On the 3D distribution and size fractionation of microparticles in a serpentine microchannel [original]

Vol.:(0123456789)

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Microfluidics and Nanofluidics (2020) 24:22

https://doi.org/10.1007/s10404-020-2326-7

RESEARCH PAPER

On the3D distribution andsize fractionation ofmicroparticles

inaserpentine microchannel

SebastianBlahout1 · SimonR.Reinecke2· HamidTabaeiKazerooni3· HaraldKruggel‑Emden2· JeanetteHussong1

Received: 2 October 2019 / Accepted: 19 February 2020 / Published online: 11 March 2020

Abstract

Suitable methods to realize a multi-dimensional fractionation of microparticles smaller than

10 μm

diameter are still rare.

In the present study, size and density fractionation is investigated for

3.55 μm

and

9.87 μm

particles in a sharp-corner ser-

pentine microchannel of cross-sectional aspect ratio

h∕

=0.25

. Experimental results are obtained through Astigmatism

Particle Tracking Velocimetry (APTV) measurements, from which three-dimensional particle distributions are reconstructed

for Reynolds numbers between 100 and 150. The 3D reconstruction shows for the first time that equilibrium trajectories

do not only develop over the channel width, i.e. in-plane equilibrium positions but also over the channel height at dif-

ferent out-of-plane positions. With increasing Reynolds number,

9.87 μm

polystyrene

(

𝜌

=1.05 g cm

−3)

and melamine

(

𝜌

=1.51 g cm

−

)

particles focus on two trajectories near the channel bisector. In contrast to this, it is shown that

3.55 μm

polystyrene particles develop four equilibrium trajectories at different in-plane and out-of-plane positions up to a critical

Reynolds number. Beyond this critical Reynolds number, also these particles merge to two trajectories at different channel

heights. While the rearrangement of

3.55 μm

polystyrene particles just starts beyond

Re >140

9.87 μm

polystyrene particles

undergo this rearrangement already at

Re =100

. As the equilibrium trajectories of these two particle groups are located

at similar out-of-plane positions, outlet geometries that aim to separate particles along the channel width turn out to be a

good choice for size fractionation. Indeed, polystyrene particles of different size assume laterally well-separated equilibrium

trajectories such that a size fractionation of nearly 100% at

Re =110

can be achieved.

Keywords Size fractionation· Serpentine microchannel· Astigmatism Particle Tracking Velocimetry (APTV)· 3D

microparticle migration

1 Introduction

Microparticles with diameters below

10 μm

and well-defined

properties are increasingly used in e.g. the pharmaceutical

and chemical industry or metallurgy (BaghbanTaraghdari

etal. 2019; Li etal. 2019; Kumar and Venkatesh 2019).

Thus, multi-dimensional fractionation processes are essen-

tial to fabricate intermediate particle products with e.g.

monodisperse size, shape or surface properties. Microfluidic

systems allow to utilize particle migration effects in a lami-

nar flow, driven by high local velocities and velocity gra-

dients without suffering from undesirable transient flow

effects. Therefore, microfluidic fractionation methods have

the potential to address different particle properties, such as

size or shape (Alfi and Park 2014; Jiang etal. 2015).

Microfluidic particle fractionation has been realized by

means of active as well as passive approaches. While active

approaches utilize external acoustic, magnetic or electrical

force fields (Nilsson etal. 2004; Suwa and Watarai 2011;

Podoynitsyn etal. 2019), passive approaches only rely on

hydrodynamic forces. Specifically, inertial effects may be

utilized to force particles on certain equilibrium trajecto-

ries, whose position depends on both the particle and fluid

flow properties. Inertial effects include particle centrifugal

forces, as well as inertial particle migration, which is present

if the particle diameter is large compared to the channel

dimensions (DiCarlo 2009). First experimental studies of

* Sebastian Blahout

[email protected]mstadt.de

1 Fluid Mechanics andAerodynamics, Technical University

ofDarmstadt, 64287Darmstadt, Germany

2 Mechanical Process Engineering andSolids Processing,

Technische Universität Berlin, 10587Berlin, Germany

3 Institute ofThermodynamics andFluid Mechanics,

Technische Universität Ilmenau, 98684Ilmenau, Germany

Microfluidics and Nanofluidics (2020) 24:22

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22 Page 2 of 10

the migration of

Dp=O(1 mm)

particles in a Poiseuille tube

flow were performed by Segré and Silberberg and showed

that particles focus at an annulus of approximately 0.6 times

the tube radius (Segré and Silberberg 1962a, b). This inertial

migration effect could later be explained to result from an

equilibrium of shear gradient and wall lift forces (Ho and

Leal 1974; Asmolov 1999). During the last decades, par-

ticle migration and underlying hydrodynamic drag and lift

forces have been investigated. Maxey and Riley assumed

infinitely small particles and derived a set of equations

considering shear gradient and Stokes drag but also time-

dependent contributions, i.e. virtual mass and history forces

(Maxey and Riley 1983). Later studies aimed to extend the

Maxey–Riley equations to develop a model that is also valid

for finite particle sizes and Reynolds numbers, as well as for

non-uniform, unsteady flow conditions (Loth and Dorgan

2009). Loth and Dorgan provide a model to calculate the

net lift force on suspended particles if the detailed flow field

around a particle is unknown. This is usually the case not

only in experimental investigations, but also in numerical

simulations, if the particle size is of similar order of magni-

tude as the numerical mesh. In the last years, direct numeri-

cal simulations have been performed where the flow field

in the vicinity of submerged particles is fully resolved. In

laminar and turbulent straight duct flow, such fully resolved

simulations revealed that the particle migration dynamics

strongly depends on the bulk Reynolds number (Kazerooni

etal. 2017; Fornari etal. 2018).

Common passive microfluidic fractionation approaches

that utilize inertial effects are e.g. the Multi-Orifice Fluid

Fractionation (MOFF) and fractionation in a spiral or ser-

pentine microchannel (Zhang etal. 2016). These approaches

aim to enhance particle migration such that a decrease in

particle focusing length is achieved. A geometrical key

parameter for particle focusing is a cross-sectional change

in streamwise direction (orifices), inducing strongly curved

streamlines and thus centrifugal forces on suspended parti-

cles. Therefore, MOFF channels have been shown to achieve

size fractionation performances of up to 100% (Park and

Jung 2009; Sim etal. 2011; Fan etal. 2014). Studies, e.g. on

hydrodynamic blood cell separation also show the practical

use of MOFF channels (Kwak etal. 2018).

Spiral and serpentine microchannels have also been suc-

cessfully utilized for particle fractionation (Bhagat etal.

2008; Zhang etal. 2015). In these channels, Dean vortices

are induced due to the channel curvature. In combination

with centrifugal forces and hydrodynamic shear forces,

particles develop Reynolds number-dependent equilibrium

trajectories (DiCarlo 2009; Zhang etal. 2014).

Previous experimental studies utilized fluorescence imag-

ing to investigate the focusing behavior of particles in curved

microchannels (Bhagat etal. 2008; Johnston etal. 2014).

With that, it was shown that in asymmetrical serpentine

channels, the number of equilibrium positions reduces in

comparison to straight duct channel flows and that the width

of the resulting focusing streak is particle Reynolds num-

ber dependent (DiCarlo etal. 2007, 2008). From studies of

DiCarlo etal. (2008), it becomes evident that equilibrium

trajectories narrow with increasing particle Reynolds num-

ber and start to blur again above a critical Reynolds number.

This effect was assumed to result from an increased magni-

tude of the secondary flow enhancing the particle mixing.

For investigations of mixing effects in curved microchannels

an APTV approach could already be successfully applied by

Rossi etal. (2011). Nevertheless, it should be noted that the

mixing strength also depends on the microchannels cross-

sectional aspect ratio (Zhang etal. 2014).

To obtain deeper insight into the development of particle

equilibrium trajecories, DiCarlo etal. (2009) performed

experimental and numerical studies to investigate the rela-

tion between particle size and microchannel dimensions. As

a result, they showed that the location of particle equilibrium

positions scales with the ratio of particle to channel dimen-

sions, which emphasizes the need of models that go beyond

point-particle descriptions. Further studies investigated the

focusing behavior of single particles in a single turn of a

curved microchannel (Gossett and DiCarlo 2009). There,

fluorescence imaging was combined with high-speed parti-

cle tracking, indicating that out-of-plane particle motion is

present during the development of equilibrium trajectories

in a single turn microchannel. However, no detailed studies

on the particles out-of-plane distribution and motion have

been performed in their study.

Further experimental studies aimed to improve the focus-

ing concept to make it usable for practical applications.

Oakey etal. (2010) showed that a combination of straight

duct and serpentine microchannel stages can lead to a single

particle equilibrium trajectory over the channel cross-section

at a particle Reynolds number determined in the straight

channel section of

Rep=6

. Furthermore, it was experi-

mentally shown that the focusing streak width decreases

with increasing particle concentration, which is assumed

to result from hydrodynamic particle–particle interactions

(Oakey etal. 2010). A more recent investigation utilized

a sharp corner serpentine microchannel to investigate the

importance of particle centrifugal forces and, therefore, of

the density difference between the fluid and particles on the

development of equilibrium trajectories (Zhang etal. 2014).

Zhang etal. (2014) provided also a scaling factor as a meas-

ure for the influence of the centrifugal force relatively to

the Dean drag force on the development of particle equilib-

rium trajectories under the assumption of negligible inertial

migration effects. As they also experimentally observed par-

ticle focusing for negligible inertial migration effects, this

indicates that the density difference between particles and

fluid, i.e. centrifugal forces play an important role for the

Microfluidics and Nanofluidics (2020) 24:22

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Page 3 of 10 22

development of particle equilibrium trajectories. Based on

the knowledge that equilibrium trajectory locations are size

and Reynolds number dependent in sharp corner serpentine

microchannels, a spatial separation of particles with

3μm

and

10 μm

diameter was reported by Zhang etal. (2015).

Extending these investigations, it was recently shown that

the usage of an elastic carrier fluid can further accelerate

particle focusing and, thus, allows a reduction of the total

amount of serpentine loops (Yuan etal. 2019).

Although it was already shown that sharp corner serpen-

tine microchannels are able to focus particles below

10 μm

diameter, detailed experimental investigations that are able

to explain the lateral migration of particles in such a flow

are still subject of ongoing research. That is also because

previous studies were only able to determine in-plane par-

ticle trajectories using fluorescence imaging, where usually

long exposure times are used during the recording of parti-

cle fluorescence signals (Gossett and DiCarlo 2009; Zhang

etal. 2015). This makes it impossible to determine details

of the three-dimensional particle dynamics that is essential

to understand the particle migration and focusing behav-

iour, as was also assumed in previous studies (Gossett and

DiCarlo 2009). Therefore, the present study is a first step to

close this gap, by reconstructing three-dimensional particle

trajectories, i.e. particle distributions not only in-plane but

also over the channel height, utilizing an Astigmatism Par-

ticle Tracking Velocimetry (APTV) (Cierpka etal. 2010)

approach. With this approach, we are testing a sharp corner

serpentine microchannel to investigate not only size but also

density fractionation at different Reynolds numbers.

2 Experimental set‑up

To reconstruct three-dimensional particle distributions,

an EPI-fluorescence microscope (Nikon Eclipse LV100)

is used for measurements. For illumination, the beam of

a double-pulsed, dual-cavity laser (Litron Nano S 65-15

PIV) is coupled into the system. Images are recorded with

a dual-frame CCD camera (LaVision Imager pro SX) of

2058 px ×2456 px

resolution. For APTV measurements, a

cylindrical lens with a focal length of

f=200 mm

is posi-

tioned in front of the CCD camera. All images are magnified

with an infinity corrected objective lens of

M=20

and a

numerical aperture of

NA =0.4

(20X Nikon CFI60 TU Plan

Epi ELWD) resulting in a field of view of

1.1 mm ×0.7 mm

The commercial software DaVis 8.4 (LaVision GmbH) is

used for image acquisition.

In the present study, a sharp corner serpentine channel

with rectangular cross-sectional area is utilized that has been

manufactured by Micronit GmbH. The channel assumes a

height of

h=50 μm

and a width of

w=200 μm

. A sketch

of the channel geometry including the region of interest

(marked red) is shown in Fig.1a, b, respectively.

Measurements are performed inside the 24th serpentine

loop, as depicted in Fig.1a. As Zhang etal. (2014) showed

in a similar sharp corner serpentine channel that equilibrium

trajectories of PS particles with diameters in the order of

magnitude of

Dp=O(10 μm)

already focus after six ser-

pentine loops, fully developed particle equilibrium trajecto-

ries are expected here in the present investigation. To study

size and density fractionation, polystyrene (PS) particles

3.55 ±0.07 μm

diameter, as well as

9.87 ±0.28 μm

(

𝜌

=1.05 g cm

−3)

and melamine

(

MF, 𝜌

=1.51 g cm

−3)

particles (microParticles GmbH) are suspended in distilled

water.

3 Calibration procedure andexperimental

execution

The presence of the cylindrical lens in the optical path of the

microscope creates an astigmatism that allows to determine

out-of-plane particle positions, i.e. their distribution over

the channel height.

The astigmatism is characterized by the presence of two

focal planes

and

, which are spatially separated. This

results in particle images, whose shape is rather circular near

the middle plane between both focal planes. Please note that

the particle is defocused at this location. In Fig.2 the char-

acteristic light path of the fluorescence signal of a particle

is shown.

The light emitted by the particle is first parallelized by

an infinity corrected objective lens. Then, the parallelized

Fig. 1 a Sharp corner serpentine microchannel and field of view at

the end of the 24th serpentine, b enlarged detail at the end of the

24th serpentine (see a with field of view marked with a red rectangle)

(color figure online)

Microfluidics and Nanofluidics (2020) 24:22

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22 Page 4 of 10

beam passes a focusing and a cylindrical lens. Due to the

presence of the cylindrical lens, the refracted light focuses

on two spatially separated, perpendicularly aligned focal

planes

and

. This leads to an elliptical image shape, as

shown in Fig.2.

To reconstruct the out-of-plane position of a particle

from its aberrated image, an insitu calibration has to be

performed before each measurement. For this, images of

sedimented particles are recorded in a scanning procedure

with

𝛿z=1μm

step size. Then, all recorded images are

pre-processed before evaluation. Following Cierpka etal.

(2010), a median filter with

5 px

filter length is applied to

spatially reduce image noise, in the first step. In the second

step, a bandwidth filter is used to nearly completely elimi-

nate background noise, while keeping structures between

3 and

70 px

diameter. In the last step, images are dewarped

on the basis of the recording of a calibration grid. All pre-

processing steps are performed in the commercial software

DaVis 8.4 (LaVision GmbH).

Separate calibration curves are generated for both cam-

era frames individually to avoid systematic errors that may

result from slightly differing illumination intensities between

both frames. Seven to eight sedimented particles distributed

over the field of view are used for each calibration curve. To

quantify the aberration, the major and minor axis lengths of

the auto-correlation peaks resulting from aberrated particle

images are evaluated. For this, the major and minor axis

lengths of the auto-correlation peak are taken at a certain

contour level height, e.g. at

R∕Rmax =0.2

for

3.55 μm

particles. The axis length ratio is a measure for the particle’s

out-of-plane position (Cierpka etal. 2011; Rossi and Kähler

2014). Exemplary auto-correlation maps for PS particles of

Dp=3.55 μm

diameter are given in Fig.3a–c, where par-

ticles are located at distances

𝛿z=40 μm

𝛿z=0μm

and

𝛿z=−40 μm

from the defined zero position. This zero posi-

tion is defined where

ax∕ay≈1

. The corresponding cali-

bration curve is shown in Fig.3d. Here, the color coding

Objective lens

Particle

Cylindrical

lens

Focusing

lens

Fig. 2 Light path of a fluorescent particle through a microscope with

implemented cylindrical lens causing aberration that results in two

spatially separated and perpendicularly aligned focal planes, leading

to elliptical particle images

Fig. 3 Sample auto-correlation maps of a single

3.55 μm

PS particle

at different out-of-plane positions

𝛿z

relatively to the zero position. a

𝛿z=40 μm

, b

𝛿z=0μm

, c

𝛿z=−40 μm

. The axis lengths

and

of the auto-correlation peak

are taken at a correlation peak height

R∕Rmax =0.2

, d corresponding calibration curve. Red markers

refer to the auto-correlation results shown in a–c. Measurement data

3.55 μm

PS particles at

Re =100

are indicated with black dots

(color figure online)

Microfluidics and Nanofluidics (2020) 24:22

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Page 5 of 10 22

indicates the position of the particle along the z axis, i.e. the

out-of-plane position relatively to the zero position. Fur-

thermore, red dots indicate data points which result from

the auto-correlation maps shown in Fig.3a–c. Black dots

indicate measurement data of PS particles of

Dp=3.55 μm

diameter at

Re =100

, resulting from 500 double-frame

images. For data evaluation, an in-house code is developed

based on the Euclidean method, as proposed by Cierpka

etal. (2011).

To determine the measurement uncertainty with which

the out-of-plane particle position can be reconstructed, abso-

lute z-positions of the centers of sedimented particles are

reconstructed from the calibration recordings and are com-

pared to their theoretical position at

p∕2

. The resulting

standard deviations between reconstructed and theoretical

z-positions are smaller than

2.7 μm

for all particle groups.

Additional analyses also confirmed that particle z-positions

can be reconstructed over the complete microchannel height

with the pre-described procedure.

After the insitu calibration, measurements are performed

inside the 24th serpentine loop at different Reynolds num-

bers of

Re =100, 110, …, 150

. DiCarlo etal. (2007) and

Zhang etal. (2015) define this as

(

max

⋅D

h)∕ν

There, the maximum fluid velocity

umax

is calculated under

the assumption of a plane Poiseuille flow as

umax =1.5

⋅

u

with

u

being the bulk velocity that can be calculated with the

help of the volume flow rate

Q=u

⋅

. As the flow profile

inside the microchannel used in the present study differs

from a plane Poiseuille profile due to finite aspect ratio and

channel curvature, we provide corresponding bulk Reynolds

numbers

Reb

(

u⋅D

∕

𝜈

as well as volume flow rates in

Table1. However, it may be noted that further results are

given as a function of the Reynolds number based on

umax

in order toensure comparability with the studies mentioned

above.

For both, calibration and measurements, a mixture of

3.55 μm

PS particles and either

9.87 μm

PS or MF particles

is injected into the serpentine microchannel. Small and large

particles are distinguished during the evaluation, due to dif-

ferent astigmatism characteristics. These result in individual

calibration curves for small and large particles. Thus, size

fractionation results of

3.55 μm

and

9.87 μm

PS particles can

be obtained simultaneously. Using an outlier criterion, par-

ticle images that exceed a certain Euclidean distance from

the calibration curve are excluded from the evaluation. In a

second measurement, particle distributions of

9.87 μm

particles are obtained and are superimposed with results of

9.87 μm

PS particles at the same Reynolds numbers. This is

done as both, PS and MF particles, assume the same image

diameter and are, therefore, not differentiable through

image processing. As the particle volume concentration in

all experiments is low, that is in the order of magnitude of

𝜑=

(10−4)

, no particle–particle interactions are expected

in the experiments and the flow can be assumed to be unper-

turbed. Assuming negligible particle–particle interactions,

particle locations of both measurement series are assumed

to be independent from each other and representative for the

hydrodynamical behavior of the individual particle group.

Thus, a superposition of particle locations of both measure-

ment series is done here.

During all measurements, 500 double-frame images

are recorded at three reference plane positions, which are

located

𝛿z=20 μm

apart from each other. The time delay

between the first and second frame is adjusted, depending

on the Reynolds number to obtain similar particle image

displacements throughout all measurements. Thus, each

particle appears once in each frame. This measurement

procedure results in a total number of evaluated parti-

cles of

6899 ±1658

(PS,

Dp=3.55 μm

2518 ±615

(PS,

Dp=9.87 μm

) and

2461 ±419

(MF,

Dp=9.87 μm

), respec-

tively, for measurements at a constant Reynolds number.

4 Results

To realize microparticle fractionation, each particle group

has to assume equilibrium particle trajectories that are spa-

tially separated across the flow such that each particle group

may be separated through a flow branching downstream the

fractionation channel. Thus, to validate the capability of the

serpentine microchannel to fractionate particles according to

size and density, three-dimensional particle distributions are

determined in a first step. Afterwards, a virtual flow branch-

ing is performed, i.e. the serpentine channel is divided virtu-

ally into individual segments in the post-processing and frac-

tionation efficiencies are analyzed with respect to varying

virtual partitioning wall positions. Size fractionation results

are analyzed in Sect.4.1; a discussion with regard to density

fractionation can be found in Sect.4.2.

4.1 Size fractionation

An APTV approach is utilized to reconstruct 3D distribu-

tions of polystyrene (PS) particles with diameters of

3.55 μm

Table 1 Flow parameters

at which measurements are

performed

[–]

100

110

120

130

140

150

Reb

[–]

66.6

73.3

86.6

93.3

100

[

ml min

−

]

0.50

0.55

0.60

0.65

0.70

0.75

Microfluidics and Nanofluidics (2020) 24:22

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22 Page 6 of 10

and

9.87 μm

. Figure 4 shows exemplary distributions of

both particle groups at a Reynolds number of

Re =110

The in- and out-flow direction is in negative x-direction and

the channel walls are sketched with black dashed lines. The

region of interest, for which the fractionation performance

is investigated, is indicated with blue dashed lines. Please

note, that Fig.4 shows a superposition of both particle frac-

tions including all reconstructed particle centroid positions

of a time series of 1500 double-frame images. Both particle

fractions were recorded simultaneously.

As shown in Fig.4, each particle fraction forms spatially

separated equilibrium trajectories. In contrast to Zhang

etal. (2014, 2015) that used fluorescence imaging to detect

in-plane particle trajectories in a similar sharp corner ser-

pentine microchannel, the 3D reconstruction reveals that

particle separation takes also place over the channel height

(z-direction) leading to four and two equilibrium positions

for

3.55 μm

and

9.87 μm

particles, respectively. Furthermore,

a slight twist between in-plane equilibrium trajectories of

3.55 μm

particles becomes visible.

The fractionation performance is analyzed in the region

of interest that is sketched with blue dashed lines in Fig.4.

The spatial distribution of particles is given in Fig.5a–f. An

increase of the Reynolds number from

Re =100

Re =150

reduces the spread of the equilibrium trajectories of

9.87 μm

PS particles in y-direction. In contrast to this,

3.55 μm

par-

ticles stay in a four equilibrium trajectory configuration up

to a Reynolds number of

Re =130

with centroid positions

y∕w≈0.33

and

y∕w≈0.8

, respectively. Therefore, they

are located closer to the channel side walls compared to the

larger particles. At

Re =150

3.55 μm

and

9.87 μm

particles

are not spatially separated over the channel width any more

3.55 μm

particle bands are merged, spanning a region of

0.2

y∕w

0.8

. It may be noted that all equilibrium trajec-

tories are displaced towards the bottom of the channel. We

assume that this is due to particle sedimentation.

To evaluate the fractionation performance, the cross-sec-

tional particle distributions of Fig.5a–f are virtually divided

Fig. 4 Measured particle centroid positions of polystyrene (PS) par-

ticles of

Dp=3.55 μm

(red) and

Dp=9.87 μm

(black) diameter at

Re =110

. The particle distribution inside the flow volume between

−200 <x<0

and

200

(sketched in blue) is analyzed (color

figure online)

Fig. 5 Projected view of particle positions of fractions with poly-

styrene (PS) particles of diameter

Dp=3.55 μm

(red) and poly-

styrene (PS) particles of diameter

Dp=9.87 μm

(black) inside the

region of interest. The size fractionation performance is investigated

for segments, which divide the test volume at

y∕

𝛥

, as well as

y∕

−

𝛥

. Here

𝛥y

=0.4

, which corresponds to the optimum size

fractionation performance at

Re =110

. Results are shown for Reyn-

olds numbers of a

Re =100

, b

Re =110

, c

Re =120

, d

Re =130

, e

Re =140

and f

Re =150

(color figure online)

Microfluidics and Nanofluidics (2020) 24:22

1 3

Page 7 of 10 22

into three segments, indicated by two vertical black lines.

These virtual partitioning walls are placed symmetrically

around

y∕w=0.5

at different distances

𝛥y

to the channel

side walls, as denoted in Fig.5a. Here,

𝛥y

is varied between

0.05 <𝛥

y<0.45

in steps of 0.025. This corresponds to a

step size of

5μm

. In each of these three segments, particle

concentrations of both fractions are evaluated by means of

a separation degree

9.87

∕

(

9.87

3.55)

(Stiess 2009).

Here,

denotes the number of PS particles of diameter

becomes zero in regions, where only

3.55 μm

particles are

located and one if only

9.87 μm

particles are included in a

segment. Thus, we obtain three values for

, i.e. one for each

segment. The difference in two neighbouring segments is

defined as selectivity

𝜅y=|Ti+1−Ti|

. In the following, the

selectivity for different

𝛥y

and

is evaluated.

𝜅y

assumes

one if both particle fractions are completely separated. Due

to the fact that particles are almost symmetrically distributed

around

y∕w=0.5

, both values for

𝜅y

are averaged and mean

values

𝜅 y

are plotted as a function of the Reynolds number

and the partitioning wall position

𝛥y

in Fig.6. Here, red

crosses indicate

𝜅 y

values that are calculated for the parti-

cle distributions and partitioning wall positions shown in

Fig.5a–f.

From this representation, a maximum mean selectivity of

𝜅 y=0.981 ±0.010

is found for

Re =110

and

𝛥y

=0.4

, as

also indicated with a bold red cross in Fig.6. This situation

is displayed in Fig.5b. Here,

3.55 μm

particles are located

in the lateral channel sections, i.e. at

y∕w

≤

0.4

and

0.6 <y∕w

≤

, respectively, while the majority of

9.87 μm

particles is located in the channels middle section, i.e. at

0.4 <y∕w

≤

0.6

. The corresponding values for

are shown

in Fig.7.

As already expected from the corresponding particle

distribution shown in Fig.5b,

values switch from a low

level of

T1=0.003

0<y∕w

≤

0.4

to a high value of

T2=0.994

0.4

y∕w

≤

0.6

and vice versa (

T3=0.023

0.6 <y∕w

≤

). This corresponds to a nearly ideal separa-

tion situation.

Velocity measurements at this Reynolds number by

means of APTV reveal that

3.55 μm

particles flow at a

median speed of

(1.25 ±0.21)ms−1

. This is similar to

the median velocity of

9.87 μms

particles, which assumes

(1.27 ±0.10)ms−1

. Interestingly, both particle groups appear

to be faster than the fluid bulk velocity of

u=0.92 ms−1

A further investigation of

7.76 ±0.12 μm

and

9.87 ±0.28 μm

PS particles revealed that if the Reynolds

number, i.e. the volume flow rate in the experiment, can be

controlled accurately enough, size fractionation can be real-

ized even for a size difference as small as

≈2μm

. Specifi-

cally, a mean selectivity of

𝜅 y≈0.9

is obtained at a Reynolds

number of

Re =102

for the particle groups described above.

Higher selectivities could not be reached here, because

9.87 μm

particles are still in a transition state while

7.76 μm

particles are located near the microchannel sidewalls. At

higher Reynolds numbers, when

9.87 μm

particles have

focused,

7.76 μm

particles are in return in a transition state.

4.2 Density fractionation

Up to date, the sharp corner serpentine channel has been only

utilized for size fractionation of bidispers particle systems

(Zhang etal. 2015). In the second part of this study, we use

the serpentine microchannel to test its capability to separate

particles with the same size and different densities. In par-

ticular, polystyrene (PS) and melamine resin (MF) particles

with densities of

𝜌PS =1.05 g cm−3

and

𝜌MF =1.51 g cm−3

and equal diameters of

Dp=9.87 μm

have been chosen.

Particle trajectories of both particle groups were recorded

Fig. 6 Mean selectivities

𝜅 y

3.55 μm

and

9.87 μm

PS particles as a

function of the Reynolds number

and the partitioning wall posi-

tion

𝛥y

. The maximum mean selectivity of

𝜅 y=0.981 ±0.010

reached at

Re =110

and

𝛥y

=0.4

. Red crosses show the fractionation

performances for the particle distributions and partitioning wall posi-

tions shown in Fig.5a–f (color figure online)

Fig. 7 Separation degree

in three segments at a Reynolds number

Re =110

and a partitioning wall position of

𝛥y

=0.4

Microfluidics and Nanofluidics (2020) 24:22

1 3

22 Page 8 of 10

sequentially, while in both recordings

3.55 μm

PS particles

were included as reference. Thus, resulting equilibrium tra-

jectories of both measurements are superimposed for the

same Reynolds number. Particle trajectories for

Re =110

are shown in Fig.8.

Obviously, both PS and MF particles are already in the

transition process from four to two equilibrium trajectories.

In contrast to the size fractionation results, where

3.55 μm

and

9.87 μm

PS particles assume spatially separated in-plane

equilibrium trajectories, PS and MF particles seem to follow

identical trajectories. Figure9a–f display results for Reyn-

olds numbers between

100 <Re <150

Here, both particle groups focus on two trajectories with

increasing Reynolds number, located around

y∕w=0.6

Hence, only low separation degrees

∕

(

PS)

and selectivities

𝜅y=|Ti+1−Ti|

are reached for the Reyn-

olds numbers investigated here. The resulting mean selec-

tivities

𝜅 y

are shown in Fig.10 as a function of the Reynolds

number

and partitioning wall position

𝛥y

Maximum values of

𝜅 y≈0.5

are reached, confirming

that only poor density fractionation can be realized. From

Fig.8a–f, it becomes visible that this value is only reached

due to a spread of single particles located closer to the side

walls. From the present results, we conclude that a particle

density increase of

𝜌MF∕𝜌PS =1.44

is not sufficient to force

particles on spatially separated equilibrium trajectories

after 24 serpentine loops at the investigated Reynolds

numbers for the present serpentine channel. It is known

that particle equilibrium trajectories develop due to iner-

tial migration, i.e. at finite particle Reynolds numbers. If

significant inertial migration effects are present, small

deviations of the force balance will not lead to a significant

shift of equilibrium positions. Such a deviation of the force

balance can only be induced by centrifugal forces for par-

ticles of identical geometry but different density. In the

present study, the particletomicrochannel diameter ratio

assumes

Dp∕

0.123 >

0.07

(DiCarlo etal. 2007) and

the particle Reynolds number can be estimated as

Fig. 8 Scatter plot of the three-dimensional distribution of polysty-

rene (PS, black) and melamine resin particles (MF, cyan) of diameter

Dp=9.87 μm

at a Reynolds number of

Re =110

. The density frac-

tionation performance is investigated inside the flow volume between

−200 <x<0

and

0<y<200

(sketched in blue)

Fig. 9 Projected view of polystyrene (PS, black) and melamine resin

(MF, cyan) particles of diameter

Dp=9.87 μm

inside a

200 μm

vol-

ume as indicated in blue in Fig.8. Results are shown for Reynolds

numbers of a

Re =100

, b

Re =110

, c

Re =120

, d

Re =130

, e

Re =140

and f

Re =150

(color figure online)

Microfluidics and Nanofluidics (2020) 24:22

1 3

Page 9 of 10 22

p=Re ⋅

(

p∕D2

)

for the whole investigated Reyn-

olds number range. As no spatial separation of particle

equilibrium trajectories can be observed in the present

study, we assume that centrifugal forces play a negligible

role compared to inertial migration effects.

5 Conclusion

In the present study, the size and density fractionation per-

formance of a sharp corner serpentine channel was inves-

tigated for defined particle combinations. For this, particle

trajectories in dilute suspension flows were recorded at

different Reynolds numbers after 24 serpentine loops. An

Astigmatism Particle Tracking Velocimetry (APTV) algo-

rithm was utilized to reconstruct three-dimensional particle

positions.

Distinct equilibrium trajectories were observed for all

investigated particle fractions. While

3.55 μm

PS particles

undergo a transition from four equilibrium trajectories at

Re =100

to two equilibrium trajectories that are spread over

the channel width at

Re =150

9.87 μm

PS and MF particles

stay in a two equilibrium trajectories configuration that nar-

rows with increasing Reynolds number. Thus, size fractiona-

tion could be achieved, due to the spatial separation of differ-

ently sized polystyrene particles. Furthermore, in the present

study, the capability for density fractionation in a serpentine

microchannel was investigated. It was found that equilibrium

trajectories of particles with equal size but different densities

do not develop significantly spatially separated equilibrium

trajectories within the investigated parameter range.

The fractionation performance was evaluated in terms

of the separation degree and the selectivity, calculated over

the channel width. For investigated particles of

3.55 μm

and

9.87 μm

diameter, a maximum mean selectivity of

𝜅 y=0.981 ±0.010

was found at

Re =110

utilizing two ver-

tical partitioning walls located at

y∕w=0.4

and

y∕w=0.6

Overall, in the present study, the 3D distribution of micro-

particle equilibrium trajectories is investigated to perform

for the first time a quantitative evaluation of the fractiona-

tion performance in a sharp corner serpentine microchan-

nel. With this methodology, new means of size and density

fractionation may be explored more easily in the future.

Acknowledgements Open Access funding provided by Projekt DEAL.

Financial support from the German research foundation (DFG-SPP

2045) is gratefully acknowledged [HU 2264/3-1

KR 3446/14-1].

Open Access This article is licensed under a Creative Commons Attri-

bution 4.0 International License, which permits use, sharing, adapta-

tion, distribution and reproduction in any medium or format, as long

as you give appropriate credit to the original author(s) and the source,

provide a link to the Creative Commons licence, and indicate if changes

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