C H A P T E R

1
Introduction to microfluidics
Jonathan Cottet and Philippe Renaud
Ecole Polytechnique F
ed
erale de Lausanne (EPFL), Lausanne, Switzerland

Nomenclature kB Boltzmann 1.38
1023 J K1

constant
NA Avogadro 6.022
1023 mol1
CM Clausius-Mossotti factor number
DEP Dielectrophoresis E0 Vacuum 8.854
1012 F m1
EDL Electrical double layer permittivity
EOF Electro-osmotic flow
LOC Lab-on-a-chip
MEMS Microelectromechanical systems
nDEP Negative dielectrophoresis
pDEP Positive dielectrophoresis Variables
Pe Peclet number
Re Reynolds number
Symbol Name Units

A Cross-sectional area m2
a, b Dimensions m
C Capacitance F
List of symbols c0,i Bulk concentration of the mol L1
ion i
Constants Ci Concentration of solute mol m3
d Thickness of dielectric m
layer
Symbol Name Value Units D Diffusion coefficient m2 s1
E Electric field V m1
e Elementary 1.6
1019 C
f Frequency Hz
charge
h Height m
g Gravitational 9.81 m s2
I Electric current A
acceleration
pffiffiffiffiffiffiffi L Length m
j Imaginary 1 –
LD Diffusion length m
number

Drug Delivery Devices and Therapeutic Systems 3 # 2021 Elsevier Inc. All rights reserved.
https://doi.org/10.1016/B978-0-12-819838-4.00014-6
4 1. Introduction to microfluidics

biodefence, molecular biology, and microelec-

P Perimeter m
tronics [1]. With the advent of photolithography
p Pressure Pa
for microelectronics and later on, microelectro-
Pc Capillary pressure Pa
mechanical systems (MEMS), the possibility to
Q Flow rate m3 s1
reduce the size of structures guiding liquids pro-
Rh Hydrodynamic resistance Pa s m3
vided a way forward to successfully displace
rh Hydrodynamic radius m
and analyze molecules as well as cells. Today,
R, r, rext Radius m
the original hope to reduce both the size of lab-
Re{z} Real part of a complex –
oratories and the amount of the sample needed
number z
have been translated into reality with the “lab-
t Time s
on-a-chip” (LOC) devices were both the han-
T Temperature K
dling of fluids and analyses are performed
tD Diffusion time s
directly on chip as presented in Chapter 4.
v Velocity m s1
Microfluidics is defined as the “Handling of
V Voltage V
fluids in technical apparatus having internal
w Width m
dimensions in the range of micrometers up to
zi Valence of the ion –
a few millimeters” [2]. More generally, the term
β Widening angle °
microfluidics is used if one of the dimensions of
γ Surface tension N m1
the microfluidic structure is smaller than 1 mm.
γ Shear rate –
Microfluidics can be distinguished from
Em , Er Relative permittivity –
“macrofluidics” by the fact that some effects vis-
ζ Zeta potential V
η Dynamic viscosity Pa s
ible at the microscale start to be predominant.
θ Contact angle °
Particularly, viscosity starts to play an important
Λ Darcy friction factor –
role and the so-called laminar flow regime,
μi Electrophoretic mobility m2 V1 s1
where adjacent layers of fluids experience little
to no mixing, is a typical characteristic.
ν Kinematic viscosity m2 s1
If one of the dimensions is below 1 μm, the
ρ Density kg m3
term nanofluidics is normally used and corre-
γ_ Shear stress N m2
σi Electrical conductivity S m1
sponds to a world where discrete effects such
ω Angular frequency rad s1
as surface effects have an important impact on
the behavior of fluids and suspended species.
The typical orders of magnitude of parame-
ters in a microfluidic chip are:
- Channel width below 100 μm and height from
1 What is microfluidics? A brief definition 1 to 100 μm
and history - Flow rates from nL s1 to μL s1
- Flow velocity from μm s1 to mm s1 and m s1
- Volumes in the order of pL to μL.
Nowadays the general tendency is to scale
down the sample volume needed for all biosen- Historically the first microfluidic devices
sing or chemical analyses as well as to increase were built in the 1940s for improving separation
throughput and reduce cost with the use of par- in electrophoresis when J. St. L Philpot proposed
allelization. Such developments were enabled to combine laminar flow and an orthogonal elec-
by many fields such as molecular analysis, tric field to separate proteins [3].

I. Microfluidics
 1 What is microfluidics? A brief definition and history 5
Since then, the field of microfluidics has con- 0 1
siderably grown and its many advantages, as vx
presented by Albert Folch [4], can be listed: v ¼ @ vy A (1)
vz
- Flow is often laminar and can be
mathematically modeled. In the field of fluid mechanics and microflui-
- Microchannel sizes are of the same order of dics, vectorial operators are often used. The
magnitude as biological cells, allowing direct operators Nabla r and Laplacian Δ are defined
interactions with cells for seeding and sorting as follows:
as well as for probing a cell and its internal 0 1
∂
components.
B ∂x C
- Automation can be provided on the system by B C  
B∂C ∂ ∂ ∂
microvalves and micropumps. B C
r¼B C¼ , , (2)
- Due to the small size of the fabricated devices, B ∂y C ∂x ∂y ∂z
B C
the batch production cost should be reduced. @∂A
- The amount of reagents used is smaller as well ∂z
as the amount of sample needed.
- Analysis can be performed faster and in So:
0
1
parallel. ∂p
- The compactness of the devices allows B ∂x C
B C  
portability. B ∂p C ∂p ∂p ∂p
B C
- The increase of the surface to volume ratio grad ðpÞ ¼ — p ¼ B C ¼ , , (3)
B ∂y C ∂x ∂y ∂z
allows some processes such as thermal cycling B C
@ ∂p A
to be accelerated.
∂z
Microfluidics is, therefore, a key technology
for biotechnologies in fields ranging from health ∂vx ∂vy ∂vz
(personalized medicine, diagnosis) to biology div ðvÞ ¼ — :v ¼ + + (4)
∂x ∂y ∂z
(cell culture, bioprinting) as well as cosmetics
(emulsions and formulations) and pharmaceuti- ∂2 p ∂2 p ∂2 p
cals (drug discovery). Δp ¼ + + (5)
∂x2 ∂y2 ∂z2
This chapter aims at offering the reader a
broad presentation of the key concepts used in 0 1
∂2 vx ∂2 vx ∂2 vx
microfluidics to provide some guidance for B ∂x2 + +
understanding the forces and effects at play B ∂y2 ∂z2 CC
B C
when designing microfluidic chips. Such under- B ∂2 v C
B y ∂ vy ∂ vy C
2 2
standing is critical for the comprehension of Δv ¼ B 2 + 2 + 2 C (6)
B ∂x ∂y ∂z C
potential failure modes of microfluidics devices B C
B 2 C
and troubleshooting, considerations that will be @ ∂ vz ∂2 vz ∂2 vz A
+ +
described in the following chapters. ∂x2 ∂y2 ∂z2

In particular, the gradient of a scalar quantity p is

1.1 Important mathematical background
noted grad( p) ¼ — p and the divergence of a
and notation vectorial quantity v is div(v) ¼ r . v.
Throughout this chapter, a quantity in bold Vectorial operators expressed in the different
will correspond to a vectorial quantity (1). coordinate systems can be found elsewhere [5].

I. Microfluidics
6 1. Introduction to microfluidics

1.2 Fluid definition The right side of the equation corresponds to the
different forces applied to the fluid particle
A fluid is characterized by the property that it where:
will deform continuously and with ease under
the action of external forces [6]. Both liquids - ρg corresponds to the volumic forces, in this
and gases can be considered as fluids and their case, the impact of gravity on the fluid.
shape will be determined by the container in - — p is a surface force and corresponds to the
which they are held. The field of study of fluid impact of pressure on the fluid.
behavior is called fluid mechanics and is divided - ηΔv is a volumic equivalent of the viscosity
between fluid statics, where fluids are at rest, and forces. The dynamic viscosity is usually
fluid dynamics, where fluids are in motion. The expressed in poise (symbol P or Po) where 1
following section is dedicated to the study of Po ¼ 0.1 Pa s. At 20°C and constant
the equations governing fluids mechanics. In atmospheric pressure, ηwater ¼ 0.01 Po ¼ 1 cPo
most cases, the fluid considered is Newtonian and ηair ¼ 0.018 cPo.
(its viscosity does not depend on the stress
applied and is constant) and incompressible
(div(v) ¼ 0) as it is the case for aqueous solutions. 1.4 Reynolds number
To characterize the fluid regime in fluidics, a
1.3 Navier-stokes equation dimensionless number is often used, the Reyn-
olds number, abbreviated Re, and is defined as
Applying Newton’s second law of motion on the ratio of inertial forces to viscous forces
a small element of a Newtonian incompressible expressed in Eq. (10).
fluid with constant viscosity, we obtain Eq. (7).
ρvL vL
dv Re ¼ ¼ (10)
ρ ¼ ρg  — p + ηΔv (7) η ν
dt
where v is the typical velocity in the microchan-
where v is the flow velocity (expressed in m s1),
nel (m s1) and L is the characteristic dimension
ρ is the density of the fluid (kg m3), t is the time
of the microchannel (m), for example, the diam-
(s), g is the gravitational acceleration (g ¼ 9.81 m
eter for a pipe. Additionally ν, the kinematic vis-
s2), p is the pressure (Pa), and η is the dynamic
cosity (expressed in m2 s1 or stokes (symbol St)
viscosity (Pa s).
where 1 St ¼ 1 cm2 s1), can be defined as (9)
The Navier-Stokes equation is composed of
several terms: η
ν¼ (11)
ρ
- The left term corresponds to the variation of the
speed of a particle moving in space. As the Three different regimes can be distinguished
speed depends on time and position, v ¼ v(r, t). [7–9]:
This derivative can be rewritten as Eq. (8). - If Re < 2300, the flow is laminar.
- If 2300 < Re < 4000, a nonfully developed
dv ∂v turbulence occurs.
¼ + ðv:— Þv (8)
dt ∂t - If Re > 4000, the flow is fully turbulent.
The Navier-Stokes equation can be rewritten as In microfluidics, Re < 2300 and the flow is
Eq. (9). typically laminar. For example, if one considers
∂v a pipe of diameter 10 μm and a water flow of
ρ + ρðv:— Þv ¼ ρg  — p + ηΔv (9) 103 m/s, the calculated Re ¼ 102 ≪ 2300.
∂t

I. Microfluidics
 1 What is microfluidics? A brief definition and history 7

1.5 Flow profile in a cylinder Projecting the Navier-Stokes equation along the
z-axis and expressing the Laplacian in the cylin-
For a low Reynolds number, the Navier- drical coordinate system, we obtain (17).
Stokes equation can be simplified as (12).
 
1∂ ∂vz 1 ∂2 vz ∂2 vz 1 ΔP
∂v r + 2 2 + 2 ¼ (17)
ρ ¼ ρg  — p + ηΔv (12) r ∂r ∂r r ∂θ ∂z η L
∂t
If we consider a permanent flow of a viscous Since v and consequently vz only depend on r,
Newtonian and incompressible fluid between we obtain (18).
 
Pin and Pout with ΔP ¼ Pin  Pout in a horizontal 1∂ ∂vz 1 ΔP
pipe of radius R and length L, as presented in r ¼ (18)
r ∂r ∂r η L
Fig. 1, we obtain (13).
Integrating this equation, we obtain (19).
ηΔv ¼ — p (13)
1 ΔP 2
vz ð r Þ ¼  r + A lnðrÞ + B (19)
Due to the symmetry of the system around the 4η L
z-axis, it is interesting to consider the cylindrical
coordinate system in this example for the veloc- Since vz(r ¼ 0) should be of a finite value and
ity (14). vz(r ¼ R) ¼ 0 (no slip condition), we have (20).
1 ΔP 2
v ¼ vðr Þ ¼ vðr, θ, zÞ (14) A ¼ 0 and B ¼ R (20)
4η L
For the same reason, v does not depend on θ and Therefore (21).
is invariant along the z-axis (15).
1 ΔP  2 2 
v ¼ vðrÞ (15) vz ðrÞ ¼ R r (21)
4η L
The pressure is also constant over a slice of the The result is characteristic of the parabolic flow
pipe (16). profile in the laminar regime and is illustrated in
Fig. 2.
∂p Pout  Pin ΔP
¼ Constant ¼ ¼ (16) The flow rate over a pipe section is called the
∂z L L Hagen-Poiseuille law and is defined as (22).
ð
ð θ¼2π
r¼R

Q¼ vz ðrÞ dr dθ (22)
r¼0 θ¼0
ur
uz
Pin R Pout
r
u r z
v(r)
R
z

FIG. 1 Schematics of a cylinder of radius R and length L

oriented along the z-axis with pressure Pin at the inlet and
Pout at the oulet. FIG. 2 Parabolic flow profile in a cylinder of diameter R.

I. Microfluidics
8 1. Introduction to microfluidics

ð
r¼R
More formulas for different geometries can be
2π ΔP  2 2 
Q¼ R  r r dr found in [6] (parabola), [5] (eccentric annulus),
4η L
r¼0 [11] (triangular and trapezoidal cross-sections
  r¼R which are profiles often generated by microfab-
π ΔP R2 r2 r4
¼  (23) rication techniques), etc.
2η L 2 4 r¼0
Thanks to this analogy, Kirchhoff laws can be
applied in a microfluidic system and the follow-
Therefore (24). ing relationships can be written:
π ΔPR4 - For n resistances in series (26).
Q¼ (24)
8η L
X
n
As presented in (24), the flow rate is propor- RhTotal ¼ Rhi (26)
tional to R4, hence reducing the diameter of a i¼1
factor 2 will reduce the flow rate by a factor 16. - For n resistances in parallel (27).
!1
Xn
1
1.6 Analogy between electrical and RhTotal ¼ (27)
hydraulic circuits R
i¼1 hi

As demonstrated in the previous equation,

the flow rate Q for a pipe is proportional to
1.7 Hydrostatic pressure
the pressure drop ΔP through a proportionality
factor defined in (25) as the hydraulic resistance Hydrostatic pressure is the pressure present
or hydrodynamic resistance Rh (in Pa s m3 or within a fluid in the absence of fluid motion. If
kg m4 s1). the fluid is at rest, the Navier-Stokes equation
can be simplified as the fundamental law of
ΔP 8ηL hydrostatics described in Eq. (28).
Rh ¼ ¼ (25)
Q πR4
ρg  — p ¼ 0 (28)
This equation is completely analogous to
Ohm’s law and is often called the “fluidic This pressure should be considered for systems
Ohm’s law.” Indeed ΔP, the pressure drop, is where a height difference between the reservoir
similar to ΔV, the voltage drop along a wire, and the microchannel exists.
and Q, the flow rate, is similar to the electric
current I through a wire and depends on the
1.8 Viscosity
cross-section of the pipe. Table 1 presents
values of hydrodynamic resistances for differ- “Viscosity is a quantitative measure of a
ent classical cross-sectional shapes considering fluid’s resistance to flow. More specifically, it
a straight channel of length L. determines the fluid strain rate that is generated
Hydrodynamic resistance formula is valid by a given applied shear stress” [12]. Microscop-
only after reaching the hydrodynamic entrance ically it is linked to the cohesive forces between
length, defined as “the duct length required to molecules and it is mostly affected by tempera-
achieve a duct section maximum velocity of ture and pressure. Those forces are normally
99% of the corresponding fully developed mag- larger between liquid molecules compared to
nitude when the entering flow is uniform” [10]. gas molecules.

I. Microfluidics
 1 What is microfluidics? A brief definition and history 9

TABLE 1 Example of hydraulic resistances for different cross-sectional shapes of a straight channel The numerical
values given for Rh are given considering η ¼ 1. 103 Pa s, L ¼ 1 mm, a ¼ 50 μm, b ¼ 20 μm, h ¼ 100 μm and w ¼ 500 μm.
P is the perimeter and A the cross-sectional area
Rh Rh
Shape Figure Formula [1011 Pa s m23]

Circle 8ηL 4.07

πa4

 2
Ellipse b b 36.92
1+
a a 1
π ηL  3 a4
4
b
a

Triangle pffiffi ηL
320 295.6
a a 3 a4

a
Two plates w 12 hηL
3w
0.24
h

Rectangle w 
12  ηL 0.27
h h3 w
h 10:63
w

Square h 28:4 ηL
h4
2.84

Concentric annulus 8ηL 0 1 1 19.9

π  2 
2 2
B 4 4 a b C
a @a b  a A
ln
b b

Arbitrary  2ηL AP 3
2
–
P
A

Adapted from H. Bruus. Theoretical Microfluidics, OUP, Oxford, 2008; F.M. White. Viscous Fluid Flow, McGraw-Hill, New York, 2006.

1.9 Flow through porous media

Many applications in microfluidics require and gels. In this case, part of the fluid flows
the fluid to go through different porous mate- through the material while the rest of the fluid
rials such as paper, membranes with nanopores, is trapped in pores. The law governing the flow

I. Microfluidics
10 1. Introduction to microfluidics

of a liquid in a porous material is known as

Darcy’s law [13] and is expressed in (29). ∂Ci
J ¼ D (31)
∂x
κA
Q¼ ΔP (29) where Ci is the concentration of solute and D is
ηL
the diffusivity also called the diffusion coeffi-
where Q is the flow rate (m3 s1), κ is the perme- cient (in m2 s1 in SI units or cm2 s1 in CGS
ability of the medium (m2), A is the cross- units) and is defined by the Stokes-Einstein rela-
sectional area of the flow (m2), and η is the tion (32).
dynamic viscosity (Pa s). kB T
D¼ (32)
6πηrh
1.10 Drag forces where kB ¼ 1.380649
1023 J K1 is the Boltz-
For a particle immersed in a moving liquid, mann constant, T is the temperature (K), η is
the fluid will exert a force, called hydrodynamic the dynamic viscosity, and rh is the hydrody-
viscous drag force, on the nonmoving particle namic radius of the particle (often approximated
that will affect its velocity [14]. The fluid motion by the radius of the particle rext).
will cause this force to pull the particle along. If Fick’s second law of diffusion (33) is related to
the particle is at the fluid velocity, no force is the influence of diffusion on the change of
applied to the particle. As the flow in microsys- concentration.
tems is usually laminar since the Reynolds num- ∂Ci ∂2 Ci
ber is small, this regime is called creeping flow ¼D 2 (33)
∂t ∂x
or Stokes flow.
The expression of the drag force on a spheri- The average diffusion length LD in 1D is
cal particle, also known as Stokes law, is (30). expressed in (34).
pffiffiffiffiffiffiffiffiffiffiffi
F Drag ¼ 6πrext ηv (30) LD ¼ 2DtD (34)
where tD is the diffusion time.
where rext is the external radius of the particle, η
Another adimensional number comparing
is the dynamic viscosity of the medium, and v is
the different transport mechanisms is often
the fluid velocity relative to the particle.
used: the Peclet number, Pe, which can be
The constant term in front of v is called the
defined as the rate of advection (transport by
friction factor and depends on the particle geom-
the fluid) to the rate of diffusion (35).
etry [14, 15]. 6πrextη corresponds to the friction
factor of a sphere. More coefficients for different vL
Pe ¼ (35)
geometries can be found in Ref. [15]. Such forces D
should be considered for liquid containing The higher the Peclet number, the more advec-
particles. tion dominates over diffusion. For Pe ≫ 1, con-
vection dominates and large concentration
gradients are possible whereas for Pe ≪ 1, diffu-
1.11 Diffusion sion dominates and concentration is uniform in
Diffusion is the macroscopic result of the ran- the volume.
dom thermally driven motion of particles. It is
described by Fick laws. Fick’s first law of diffu-
sion is expressed in Eq. (31).

I. Microfluidics
 1 What is microfluidics? A brief definition and history 11

1.12 Surface tension - If θ > 90° and the liquid is water, the surface is
said to be hydrophobic.
Surface tension is “the property of the surface - If θ < 90° and the liquid is water, the surface is
of a fluid that causes its surface to be attracted to said to be hydrophilic.
another surface” [4, 16], expressed in N m1 in SI
units. For liquids, surface tension is equivalent For such a system, we can also write the law
to the surface energy. of Young-Dupr
e (38).
If we consider a drop of fluid or radius R
immersed in another fluid and P1 the pressure γ sg  γ lg cos ðθÞ  γ sl ¼ 0 (38)
inside the drop and P2 the pressure outside,
where γ sl, γ lg, and γ sg correspond, respectively, to
we obtain the Young-Laplace equation (36).
the surface energy between solid-liquid, liquid-
γ gas, and solid-gas.
P1  P2 ¼ ΔP ¼ (36)
R It is important to notice that surface tensions
where γ is the surface tension (N m1), R is the are temperature sensitive and in most cases, the
radius of the drop of fluid (m), and P is the pres- surface tension decreases with an increase in
sure (Pa). temperature.
For a three-dimensional interface of arbitrary
shape between two fluids, the Young-Laplace
equation [17] becomes (37). 1.13 Capillary action
 
1 1 Capillary action is the result of surface ten-
P¼γ + (37)
R1 R2 sion and corresponds to the ability of a liquid
to flow spontaneously through thin pipes. If
where R1 and R2 are the two radii of principal
we consider a capillary filled with liquid as illus-
curvature at the point considered and γ is the
trated in Fig. 4, the capillary pressure obtain is
surface tension.
presented in Eq. (39).
If we now consider a drop of liquid on a sur-
face surrounded by gas as illustrated in Fig. 3, 2γcosðθÞ
we obtain two possible cases depending on the ΔPc ¼ (39)
R
value of the angle between the solid-liquid inter-
face and the liquid-gas interface called “contact where γ is the interfacial tension and R is the
angle” θ: radius of curvature of the interface.

= lg
Liquid
Gas Gas
Liquid
sg sl

Solid Solid
(A) (B)

FIG. 3 Forces at the contact point. (A) Hydrophobic surface if θ > 90° and (B) Hydrophilic surface if θ < 90°. γ sl, γ lg, and γ sg
correspond, respectively, to the surface energy between solid-liquid, liquid-gas, and solid-gas, respectively.

I. Microfluidics
12 1. Introduction to microfluidics

Gas Similarly, if a rectangular channel of section w

and height h locally widens with an angle β as
R presented in Fig. 5, the external pressure needed
to overcome the pressure difference and restart
the flow, also called burst pressure, is presented
in Eq. (41) [18].
2γ w
ΔPc ¼  cos ðθÞ  cos ðθ + βÞ (41)
w h
Liquid

1.14 Head losses

For a real fluid, friction causes energy dissipa-
tion. This dissipation, called head loss, can be
divided between “major losses” caused by
FIG. 4 Illustration of the capillary rise of a liquid in a energy dissipation per length of pipe and
hydrophilic capillary with R the radius of curvature of the
interface and θ the contact angle.
“minor losses” caused by changes of cross-
section, bending, valves. They result in an equiv-
alent loss of pressure in the pipe. For more
details see Chapter 4 on LOC.
For water at 20°C, in air, γ ¼ 7.28 102N m1. Major head losses are characterized by the
Assuming a 1 μm radius glass channel with Darcy-Weisbach equation (42).
θ ¼ 45 degrees, we obtain ΔPc ¼ 1 bar.
If the contact angle is locally changed from ρv2
ΔP ¼ Λ (42)
θ1 < 90 degrees to θ2 > 90 degrees, the associated 2
variation of capillary pressure is expressed in (40). where Λ is the Darcy friction factor.
2γ gl For short pipes, minor losses can exceed the
ΔPc ¼ ðcos ðθ2 Þ  cos ðθ1 ÞÞ < 0 (40) major losses and can be estimated through tables.
R
The flow will stop at the beginning of the hydro-
phobic section: such an interface is also called a
1.15 Non-Newtonian fluids
capillary stop. External pressure should be Non-Newtonian fluids are fluids where the
applied to overcome the pressure difference assumption of constant viscosity does not apply.
and restart the flow. Such cases can be modeled by Eq. (43).

w
Liquid Gas

FIG. 5 Illustration of a capillary valve created by a widening of a rectangular microchannel of width w and height h.

I. Microfluidics
 2 Fluids in electrical fields 13

σ ¼ m_γ n (43) 2 Fluids in electrical fields

where σ is the shear stress and γ_ is the shear rate. 2.1 Electrophoresis
The apparent viscosity can be expressed by the
Ostwald-de Waele power-law model (44). Electrophoresis corresponds to the move-
ment of charged particles in a liquid under an
σ applied electric field. It is often used to separate
η ¼ ¼ m_γ n1 (44) small ions and charged molecules (small and
γ_
large such as DNA or proteins). Each charge is
attracted by the electrode with the opposite
The behavior of the fluid depends on the value of
charge. Because of friction forces, the maximum
n, as illustrated in Fig. 6.
velocity vi reached by an ion in an electrical field
If n < 1, the fluid is non-Newtonian and dis-
E is expressed in Eq. (45)
plays a shear-thinning behavior also called pseu-
doplasticity. Examples of such fluids are polymer zi e
solutions or blood where the content (molecules vi ¼ E (45)
6πri μi
or particles) is deformable and get stretched out
with an increase in shear stress, resulting in the
decreased viscosity. where zi is the valence of the ion, e is the elemen-
If n ¼ 1, the fluid is Newtonian and η ¼ m. tary charge (e ¼ 1.6
1019 C), ri is the radius of
If n > 1, the fluid is non-Newtonian with shear- the ion, and μi is the electrophoretic mobility.
thickening or dilatant behavior. Such behavior The expression of the electrophoretic mobility
is displayed by some polymer solutions. for large molecules is given by Eq. (46).
Typical values for different fluids can be
v ζD
found in Chhabra and Richarson [19]. μ¼ ¼ (46)
E 4πη

Shear stress σ (Pa)

Dilatant Newtonian
n>1 n=1

Pseudoplastic
n<1

.
Shear rate g [s–1]

FIG. 6 Illustration of the evolution of the shear stress as a function of the shear rate for the different class of fluids: Dilatant
(n > 1), Newtonian (n ¼ 1) and Pseudoplastic (n < 1).

I. Microfluidics
14 1. Introduction to microfluidics

where ζ is the zeta potential and D the diffusion and mobile layers corresponding to the shear
coefficient. plane. A more precise explanation taking into
account the solvent molecules can be found in
Pardon and van der Wijngaart [21].
2.2 Electrical double layer The Debye length is the characteristic thickness
For a surface in contact with a solution, the of the Debye layer and corresponds to the distance
charge of the surface will depend on the pH of over which the surface charges influence the distri-
the solution. For instance, for a glass surface bution of charges in the solution. For a monovalent
composed of silanol groups (-Si-O-H) immersed electrolyte, the Debye length is given by (47).
in water with neutral pH, the hydroxyl groups
will lose a proton and the surface will become sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Em E0 k B T
negatively charged (-Si-O). The charge imbal- λD ¼ X (47)
3
ance will be compensated by counter ions (also 10 NA e2 i c0, i z2i
called counterions) from the solution, in the
example positive ions, as presented in Fig. 7. where Em is the relative permittivity of the liquid,
This layer, also called the Debye layer or electri- E0 ¼ 8.854
1012 F m1 is the vacuum permittiv-
cal double layer (EDL), is composed of two sub- ity, kB is the Boltzmann constant, T is the temper-
layers as described by Barz and Ehrhard [20]. ature, NA ¼ 6.022
1023 J K1 is the Avogadro
The inner layer contains a fixed Stern layer number, e is the elementary charge, c0, i is the bulk
and an almost immobile shear layer. The outer concentration of the ion i (in mol L1), and zi is the
layer, called the diffusive layer, contains ions corresponding valence of this ion.
subjected to electrostatic interactions but still
mobile. The zeta potential is defined as the
potential at the interface between immobile 2.3 Electro-osmosis
If a pair of electrodes is used to apply an
Stern Shear external electric field, the counterion layer is
plane plane moved (toward the negative electrode for posi-
- + + tive counterions for example). Because of the
- + - - short distance between the molecules, the ions
+ + - +
- + +
Charged material

- + + viscously drag the solution with them at the

+ - - + -
- + - same speed. The flow profile, in this case, is flat
+ +
- + - and is commonly called the “plug flow profile”
+ + + - +
- + + -
+ - + - - as presented in Fig. 8 (in opposition to the para-
- -
- + + - - + bolic flow profile in pressure-driven flow
+ +
- + + + presented in Fig. 2). This principle, called
electro-osmotic flow (EOF), is commonly used
Stern layer Bulk
Shear layer in capillary electrophoresis systems.
diffusive layer It is important to notice that both electro-
Debye length D
osmosis and electrophoresis happen at the
same time. EOF pumping is strongly dependent
FIG. 7 Schematic of the double layer composed of an on the surface charges and is therefore influ-
inner layer (Stern layer and shear layer) and the outer layer
named diffusive layer. Adapted from D.P. Barz, P. Ehrhard. enced by adsorption and desorption. Typical
Model and verification of electrokinetic flow and transport in a devices using EOF pumping are presented in
micro-electrophoresis device. Lab Chip 5(9) (2005) 949–958. Chapter 3.

I. Microfluidics
 2 Fluids in electrical fields 15
- - - - - - - - - - - - - - -
+ + + + + + + + + + + + + + +

+
- + - - + -
-
- - + +
+ -

+ + + + + + + + + + + + + + +
- - - - - - - - - - - - - - -

FIG. 8 Illustration of the electroosmotic flow in a microchannel between two electrodes located at each end. The flow is
driven toward the negative electrode.

pDEP nDEP
E E

a b a b

Fb m
Fa m Fb

Fa

FIG. 9 Principle of (A) pDEP and (B) nDEP. E represents the electric field, m the dipolar moment, and Fa and Fb the Coulomb
force (F ¼ qE) on each barycenter of the charges on each side of the particle.

2.4 Dielectrophoresis where CM( f ) is the Clausius-Mossotti factor

(49).
Dielectrophoresis (DEP) relies on the differ-
ences in dielectric properties between cells and ε∗p  ε∗m
CMð f Þ ¼ (49)
their suspension liquid [22, 23]. Cells placed in ε∗p + 2ε∗m
a nonuniform electric field can, therefore, be
either attracted in the high field region (positive and ε∗i is the complex permittivity (50).
dielectrophoresis or pDEP) or repelled in the jσ i
low field region (negative dielectrophoresis or ε∗i ¼ εi ε0  (50)
ω
nDEP) as illustrated in Fig. 9.
The time average expression of the DEP force where εi is the relative permittivity, ε0 is the vac-
exerted on a polarizable particle (index p) of uum permittivity, σ i is the electrical conductivity
radius rext immersed in a suspending medium (S m1), rext is the external radius of the particle,
(index m) in a nonuniform electric field E is ω ¼ 2πf is the angular frequency with f, the fre-
expressed in (48). quency of the electric field E. Tools are available
for simulating the behavior of particles in a
FDEP ¼ 2πε0 εm r3ext RefCMð f Þg— E2RMS (48) suspending medium [24].

I. Microfluidics
16 1. Introduction to microfluidics

DEP can, therefore, be used for various appli- predictable and, at low Reynolds numbers, the
cations such as cell centering [26], cell separation laminar flow regime dominates. Modifications
[27], cell sorting [28], or cell aggregate in channel dimensions or surface properties
creation [25]. can then be used to generate or stop the liquid
flow. Furthermore, liquids as well as ions, mol-
ecules, and cells can be moved or separated by
2.5 Electrowetting the use of an electric field.
Electrowetting utilizes the electric field to Most of the phenomena that occur in microflui-
modify the wettability of an electrolyte on a sur- dics are described by the equations of fluid
face. It is defined as a “change in solid- dynamics presented in this chapter. However,
electrolyte contact angle due to an applied when at least one dimension is below 100 nm, dis-
potential difference between the solid and the crete effects are predominant and define the
electrolyte” [29]. world of nanofluidics. For example, if the Debye
If we consider a dielectric of thickness d and length—the thickness of the double layer—is sim-
relative permittivity εr covering the electrode, ilar or larger than the channel size, the flow will be
the contact angle θ0 is modified (51). mostly controlled by surface charges. Addition-
ally, as channel size becomes even smaller, sur-
CV 2 face to volume effects start to be even more
γ lg cos ðθ0 Þ ¼ γ sg  γ sl + (51)
2 significant. Surfaces, indeed, play a major role
in nanofluidics and surface energy is paramount.
where C ¼ εrdε0 is the capacitance of the interface In channels below 50 nm, forces between atoms
and V the applied voltage. and molecules—called van der Waals forces—
The principle is often used for electrowetting need to be considered. At this scale, for electroos-
on dielectrics (abbreviated EWOD) where the motic flows, the electrostatic forces between
base plate consists of an array of individually
surface charges and a fluid containing charged
addressable electrodes, coated with a hydropho- particles will influence the flow dynamics. Addi-
bic dielectric layer (e.g., Teflon thin film) and the tionally, density as well as charge distribution is
top plate consists of a single ground electrode, not constant and is affected by intermolecular
coated with a hydrophobic dielectric layer. The interactions. This field of nanofluidics is foreseen
space between the two plates is filled with an to open a new range of possibilities such as single-
oil phase in which water droplets are inserted molecule analysis and manipulation, following
and manipulated. This concept is now known rapid kinetics of chemical reactions.
as digital microfluidics [30] and allows for the
fluid motion to be completely automated but
has some disadvantages such as the cost of the
microfluidic chip.
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