the acoustic pressure amplitude. On the graphics

is represented, for different values of the acoustic pressure amplitude,

the percentage of in-plane particles as a function of time. C = 1 and

U ¼ 7 lm for every experiment

Fig. 6 Influence of the particle diameter. On the graphics is

represented, for different values of the particle diameter, the

percentage of in-plane particles as a function of time. C = 1 and

P = 5 for every experiment

Fig. 7 Each curve represents the vertical distribution of the particles

for different values of the flow rate. The reference (without acoustic

wave: P = 0) curve has been calculated for Q ¼ 2:1 ml h�1: C ¼ 1;P ¼ 10 V and U ¼ 2 lm for every experiments

Microfluid Nanofluid (2009) 7:857–867 863

123

The mean flow is stationary and goes from the left to the

right. Since the Reynolds number of the flow is low

(Reh = 0.06 \ 1), the flow in the microchannel is laminar

and fully developed. Therefore, one can expect the velocity

field to be a uniform set of identical arrows for a given

height. Two main observations can be made from Fig. 8.

The first observation is that there are more wrong vectors

(not validated and not shown on the mean PIV field) and

vectors with a direction that is not the expected horizontal

direction from left to right, in Fig. 8a than in Fig. 8b. The

second observation is that the velocity vectors are longer in

Fig. 8b than in Fig. 8a. It means that the mean velocity in

the focal plane is larger when the particle are focused

whereas the flow rate applied is exactly the same in both

cases. These two observations are actually the conse-

quences of both increasing number of in-plane particles

and decreasing number of out-of-plane particles.

The autocorrelation calculation that gives us the velocity

fields needs a minimal concentration of particles, or else it

leads to many errors. The particles density in Fig. 8a is

clearly too low for a proper PIV autocorrelation calcula-

tion. The calculation is improved in Fig. 8b, but it may be

even better by increasing the particle concentration of the

solution.

When the particles are not focused, they still can be seen

by the camera (Airy disks) and taken into account during

the auto-correlation computation. Since the Reynolds

number is very low (Reh = 0.06 \ 1), the flow in the mi-

crochannel follows the Poiseuille law, which is a parabolic

law with its maximum at mid-height. Therefore, out-of-

plane particles are slower than in-plane particles and leads

to an underestimation of in-plane velocities. Finally, the

right picture is much closer to the expected velocity field

than the left one. This result clearly demonstrates the

potential of the acoustic focusing to improve micro-PIV

measurements.

From a practical point of view, one should carry out the

PIV measurements downstream of the region where the

acoustic wave is applied, where the flow is not perturbed by

the acoustic focusing. The sheet of particles is created

under the transducer and the PIV measurements can be

carried out downstream on the unperturbed flow (like the

added vertical component due to the acoustic focusing).

Sedimentation (TS & 1340 s see Sect. 3.1) and Brownian

motion [TB & 105 s according to the Einstein formula

(Einstein 1905)] are too large to disperse the particles.

Therefore, particles stay focused as long as they are in the

micro-channel.

6 Discussion

6.1 Limitation of the technique

In the previous section the principle of acoustic focusing

has been validated and quantified. It has also been shown

that it can improve PIV velocity fields under some

hypothesis and using the proper parameters. It also high-

lighted some ways to optimize the focusing with different

parameters. As we already mentioned for the pressure

dependence (see Sect. 4.2), some phenomena described

below (see Sect. 6.2) prevents us to increase particle radius

to improve the velocity fields, in addition of auto-correla-

tion issues. Another way to improve the velocity fields

would be to use several transducers successively along the

streamwise direction. This way, the focusing time would

remain the same while the time of flight would increase.

This way the method could be used for larger freestream

velocities.

6.2 Streaming and structures

For the typical dimensions of our micro-fluidics device,

there are two main acoustic effects that can be expected,

because of the non-linearity of the acoustic wave:

– the acoustic radiation force (Gor’kov 1962), which

moves particles toward pressure nodes,

– acoustic streaming (Spengler and Jeckel 2001), which

induces motion onto the entire solvent.

The acoustic radiation force has already been discussed

in the previous sections so that we will focus on streaming

Fig. 8 Height-centered velocity

fields for stationary

Q = 0.7 ml h-1 flows.

aVelocity field obtained without

any acoustic wave. b A 2 MHz

acoustic wave is applied, clearly

improving the velocity field

obtained without acoustic wave

864 Microfluid Nanofluid (2009) 7:857–867

123

effects in the following. There are three different types of

acoustic streaming (Zarembo 1971; Nomura et al. 2000):

Eckart (if h� k), Schlichting (if h� k), and Rayleigh

streamings (if h & k). Only the latest has been observed

during the experiments without mean flow.

Indeed, Spengler and Jekel (2000) demonstrate that a

short container dimension along the sound axis effectively

minimizes large-scale streaming such as Eckart streaming.

Since the dimension along the sound axis in our experi-

ments is h ¼ k2; then the Eckart streaming is negligible

compared to other phenomena in our system. As for the

Schlichting streaming, it mainly consists in a vortex flow in

the acoustic boundary layer thickness d. In our experi-

ments, d = 0.4 lm. The Schlichting streaming happens

only in a very thin part of the micro-channel, therefore it is

not responsible for what is observed in our experiments.

In the Sect. 3.1, it was recalled that the acoustic force

was larger than acoustic streamings considering the phys-

ical parameters defining our experimental setup. One

would want to confirm this assertion. Rayleigh streaming

has a y-component and a z-component. Only the latest can

affect acoustic focusing. Since the channel width is wider

than the acoustic wavelength (l� k) and the acoustic

wavelength is longer than the acoustic boundary layer

thickness (k� d), the z-component has the following form

(Rayleigh 1894):

uzðy; zÞ ¼ �3v0

16c0

1� 3y2

ðl=2Þ2

!sin

4pz f

c0

� �ð8Þ

Figure 9 shows the streaming pattern calculated

according to Eq. 8. The flow generated by the acoustic

standing wave through Rayleigh streaming is directed

toward the nodal-plane for y 2 � l2ffiffi3p ; l

2ffiffi3p

h i: It is directed

away from the nodal-plane for y 2 � l2;� l

2ffiffi3p

h iand y 2

l2ffiffi3p ; l

2

h i: Assuming the particle would be dragged by the

streaming, which is highly likely since their properties have

been chosen so that inertia plays no role, the migration

caused by Rayleigh streaming would be different from

the acoustic focusing studied above. In particular, the

dependance on some physical parameter is different, like a

linear dependance on the particle radius. Thanks to Eq. 9

and experimental observations (Fig. 10a), we obtain

uzj j � 3 lm s�1; which gives us a characteristic focusing

time, such as defined in the Sect. 3.1, TF * 60 s. It is

approximatively 20 times longer than the focusing time

calculated with the acoustic radiation force. The acoustic

force is therefore much stronger over than the acoustic

streaming when it comes to acoustic focusing with

the parameters used in our experiments. Nevertheless,

Rayleigh streaming may contribute for a small part to the

observed focusing.

In the long time limit, most of the particles are

focused into the nodal-plane. In this case, uzj j ¼ 0 in the

nodal-plane (z = 0) and the acoustic radiation force

keeps the particles inside the nodal-plane without

modifying particle motion inside the nodal-plane in any

way. Therefore, when this situation is established, the

y-component of the Rayleigh streaming rules the dynamics

of the particles.

Actually, Rayleigh streaming also consists in a centrif-

ugal motion from directly under the transducer, as observed

on the Fig. 10a. The motion is actually applied only to the

solvent, not the particles. Through the Stokes force, the

particles follow the streaming since their inertia has been

chosen to be good passive tracers. The particles just leave

the region directly under the transducer and aggregate at

the periphery. This phenomenon happens only after a long

time, when most of the particles have already focused in

the nodal-plane, and only for the in-plane particles. The

expression for the Rayleigh velocity (Spengler et al. 2003)

is:

uR ¼3pv2

0 fac

64c20

l; with EAh i ¼ q0v20

2ð9Þ

where v0 is the acoustic velocity amplitude, fac the acoustic

frequency and l the channel width. From Eq. 9, one can see

that measuring uR gives access to one of the unknown

parameters, v0, and as a consequence, EAh i; which is very

useful for some numerical calculations about the acoustic

focusing. This is precisely the way EAh i has been estimated

in Sect. 3.1.

In the perspective of micro-PIV measurements, and then

in the short time limit, we will avoid the Rayleigh

Fig. 9 z-component uz (y, z) of the Rayleigh microstreaming between

parallel walls. This graphic corresponds to Eq. (8). y 2 � l2; l

2

and

z 2 �h2; h

2

Microfluid Nanofluid (2009) 7:857–867 865

123

streaming. This is the reason why very energetic acoustic

waves are not advised even if it maximizes the acoustic

focusing. In Fig. 10a, the order of magnitude of the

acoustic streaming is uR = 36 lm s-1. Therefore, for

P = 10 V, acoustic streaming would not affect the main

flow as long as Q� QR ¼ 0:14 ml h�1ðQR being the flow

rate corresponding to uR). The flow rates used in our

experiments are higher than QR, therefore Rayleigh

streaming does not affect the main flow. However, Ray-

leigh streaming could have an effect on the velocity field if

P was a lot higher and Q the same, or even lower.

After some times, the particles aggregate at the periph-

ery, increasing the local concentration, and some clumps

are rapidly formed. This phenomenon is the consequence

of particles interactions (Nyborg 1978). Indeed, now that

the particles are concentrated only in the nodal-plane, the

local particle concentration has strongly increased. There-

fore, Bjerknes forces (or secondary radiation forces

[Bjerknes 1906]) cause attraction between particles, which

leads to the creation of clusters of particles. Then, the

particles aggregates form spiral structures, as shown on

Fig. 10b. Of course, this process can only be observed

without mean flow and in the long time limit, after all the

particles have focused in the nodal-plane. In the perspec-

tive of PIV measurements we will always avoid this situ-

ation since an homogeneous distribution of particles is

important for the auto-correlation calculation. This aggre-

gation process could not be observed together with a

mean flow.

7 Conclusion

Acoustic focusing of particles has been validated and

quantified as a function of different parameters. The

method is based on a radiation acoustic force, generated

from a standing acoustic wave in the micro-channel, which

moves the particles toward the measurement plane and

therefore eliminates most of the out-of-plane particles that

generate background noise. It was shown that the dynamics

of the focusing depends strongly on the particles concen-

tration, the pressure amplitude and the particles radius.

Some counterparts are to be taken into account, including

some after-effects of the acoustic waves, especially

acoustic streaming and aggregation. It has been shown that

it can help improving the PIV measurements but it can also

be used, from a general point of view, to manipulate or sort

particles in a micro-channel or in Hele-Shaw cells (Hoyos

et al. 2005).

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