Download 2. Electrical forces in the bulk: Injection, Conduction and Induction

2. Electrical forces in the bulk: Injection, Conduction and Induction
EHD micropumps
The electrical body force density in a liquid is [1]
Coulomb force
Dielectric force
Electrostriction
The electrostriction term can be incorporated into the pressure for an incompressible
fluid and can be ignored (it does not have an effect in the mechanics). A simple way of
seeing that the electric field exerts forces on the fluid is to consider that the forces on
the free charges and dipoles are transmitted directly to the liquid [2]. The Coulomb
force represents the force on the free charge. It is the main term responsible for
generating pumping in EHD.
The dielectric force represents the force on the dipoles:
where P = np is the polarization (n, number density; p, dipole moment of a molecule).
The dielectric force is important in hydrostatic equilibrium. The dielectric siphon is an
example. However, the dielectric siphon is not a pump that can move fluid in a loop or
between two points at equal pressure. The dielectric force is relevant in some induction
pumps. The term responsible for generating pumping in Electrohydrodynamics is
primarily the Coulomb force ρE. There are three main mechanisms of generating free
charge in the bulk of a dielectric liquid: (a) injection of ions in a very insulating liquid
from electrodes; (b) generation of nonequilibrium charged layers close to the electrodes
because the rate of dissociation of neutral molecules exceeds the rate of recombination
of ions [3]; and (c) induction of charges in a conducting liquid due to gradients of
electrical conductivity. These three mechanisms give rise to three kinds of pumps.
2.1 Injection Pump
The injection pump, also known as ion-drag pump, uses the interaction of an electric
field with electric charges injected into a dielectric fluid. The electric field is imposed
between an electrode called emitter and another called collector. The ions, travelling
from the emitter to the collector, drag by friction the working fluid. This pumping
mechanism has been known for quite some time. Stuetzer [4] and Pickard [5] were
among the first who studied theoretically and experimentally this EHD pump. In
microsystems, Richter and Sandmaier [6] and coworkers (1990,1991) fabricated an iondrag EHD micropump, consisting of pairs of metallic planar grids through which the
pumped fluid moves. Ahn and Kim (1997) [7] experimented with an ion-drag pump that
consisted of pairs of coplanar microelectrodes. Darabi et al. (2002) [8] tested different
designs of coplanar microelectrodes for ion-drag pumping: emitter electrodes of planar
shapes, of saw-tooth shapes or of saw-tooth shapes with bumps.
D
X+
u
E(x)
X+
u
X
X
emitter
collector
x=0
x=L
2.1.1 Pump Principle
Let us consider two plane-parallel perforated electrodes placed perpendicular to the axis
of a pipe of circular cross section along which the liquid is pumped. A potential
difference V is applied between electrodes. One electrode is the emitter and the other is
the collector. The left-right symmetry can be broken by: 1- the difference between
molecule-ion energy barriers at emitter and collector, 2- different geometry between
emitter and collector. Simplifications: 1D problem, diffusion negligible, unipolar
injection (only one kind of ions is present). The force per unit area generated by the
pump is
The 1D current density is, neglecting diffusion,
where u is averaged liquid velocity in the cross-section of the pipe. This velocity is a
result of Δpg and the hydraulic resistance of system.
The electric field can be integrated
and the charge density is
For these equations u>0, that is, the flow is generated by the pump. From the emitter to
the collector, the charge density decreases while the field strength E increases. The
injection of charge decreases the electric field at the emitter, since each additional
unipolar charge already present in the pipe repels the incoming charge. When the field
E0 is too low, charge can no longer be removed from the emitter, and the pump reaches
its space charge limit (SCL). In this situation, the charge density at the emitter is quite
large.
ρ(x)
E(x)
x=0
x=L
For SCL emission, E0 =0 and the pressure generated is maximised Δpg
Let us analyse two limits: fluid velocity much smaller than ion-migration velocity and
fluid velocity much greater than ion-migration velocity.
u << μ E
(E0=0, or << V/L)
Two limits:
u >> μ E
We can determine the current density j from
Case u<<μE:
The generated pressure (or maximum pressure) is:
And the maximum flow-rate and maximum velocity are:
Case u >> μE:
The generated pressure is
And the maximum flow-rate is
(E0=0, or << V/L)
The maximum pressure that the pump can delivered but at negligible Q is
For the general case (u ≠ 0 ) and under SCLE condition, E0= 0 , the equations provide j
and V as functions of EL and u.
and the generated pressure as
Generated pressure is
function of V and Q
The generated pressure by the pump is used in overcoming the internal hydrodynamic
resistance of the pump plus the external load: Δpg=RinQ+Δpout. It is now possible to
obtain the flow rate Q versus the external back pressure +Δpout at different applied
potentials V.
2.1.2 Energy Efficiency
The power consumption must take into account that, in practice, there are always other
ions, different from those injected, that come from dissociation of impurities or of the
liquid itself. According to Pickard [5] dissociation will produce equal amounts of
positive and negative ions and their contribution to the space charge should be
negligible. The pressure heads can be relatively well accounted for by emission theory,
while the total current is not so well explained.
Power
consumption
as given by
unipolar injection
residual conductivity
The energy efficiency defined as eff=(maximum hydraulic power at V=Const)/(power
consumption ) for the case u << μE with no residual conductivity is
The energy efficiency for the case u >> μE with no residual conductivity is
For general u, we can write
There are two origins for α in this expression. For high efficiency α should go to zero.
2μV L
For high efficiency, fluid velocity should be much greater than ionumax
migration velocity.
σL ε
α2 =
Also, for high efficiency, fluid must go from emitter to collector before
umax
charge has time to relax.
α1 =
2.1.3 An Example
Darabi and Wang (2005) [9] working with refrigerant HFE-7100 at 180 V obtained:
- pressures of the order of 300 Pa; - flowrates of the order of 30 mm3/s; - a maximum
energy efficiency of around 0.0015
They used sharp edges in order to reduce voltage required for ion injection, helping to
break left-right symmetry. If we remind the current-voltage characteristic:
1. For applied fields up to around 1 kV/cm current is linear with voltage. The
simplest model is that of dissociation/recombination of ion pairs : [A+B-] <=>
A++B- and neutralization of ions at the electrodes.
2. Up to about 10 kV/cm the current is sub-linear due to saturation of ion
generation.
3. At high fields the current increases rapidly. The current can increase rapidly by
enhancement of dissociation by field effect. There is a certain voltage at which
ion injection starts to be important.
4. At 40-100 kV/cm or beyond, electrical breakdown occurs. Darabi and Wang
observed electric breakdown at about 11 kV/cm.
2.1.4 Working Liquids
Typically, the liquids that are candidate to be pumped by ion-drag micropumps are
highly insulating liquids. An upper bound for the conductivity of the liquid to be
pumped comes from comparison between the time needed for the ion to go from the
emitter to the collector and the charge relaxation time ε/σ. If the charge relaxation time
is too short, the charge is screened before enough force is transmitted to the liquid.
Therefore, we do not expect that ion-drag pumps can actuate liquids with conductivities
much greater than σ >> vε/L, where v is the ion velocity. Liquids with σ ≤ 10-7 S/m have been
pumped (alcohol, mineral oils, CFC refrigerant...). The highest fluid velocities are predicted for
liquids with high dielectric constant and low viscosity. However, high dielectric constant also
means that they have conductivities too large to allow efficient operation. Liquids of high
dielectric constant should be deionized, in order to be used for ion-drag pumping.
2.1.5 Problems
•
•
•
High electric field strengths can change liquid electrical properties, and can
cause corrosion of the electrodes. These two effects can affect run-to-run
repeatability and reduce pump life.
However, some liquids like liquid nitrogen can be ideal working fluids for ion
drag pumping. According to Darabi and Wang slight electrode degradation was
observed using the HFE-7100 fluid; however, no noticeable electrode
degradation was observed using liquid nitrogen. It seems that the good
performance of liquid nitrogen is due to its very stable molecular structure.
According to Zhakin [10], it is possible to add ionizing admixtures in order to
have regenerative red-ox reactions.
2.1.6 Applications
•
•
Potential application is its use in pumping fluids in cryogenic cooling
microsystems (Darabi Wang 2005, Foroughi et al 2005, Lee et al 2007).
Cryogenic cooling has become a widely adopted technique to improve the
performance of electronics and sensors. Super-conducting devices, which are
used to increase signal-to-noise ratio in communications, must also be
maintained at cryogenic temperatures. Micropumps capable of pumping liquid
nitrogen would enable the development of compact and lightweight chip
integrated cooling systems.
Other promising applications for ion-drag micropumps include fuel injection
loops, and gas and liquid pumping where small quantities of dielectric fluids
need to be pumped.
2.2 Conduction Pump
Under the weak electric field regime (field much less than 107 V m-1), the conduction in
a dielectric liquid is carried out by ions generated by dissociated molecules. The ions,
that reach the electrodes, are assumed to discharge there. Electric conduction in a pure
dielectric liquid can be explained with a simple model, which considers a reversible
process of dissociation-recombination of a neutral species (denoted by XY) into
univalent positive X+ and negative Y− ions. These ions generated in the bulk move by
migration towards electrodes. Near them, charged layers appear due to ion generation
not balanced by recombination. The charged layer sign is opposite to that of the
adjacent electrode. The electrical force on these layers is responsible of the fluid
motion. Because the typical thickness of these charged layers is much greater than the
Debye length, we include this kind of pump among those whose mechanism is due to
electric forces acting in the bulk. This kind of pump has been realized experimentally in
millimetre dimensions [3, 11]. Pearson and Seyed-Yagoobi (2009) [12] reports a microscale EHD conduction pump (2009).
V
E(x)
X+
Y
u
D
X-
XYÙX++Y-
u
X
x=0
x=L
2.2.1 Pump Principle
The EHD conduction pumping mechanism can be illustrated considering two parallel
perforated electrodes immersed in a dielectric liquid. We are going to consider that there
is no charge injection in our analysis. This is based upon the model of J.J. Thompson for
gases. Simplifications: 1) 1D problem; 2) bipolar conduction model with n0 >> n+ + n- ;
3) negligible diffusion and convection; 4) ions discharge at electrodes, no injection.
equations
At equilibrium equilibrium, kdn0=kr n+ n- and n+= n-=neq. The boundary conditions
for the previous equations are the following:
n+ = 0 at x=0
n- = 0 at x=L
Coulomb repulsion
between electrodes and
co-ions
These equations were solved by Thomson and Thomson [13]. They provided an
approximate solution valid for the quasi-ohmic regime, i.e., when the thickness of the
heterocharge layer, λ, is much smaller than the distance between electrodes L. In this case, the
liquid is electroneutral in the bulk except for the heterocharge layers. The approximation
consists on neglecting the recombination in the heterocharge layers (to neglect krn+n-). In this
situation, the conductivity σ=e(μ++μ-)neq and electric field Eb are constant in the bulk.
Near the electrode at x=0 we have:
The electrical current is constant along x
For x>λ+ the liquid is assumed to be electroneutral
and equivalently
The thickness of the charged layer can be seen as the distance an ion travels without
recombination, taking into account that tr=1/krneq=ε/σ is the charge relaxation time
(Langevin [14]).
Gauss’s law can be integrated for x<λ+:
using
and
E(x)
n-(x)
x=0
λ+
thickness layers:
n+(x)
x=L
λ-
There is a Coulomb force directed towards the electrodes:
For λ<<L, we have at layer near x=0:
and at layer near x=L:
The total generated pressure is:
In experiments, left-right symmetry is broken using geometry. It works even for μ+=μV
perforated
electrode
ring electrode
E(x)
D
u
u
heterocharge
layers
x=0
(Jeong and
Seyed-Yagoobi,
2004)
x=L
The Coulomb force in the charge layer adjacent to the perforated disc electrode is mainly axial
while the electric force in the charge layer adjacent to the ring electrode is mainly radial. The
radial force is balanced by the pipe wall. Experimentally in the point-plane configuration, the
conduction pump drives the liquid in the opposite direction to the ion-drag pump. Therefore,
increasing the voltage flow reversal is observed when ion injection starts to play a role.
For the configuration of perforated and ring electrodes [11], the pressure generated at zero flow
rate is
which is of the order of the expression previously obtained for the pressure generated at only
one electrode. For the maximum flow-rate, if we can neglect the convection current in front of
the conduction current, the expression is
2.2.2 Energy Efficiency
For λ << L, we can estimate the electric input power IV ≈ σV 2S/L . The energy
efficiency is then
The combination of parameters uε/σL is called the electric Reynolds number. In the
conduction regime, the convection of current is usually negligible; therefore, this
number is typically very small. The efficiency decreases with ionic concentration.
For the experiments presented in [11] at V = 10 kV, the maximum efficiency was
eff~10-3.
2.2.3 Examples
Feng and Seyed-Yagoobi (2004) working with refrigerant HCFC-123 at 10 kV
(millimeter scale micropump) obtained:
- Maximum pressure around 270 Pa.
- Maximum flow velocity around 5 cm/s, maximum flow-rate around 4000 mm3/s.
- Power consumption around 0.25 watts.
- Energy efficiency around 0.001.
- λ ~ 100 μm (for 10 kV) while λD ~ 1 μm
Pearson and Seyed-Yagoobi (2009) working with refrigerant HCFC-123 at 4 kV
(micrometer scale micropump) obtained:
- Maximum pressure around 500 Pa.
- Power consumption around 90 mW.
2.2.4 Working Liquids
Liquids of low conductivity can be used as working fluids. An upper bound for σ comes
from the thickness of the heterocharge layer as compared to the Debye length. If the
conductivity is high, both layers (diffuse and heterocharge layers) can have similar
thicknesses. In this case, the diffusion processes are not negligible and the
approximations are not applicable.
For conductivities much smaller than this, we cannot speak about Conduction Pump.
A lower bound for σ comes from the comparison between the heterocharge layer
thickness and the distance between electrodes L [3]. If λ >> L recombination is
negligible in the bulk, the current saturates and the electric field does not vary
appreciably between electrodes. The generated pressure is small because it comes from
increments of εE2. Tentatively, we can say that a lower bound for σ is 0.01μEε/L.
The range of conductivities that we obtain for given conditions is
L=10-4 m, V=10 volt
3 x 10-11 S/m
10-6 S/m
2.2.5 Problems
The voltages can not exceed the threshold voltage for charge injection, which may limit
the pressure and flow generated. To limit the charge injection, the use of electrodes with
small radii of curvature should be avoided. As an advantage, the working fluid and
electrodes are not subjected to the degradation that ion injection usually generates.
2.2.6 Applications
As for the ion-drag micropump, a potential application is its use in pumping fluids for
heat transfer in microsystems (Pearson and Seyed-Yagoobi (2009) [12])
2.3 Induction Pump
In this way of actuation, the liquid is quasi-electroneutral, meaning that n+-n-<< n++n-.
Charge is induced in inhomogeneous liquid by the electric field, and the electric force
upon this induced charge is transmitted to the liquid. Although the charge density is
very small, the force acting on this residual charge is sufficient to drive the liquid.
Usually, spatial variation of temperature is imposed in order to have gradients in
conductivity and permittivity. The common induction pump consists of an array of
coplanar electrodes subjected to a travelling wave potential together with a vertical
gradient of temperature. This temperature gradient can be imposed externally or induced
by Joule heating.
Melcher and Firebaugh (1967) [15] studied it first for semi-insulating liquids. In
microfluidics, Fhur et al (1992) [16] employed Joule heating in order to generate the
temperature gradient. Felten et al (2006) [17] realized an induction micro-pump that
employed external gradient of temperature.
typical
experimental
geometry
Liquid
—T
u
w
Glass Substrate
y
0º
90º 180º 270º
0º
90º 180º 270º
h
u
z
x
Unidirectional motion can also be obtained with an array of electrodes subjected to a single
phase signal with an imposed longitudinal temperature gradient.
2.3.1 Pump Principle
Let us consider a simple geometry: two parallel perforated electrodes subjected to a
potential difference inside a circular capillary. The electrode at x = 0 is at temperature
T0 and the electrode at x=L is at temperature TL. Solving the diffusion equation for
temperature, a linear temperature profile is achieved between electrodes in the steady
state, for negligible convection of temperature.
V
T =T0
D
T =TL
E(x)
u
u
j=σE
x=0
x=L
Simplifications: 1) 1D problem; 2) quasi-electro-neutrality, this leads to j =σE, with
σ =e(μ++μ-)n0 equilibrium conductivity; 3) negligible convection, both of charge
and temperature; 4) linear profiles of conductivity and permittivity σ(x)=σ1+ax,
ε(x)= ε1+bx.
In this case that the permittivity varies spatially, the electric force expression has two
contributions, the Coulomb term and the dielectric term
The generated pressure is
The current density is constant, j =σE=Const.
The induced charge is
The generated pressure can be written as
When the variation of temperature is small (Δσ/σ and Δε/ε <<1), we can approximate
this expression to
where α = (1/σ)(dσ/dT) and β = (1/ε)(dε/dT). For water-saline solutions, α = 0.02 K-1
and β = -0.004K-1. If T0>TL the liquid moves to the right in the figure (notice that the
induced ρ > 0).
The liquid motion modifies the linear profile of temperature because of heat convection.
To neglect heat convection in front of heat diffusion, the Péclet number should be small,
i.e. uL/χ <1, where χ is the thermal diffusivity. For a fluid velocity of 1 mm/s, length
L = 10−4 m and water thermal diffusivity χ =1.4×10−7m2s−1, the Péclet number is close
to 0.7, and the previously calculated expressions for T and Δp are approximately
correct.
2.3.2 Mechanical Properties
The maximum pressure and flow-rate are
We can see that the induction pump generates much smaller flow rate and pressure than
the conduction or injection pumps for the same applied voltage and geometry (between
5 and 10 times less). In effect, ΔT will be of the order of 10 K or less, and the generated
pressure is of the order of Δp ~ ε(V2/L2)α ΔT< 0.2 ε(V2/L2) for water.
To avoid Faradaic reactions, AC voltages of high enough frequency are usually applied.
The time-average force is different from zero because it is quadratic with voltage. The
expressions are the same substituting V by Vrms for frequencies ω<<σ/ε. For ω>>σ/ε the
liquid behaves as a perfect dielectric and only the dielectric force is important. It can be
2
shown that for small ΔT and ω>>σ/ε, Δpmax ≈ 12 ε1βVrms
L−2 and the pumping is in
opposite direction than for the case ω<<σ/ε.
electric
Reynolds
number
The energy efficiency is
where q is the extra power required to obtain the temperature gradient. If this is
obtained by Joule heating q =0 because it is already included in IV. We have neglected
β since for water-saline solutions, α=0.02 K-1 and β=-0.004K-1. In the case that Joule
heating is the source of temperature gradient ΔT~σV2/κ, which comes from the equation
for temperature:
That means that pressure and flow-rate are proportional to V4
2.3.3 Travelling-wave Induction Pump
The travelling electric field induces a travelling-wave charge in the bulk due to the
vertical gradients in conductivity and permittivity. This wave of induced charge lags
behind the voltage wave. Maximum longitudinal force is obtained at frequency ω=σ/ε.
At higher frequencies negligible charge is induced, at lower frequencies, there is
negligible delay between induced charge and applied voltage which implies negligible
longitudinal force.
F
F
E
F
E
F
tw induced charge
E
tw voltage
In this figure, positive charge and voltage are blue and negative are red. The induced
charge density wave has a phase with respect to the voltage wave, which is the origin of
the longitudinal force.
Let us see the solution for the case w >> h >> 1/k (k=2π/λ), small ΔT, Δε, Δσ, and
imposed vertical ΔT.
conservation of
charge
induced charge
Gauss’s law
For an applied voltage
bulk is
longitudinal force is
, the solution to Laplace’s equation in the
. The Coulomb force density is
. The
with τ=ε/σ. The maximum force is for ωτ=1.
The integration of the longitudinal velocity is (no applied pressure, i.e. maximum flowrate)
(ωτ =1)
if dT/dy>0 u>0
V0=10 V
1 10 -4
attraction mode,
8 10 -5
if dT/dy<0 u<0
repulsion mode
ux
dT/dz = 103 K/m
6 10 -5
σ=0.002 S/m
4 10 -5
2 10 -5
3
The maximum flow-rate is
The maximum pressure is
4
5
6
log(f)
7
8
9
The dissipated power is
The energy efficiency is
2.3.4 Some Examples
Felten et al (2006) [17] employed a 4-phase travelling-wave applied to a microelectrode
array with an imposed vertical gradient of temperature. The frequency range of applied
signal was f =1–10 MHz and the voltage range was V =4–10 V. The conductivity of the
water saline solution was σ=0.01 S/m. Characteristic dimensions: h =50 μm, w=100
μm, λ=80 μm, L=240 μm. They obtained typical velocities around u =100 μm/s, for
Vpp=10 V and f =2 MHz. They applied temperature increments of around 40 K to the
microsystem, but only a small fraction of this was the temperature increment inside the
channel.
Iverson and Garimella (2009) [18] employed a 3-phase travelling-wave applied to their
microelectrode array. The signal frequency was f =10–300 kHz at V =10–30 V. The
conductivity of the working fluid was σ=6·10-4 S/m. Characteristic dimensions:
h =50 μm, w =5 μm, λ=72 μm, L=1 mm. They obtained typical velocities around u
=100 μm/s, for V=30 V and f =122 kHz. The temperature field was generated by Joule
Heating.
2.3.5 Working Liquids
Liquids should have some conductivity. A lower bound for σ can be obtained from
conditions of electroneutrality.
This expression can be seen as the ratio between the time that an ion takes to travel the
typical system distance by migration and the charge relaxation time. In order to consider
a liquid to be electroneutral, any charge that could be injected from the electrodes
should relax in a time (given by ε/σ) much shorter than the time to travel the typical
length. This leads to
An upper bound for σ comes from the generated temperature due to Joule heating.
Typically the voltages needed to produce enough flow in microsystems are in the range
from 10 to 50 V peak-to-peak. For these voltages, we expect that the increment of
temperature will be excessive (reaching boiling temperature) for conductivities σ ≥ 1
S/m [19]. Therefore, a wide range of conductivities can be used for this micropump:
10−9 < σ < 1 S/m.
2.3.6 Problems
Voltages cannot be so high that Joule heating is excessive. This may be a limiting factor
if we are interested in pumping liquids with bio-particles (cells, DNA, ...) where the
temperature must be controlled. Instabilities also limit the maximum voltage, destroying
by convection the pattern of σ created by the gradient of temperature. Because electric
fields are not so high, the electrodes and liquids are not subjected to degradation
comparable to ion injection. The performance of the pump can be very reproducible
using some coating on electrodes in order to avoid Faradaic reactions: accurate pumping
over eight months was observed by Fuhr and co-workers with a microelectrode array
protected with a coating.
2.3.7 Applications
Since σ can be as high as 1 S/m, pumping of biofluids for the Lab-on-Chip is a possible
application. Generated pressure is usually very low. However, localized flows (for
mixing, cooling of special elements, or diversion of cells) do not require high pressure.
Energy efficiency of this pump can be quite low. This, however, may not be a limitation
because total consumed power is, in general, low.
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