Supporting Information for:
Numerical modeling of Joule heating effects in insulatorbased dielectrophoresis microdevices
Akshay Kale, a Saurin Patel,a Guoqing Hub and Xiangchun Xuana
a
Department of Mechanical Engineering, Clemson University, Clemson, SC 29634-0921, USA
b
LNM, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100190, China
Correspondence: All correspondences should be addressed to [email protected] (Dr. Xuan) or
[email protected] (Dr. Hu).
S-1. Effects of constriction length
The Constriction length increases the geometry of the constriction along the direction of the
electric field. Hence, there is no change in the amplification ratio of the electric field, thereby
also maintaining the electroosmotic flow to be constant. However, since both Joule heating and
electrothermal flow are volumetric phenomena, the total amount of heat generated in the
constriction and the electrothermal flow are expected to increase with the length of the
constriction. However, the real increase is negligible, and the fluid temperature and flow fields
intensify only slightly. Hence even the flow circulations do not exhibit a significant change when
the constriction length is varied.
S-2. Effects of corner radius
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Since we provide a slip velocity boundary condition at the channel walls, we obtain a potential
flow solution to the flow field. Hence fillets are created for avoiding singularity at the
constriction necks. Just like the constriction length, the fillet radius does not have a significant
effect on the temperature field. However, the fillet radius does affect the local electric field at the
neck of the constriction. Fig. S-1 represents the velocity fields in the center plane of the
constriction region for variations in the fillet radii. The local electric field is amplified for sharper
radius, and hence increases the local electroosmotic flow at the necks of the constriction. This
alters the overall magnitude of the flow field. But since the bulk region of the constriction
remains unaltered in geometry, the temperature field and electric field, and hence the
electrothermal flow, also remain constant on the whole. So the flow circulations also are not
affected by the fillet radius and the net flow field increases for a sharper radius solely due to
electroosmotic flow.
5 µm radius
20 µm radius
35 µm radius
Fig. S-1: Fluid velocity magnitude (in the unit of m/s) in the microchannel constriction of
different corner radii. Other conditions are referred to the text in section 3.1
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S-3. Effects of electric conductivity
In order to investigate the effect of electric conductivity on the temperature and flow fields, we
worked on two different situations. In the first one, we simply increased the electric conductivity,
and maintained the electric field constant. At higher conductivity the Joule heating, the
temperature field and the flow field, and hence the flow circulations are stronger. Since these
effects are quite obvious, they are not elaborated here. In the second situation, effects of higher
conductivity are studied by proportionally reducing the electric field, and hence maintaining the
Joule heating constant. Fig. S-2 represents the effects of solution conductivity in such a situation
for 2 cases, one with the reference conductivity and electric field, and the other one with a
quadrupled conductivity and a halved electric field. The Joule heating being fixed, the
temperature field, and hence the temperature gradients remain unaltered over the entire system.
But it’s observed that, at a higher conductivity, the electrothermal flow and the flow circulations
are smaller. This can be explained easily. The magnitude of this Force (Equation 8) directly
depends upon the electric field gradients and the temperature gradients. Since the temperature
gradients are constant for both the cases, the electric field gradients solely govern the
electrothermal flow field. For a higher conductivity, the electric field is weak enough to generate
high gradients required for establishing strong electrothermal flow and flow circulations.
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σ = 0.188 S/m
E = 30 kV/m
σ = 0.047 S/m
E = 60 kV/m
Fig. S-2: Fluid streamlines in the microchannel constriction at two different values of fluid
electric conductivity. The total Joule heating, E2 is maintained between the two cases.
S-4. Modified expressions for DC-biased AC voltages in terms of the AC to DC ratio
The AC to DC ratio is defined mathematically as:
r=
VAC
EAC
=
VDC
EDC
Here the AC voltage is the rms value of the voltage, and the total electric field is the sum of the
DC field and the rms AC field. However, since the Joule heating, Electro-thermal Force as well
as the DEP velocity field is not linearly dependent on electric field, their value is the net sum of
the contributions of the individual DC and AC components.
Current Conservation Equation:
∇. ((𝐄DC + 𝐄AC )σ +
∂(𝐃DC + 𝐃AC )
)=0
∂t
Using the definition of AC to DC ratio and simplifying, we get
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∇. (𝐄DC σ +
∂𝐃DC
)=0
∂t
Similarly we have,
Heat Source for Joule heating:
q′′′ = σ(< 𝐄DC >2 + < 𝐄AC >2 )
= (1 + r 2 )σ < 𝐄DC >2
Electrothermal Volume Force:
𝐟 ′′′ = 𝐟′′′DC + 𝐟′′′AC
= (1 + r 2 )[ (∇. EDC )𝐄DC −
1
< 𝐄DC >2 aϵref 𝛁T ]
2
DEP Velocity:
𝐔DEP = 𝐔DEP DC + 𝐔DEP AC
= (1 + r 2 )[
1 2
d ε(𝐄DC . 𝛁EDC )Re(fCM ) ]
6η
S-5. Heat transfer coefficients for various surfaces
The commonly used correlations for heat transfer coefficients are defined as functions in
COMSOL and can be directly used for the boundary conditions. The equations are as below:
Top Surface and the Free Surface of the Reservoir:
k
h = 0.54 RaL1/4 for RaL upto 107
L
k
h = 0.15 RaL1/3 for RaL > 107
L
Bottom Surface:
k
h = 0.27 RaL1/4
L
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Side Walls:
k
h=
0.68 +
L
(
0.67RaL1/4
0.492
(1 + ( Pr )
for RaL upto 107
9/16 4/9
)
)
2
k
h=
0.825 +
L
(
0.387RaL1/6
for RaL > 107
9/16 8/27
0.492
(1 + (
)
Pr
)
)
Here Pr is the Prandtl Number and RaL is the wall Raleigh Number based on the characteristic
length L, which is the ratio of surface area to perimeter for horizontal walls and the height for the
vertical walls. This value is a user input to COMSOL based on the geometry of the surface. The
thermal conductivity k and the Prandtl number are that of the ambient air and not the medium.
S-5. Rate of change of electric field with conductivity
We have the current conservation law for a steady state as:
∇ ∙ (σ𝐄) = 0
For the electrolyte to obey the same, it must follow that, at any point in the spatial domain,
E∝
1
σ
Differentiating with respect to conductivity we get,
dE −1
∝ 2
dσ
σ
The negative sign indicates the opposite trend of the conductivity and the electric field. As the
temperature rises, the conductivity also increases, and hence magnitudes of the rates of change of
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electric field with respect to conductivities reduce. Hence, the gradients of electric field are also
smaller in magnitude at higher temperatures.
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