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Highly Nonlinear Electrokinetic Simulations Using a Weak Form
Gaurav Soni, Todd Squires, Carl MeinhartUniversity Of California Santa Barbara,
USA
Presented at the COMSOL Conference 2008 Boston
Induced Charge Electroosmosis
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KCl + H20
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Induced Charge Electroosmosis
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Induced Charge Electroosmosis
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Induced Charge Electroosmosis
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Effective Boundary PDE
• Conservation of double layer surface charge
q Dut y x x
φ φ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂
2
2sinh( 2)4 (1 )sinh ( / 4)
qDu m
ζ
ε ζ
= −
= +
Non-linear Capacitance
Transformation for Stability• Double layer charge grows
exponentially and causes numerical errors.
• To make the solution stable, a variable transformation is required.
• Stability is improved by creating diffusion without losing accuracy.
q Dut y x x
φ φ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂
2sinh( 2) cosh2
dq dqdt dt
ζ ζζ = − ⇒ = −
φ ζ= −
⇓
1cosh( 2)
Dut y x xζ φ ζ
ζ ∂ ∂ ∂ ∂ = − − ∂ ∂ ∂ ∂
Nonlinear Electrokinetic Model
0u =
0u =
HSu u=
2 0φ∇ =
ˆ 0,n φ⋅∇ =
q Dut y x x
φ φ∂ ∂ ∂ ∂ = + ∂ ∂ ∂ ∂
2 0u pη∇ −∇ =0u∇⋅ =0u =
( )2L f ta
φ α= ( )2L f ta
φ α= −
φ ζ= −
2
2sinh( 2)4 (1 )sinh ( / 4)
qDu m
ζ
ε ζ
= −
= +
22HS
D
u xaα ζ φ
ε λ= ∂ ∂
=
Bulk b.c.
Boundary PDE
L
a
Non-Dimensional Simulation Parameters
zeta magnitudethermal voltage ext
ze a VkT L
α = =
debye lengthelectrode length
D
aλε = =
Voltage Scaling
Length scale ratio
Weak Form in COMSOL
• Weak form– An integral form equivalent to the original PDE.– Derived by multiplying the PDE with a test
function and then integrating over the domain.
• Weak form advantages– Create custom equations on with dimension– Implement PDEs on boundaries– Mixed time and space derivative
Weak Form: Example
( )Cv d v D C d vRdtΩ Ω Ω
∂Ω = ∇⋅ ∇ Ω+ Ω
∂∫ ∫ ∫
( )C D C Rt
∂= ∇ ⋅ ∇ +
∂
( )Cv d vD C d D v Cd vRdtΩ Ω Ω Ω
∂Ω = ∇⋅ ∇ Ω− ∇ ⋅∇ Ω+ Ω
∂∫ ∫ ∫ ∫
Consider Diffusion Equation
Weak Form: multiply with test function v and integrate over domain
Simplify
( )ˆCv d vn D C d D v Cd vRdtΩ ∂Ω Ω Ω
∂Ω = ⋅ ∇ ∂Ω − ∇ ⋅∇ Ω + Ω
∂∫ ∫ ∫ ∫
Apply Green’s Theorem
Time dependent term Boundary fluxes Diffusive term Source term
Weak Form Contributions
weak terms for Ω
Cv dtΩ
∂Ω
∂∫
weak terms for ∂Ω
Time dependent or dweak terms:
D v Cd vRdΩ Ω
− ∇ ⋅∇ Ω + Ω∫ ∫
( )ˆvn D C d∂Ω
⋅ ∇ Ω∫
Double Layer Boundary PDE∂Ω
1cosh( 2)
Dut y x xζ φ ζ
ζ ∂ ∂ ∂ ∂ = − − ∂ ∂ ∂ ∂
, BoundaryΩ
cosh( 2) cosh( 2)v vv d d Du d
t y x xζ φ ζ
ζ ζΩ Ω Ω
∂ ∂ ∂ ∂ Ω = − Ω+ Ω ∂ ∂ ∂ ∂ ∫ ∫ ∫
, Points∂Ω
Boundary PDE
Weak Form
2
ˆ tanh( / 2)cosh( / 2) cosh( / 2) 2
Du d Du v vK v nd ddx x x xζ ζ ζζ
ζ ζ∂Ω Ω
∂ ∂ ∂ = ⋅ ∂Ω− − Ω ∂ ∂ ∂ ∫ ∫
K
Ω ContributionContribution∂Ω
Simplify
Weak Form Contributions
weak terms for boundary Ω
v dtζ
Ω
∂Ω
∂∫
weak terms for Points ∂Ω
Time dependent or dweak terms:
ˆcosh( / 2)
Du dv nddxζ
ζ∂Ω
⋅ ∂Ω
∫
2
tanh( / 2)cosh( / 2) 2 cosh( )
v Du Du vd dy x a x xφ ζ ζζ
ζ ζΩ Ω
∂ ∂ ∂ ∂ − − Ω− Ω ∂ ∂ ∂ ∂ ∫ ∫
DiffusionSources
Effect of Surface Conduction
α=10 ε=0.0001
α=31.6
Uslip/Ulinear
x/a
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
α=10
α=3.16ε=0.000316
α=31.6
x/a
Uslip/Ulinear
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
α=10
α=3.16ε=0.001
α=31.6
x/a
Uslip/Ulinear
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
α=31.6
α=10
α=3.16α=1
ε=0.00316
x/a
Uslip/Ulinear
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
α=31.6
α=10
α=3.16
α=1ε=0.01
x/a
Uslip/Ulinear
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
α=31.6
α=10
α=3.16
α=1 ε=0.0316
x/a
Uslip/Ulinear
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
Normalized Velocity: Nonlinear Effects
10-4
10-3
10-2
10-1
100
0
0.2
0.4
0.6
0.8
1
ε
Um
ax
α=31.6
α=10
α=3.16
α=1
α=0.316α=0.1
ε=λD/a
As double layer gets thicker, the normalized velocity decreases due to higher surface conduction.
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
Normalized Velocity: Nonlinear Effects
10-1
100
101
102
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
α
Um
ax
ε=0.1
ε=0.0316
ε=0.01
ε=0.00316
ε=0.001
ε=0.000316
ε=0.0001
As zeta potential increases, the normalized velocity decreases due to higher surface conduction.
zeta magnitudethermal voltage
debye lengthelectrode length
ext
D
ze a VkT L
a
α
λε
= =
= =
Conclusions• An effective boundary PDE was used for simulation
electroosmosis.• Nonlinear effects such as exponential capacitance and surface
conduction were simulated.• The boundary PDE was made stable by variable
transformations.• Results for high zeta potentials (~30x thermal voltage) were
obtained.• The dimensionless velocity decreases with increasing zeta
potentials and increasing double layer thickness, because surface conduction effects become prominent for high zeta potentials and thicker double layers.
