Highly Nonlinear Electrokinetic Simulations Using a Weak Form

Transcription

1 Presented at the COMSOL Conference 2008 Boston Highly Nonlinear Electrokinetic Simulations Using a Weak Form Gaurav Soni, Todd Squires, Carl Meinhart University Of California Santa Barbara, USA

2 Induced Charge Electroosmosis KCl + H

3 Induced Charge Electroosmosis + -

4 Induced Charge Electroosmosis + -

5 Induced Charge Electroosmosis + -

6 Effective Boundary PDE Conservation of double layer surface charge q φ φ = + Du t y x x Non-linear Capacitance q = 2sinh( ζ 2) Du = ε + ζ 2 4 (1 m)sinh ( / 4)

7 Transformation for Stability Double layer charge grows exponentially and causes numerical errors. q φ φ = + Du t y x x To make the solution stable, a variable transformation is required. Stability is improved by creating diffusion without losing accuracy. q dq ζ dζ = 2sinh( ζ 2) = cosh dt 2 dt φ = ζ ζ 1 φ Du ζ = t cosh( ζ 2) y x x

8 Nonlinear Electrokinetic Model L nˆ φ = 0, u = 0 u = 0 L φ = α f () t 2a 2 φ = 0 2 η u p= u = 0 u = u HS φ = ζ a 0 Bulk b.c. u = 0 L φ = α f () t 2a q = 2sinh( ζ 2) Du q φ φ = + Du t y x x 2 4 (1 m)sinh ( / 4) = ε + ζ u HS ε = λ Boundary PDE 2 = 2 α ζ φ D a x

9 Non-Dimensional Simulation Parameters Voltage Scaling zeta magnitude α = = thermal voltage ze a Vext kt L Length scale ratio debye length ε = = electrode length λd a

10 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

11 Weak Form: Example Consider Diffusion Equation C t ( D C) = + R Weak Form: multiply with test function v and integrate over domain Simplify Apply Green s Theorem C v dω= v ( D C ) dω+ vrdω t Ω Ω Ω C v dω= ( vd C ) dω D v CdΩ+ vrdω t Ω Ω Ω Ω C v dω = vnˆ ( D C ) d Ω D v CdΩ + vrdω t Ω Ω Ω Ω Time dependent term Boundary fluxes Diffusive term Source term

12 Weak Form Contributions Time dependent or dweak terms: Ω v C dω t weak terms for Ω D v CdΩ + vrdω Ω Ω weak terms for Ω Ω ˆ vn D C dω ( )

13 Double Layer Boundary PDE Ω Ω, Boundary Ω, Points Boundary PDE ζ 1 φ Du ζ = t cosh( ζ 2) y x x Weak Form ζ v φ v ζ v dω= dω+ Du d Ω t cosh( ζ 2) y cosh( ζ 2) x x Ω Ω Ω Simplify 2 Du dζ Du ζ v v ζ K = v nd ˆ tanh( ζ / 2) d Ω Ω cosh( ζ / 2) dx Ω Ω cosh( ζ / 2) x x 2 x K Ω Contribution Ω Contribution

14 Weak Form Contributions Time dependent or dweak terms: weak terms for boundary Ω Ω v ζ dω t v 2 φ Du tanh( / 2) ζ d Du ζ ζ v d Ω Ω Ω cosh( ζ / 2) y 2 x Ω cosh( aζ) x x Sources Diffusion weak terms for Points Ω Ω Du dζ v nd ˆ Ω cosh( ζ / 2) dx

15 Effect of Surface Conduction

16 α=10 ε= α=31.6 zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext U slip /U line ar x/a

17 α=3.16 α=10 α=31.6 ε= zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext U slip /U line ar x/a

18 α=3.16 α=10 α=31.6 ε=0.001 zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext U slip /U line ar x/a

19 α=1 α=3.16 α=10 α=31.6 ε= zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext U slip /U line ar x/a

20 α=1 α=3.16 α=10 ε=0.01 zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext U slip /U line ar α=31.6 x/a

21 α=1 ε= α=3.16 α=10 zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext U slip /U line ar α=31.6 x/a

22 Normalized Velocity: Nonlinear Effects U max α=0.1 α=0.316 α=1 α=3.16 α=10 zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext 0.2 ε=λ D /a α= ε As double layer gets thicker, the normalized velocity decreases due to higher surface conduction.

23 Normalized Velocity: Nonlinear Effects U max ε=0.1 ε= ε= ε=0.001 ε= ε=0.01 zeta magnitude α = = thermal voltage debye length λd ε = = electrode length a ze a V kt L ext 0.2 ε= α As zeta potential increases, the normalized velocity decreases due to higher surface conduction.

24 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.