The electro-osmotic effect (電滲效應): Interplay between the interfacial boundary

conditions and the maximum efficiency bound

Ping Sheng 沈平

Workshop on Mathematical Models of Electrolytes with Application to Molecular Biology

January 5, 2012

Collaborators

• XU Zuli • MIAO Jianying

• WANG Ning,

• WEN Weijia

* L. Onsager, Physical Review 37, 405 (1931) & L. Onsager, Physical Review 38, 2265 (1931).

3

Onsager Relation

12

21

11

22

e

f

J L V PJ V

LL PL

= − ∇ − ∇= − ∇ − ∇

12 21L L=

11eJ L V= − ∇

22fJ L P= − ∇

Ohm’s law:

Darcy’s law:

Onsager relation*:

Introduction (I)

12

21

11

22

e

f

JLL V

LL

J P∇

= − ∇

21f LJ V= − ∇

12e LJ P= − ∇

EO:

SP:

Conductivity

Permeability

4

Introduction (II) Electrical double layer at some fluid-solid interfaces (e.g. silica-water interface)

Stern layer (Fixed)

Mobile layer Po

tent

ial

Distance from the interface ζ : Zeta potential λD: Debye length

Shear Plane 0

5

21fJ L V= − ∇

Electroosmotic effect Electroosmotic flow is the motion of liquid through a porous material under the influence

of applied electric field.

Introduction (III)

Notice the charge separation effect!

6

Streaming potential The complementary effect to electroosmosis: when an electrolyte solution is forced

through a porous media by means of applied pressure difference ΔP, a potential difference ΔVstr arises between electrodes placed on different sides of the channel.

12eJ L P= − ∇

Introduction (IV)

Potential applications of micro/nano-electroosmotic pumps

Liquid Drug Delivery

Electrical Generator (tidal/wind energy)

Electronic Printing

Micro/nano- machine

for fluidics

Micro/nano-electronic

cooling

……

The Low Efficiency Obstacle • The electrokinetic effects (electro-osmosis and streaming

potential) are usually very weak. They are interfacial effects. Efficiency usually in the range of 0.05%.

• Maybe the effect can be enhanced by increasing interfacial

area by making the channels smaller?

• How far can one go in pursuing this strategy? What is the upper bound?

--The competing limitation is the hydrodynamic boundary

condition at the fluid-solid interface. --What is the condition of attaining the upper bound in

efficiency?

9

Schematic Process Flow of Silicon Wafer Etching (Substrate: Silicon wafer, 4’’, 400µm thickness with 3µm Oxide (Double Sides))

Photolithography

Oxide Etching

Photoresist Stripping

Silicon Etching (ICP-DRIE )

Oxidation Silica

Si

Photo Resist

Sample fabrication Experimental Details

Sample Fabrication (I)

10

SEM images of the etched microchannels (a)Top view of 3.5 µm triangular pores array (b) Cross-sectional view of the straight pores

Sample Fabrication (II)

SEM images of microspheres on the surface

11

Experimental measurement setup Schematic illustration of the setups for (a) EO and (b) SP measurements.

Measurement Setup

(a) (b)

12

An Advantage of the Membrane-Type EO Pump

[1] Chen and Santiago (2002) [2] Jiang et al. (2002) [3] Gan et al. (2000) [4] Guenat et al. (2001) [5] McKnight et al. (2001) [6] Tripp et al. (2004) [7] Vajandar et al. (2007) [8] Wang et al. (2006) [9] Yao et al. (2003) [10] Yao et al. (2006) [11] Zeng et al. (2001) [12] Zeng et al. (2002)

.Present work

Flow rate comparison with the literature results

High flow rate Large dynamic range Low operation voltage

13

Nonlinear EO flow rate vs. Applied DC voltage

Problems in Obtaining Accurate Measurements (I)

0 5 10 15 20 25 30 350

50

100

150

200

Fl

ow R

ate (

mL/

min

/cm2 )

Applied Voltage ( V )

Notice the threshold behavior!

14

Decay as a function of time for the EO flow rate: DC vs. voltage pulse

Problems in Obtaining Accurate Measurements (II)

DC

Pulse √ ×

Our Approach to Resolve the Problems

• To obtain accurate measurements, we implemented the pulsed approach to generate the electrokinetic effects.

• Use the Onsager relation to check the results.

• Fabricate samples with varying channel diameters.

• Check the efficiency trend as a function of channel diameter, with theory-experiment comparison.

16

Profiles of voltage pulses with different duty cycles

Digital Flow Control of Electroosmotic Pump

Amplitude: 15 V Duration: 1 ms

17

FIG. 1. Experimentally measured results on (a) flow rate; (b) pressure and (c) electrical current plotted as a function of duty cycle for the silicon membrane EO pump.

Thus, the Onsager coefficient L21:

8 221

/ 4.1 10 m /Vs/

Q ALV L

−= = ×∆

Experimental results of EO

From the slopes of FIG. 1(a) and FIG. 1(c),

10 3/ 4.8 10 m /VsQ V −∆ = ×

/ 6.5 μSI V∆ =

L: length of microchannel A: total cross-sectional area of sample

5.5 μS/cmTz

I I LE A V A

σ = = × =∆

Total electric conductivity σT:

* SSC 151, 440 (2011).

Digital Flow Control of Electroosmotic Pump—Results (I)

18

FIG. 2. Measured streaming potential vs. applied pressure

Thus, the Onsager coefficient L12:

12

8 1 1

//

3.8 10 A Pa m

str strI A V I LLP L P V A

− − −

∆ = = ∆ ∆ ∆ = × ⋅ ⋅

Experimental results of SP Slope:

/ 68 mV/kPastrV P∆ ∆ =

8 2 1 121 4.1 10 m V sL − − −= × ⋅ ⋅

Noted that the unit [A·Pa-1·m-1)] is the same as [m2·V-1·s-1)] in the SI unit.

Verification of the Onsager relation: < 10% difference between L12 and L21

Digital Flow Control of Electroosmotic Pump—Results (II)

*SSC 151, 440 (2011).

19

( ) 210

68.8 mVD r

Lf

ηζλ ε ε

= − = −

Zeta potential (ζ)

Surface conductivity (σsi)

( ) 42 3.3 10 μS

2is T b

a Gσ σ σ −= × − = ×

( )021

f rD

JL f

Vε ε ζ λ

η= = −

−∇

1

2 02 coshG rdrψ= ⋅∫

Measured bulk conductivity: σb = 1.1 µS/cm Measured total conductivity: σT = 5.5 µS/cm

The contribution of surface conductivity in the Stern layer to the total conductivity is up to 60%, which is increased with decreasing pore radius.

where

Determination of zeta potential and surface conductivity

Digital Flow Control of Electroosmotic Pump—Results (III)

≈ 1.58

*SSC 151, 440 (2011).

20

Implementation of digital fluid flow control for the EO pumps to obtain stable and accurate flow rates.

Verification of the Onsager relation.

Determination of zeta potential and surface conductivity of the silica-DI water interface from the Onsager coefficients.

Summary

Digital Flow Control of Electroosmotic Pump--Summary

21

Two types of electroosmotic pumps (EOP) were fabricated and measured: (1) Silicon membrane EOPs (diameters from 2.5 µm to 4.5 µm); (2) EOPs fabricated from anodic aluminum oxide (AAO) templates with silica-coated nanochannels (diameters: 20nm, 100nm and 200nm).

*Advanced Materials 19, 4234(2007)

EO Pumps with Different Channel Diameters

250 nm

SEM image AFM image TEM image

22

0 10 20 30 40 500

10

20

30

40

50

60(a) 2.5 µm

3.0 µm 3.5 µm 4.0 µm 4.5 µm

Flow

Rat

e (m

L/m

in/cm

2 )

Duty Cycle (%) 0 10 20 30 40 500.0

0.2

0.4

0.6

0.8 2.5 µm 3.0 µm 3.5 µm 4.0 µm 4.5 µm

(b)

Pres

sure

(kPa

)

Duty Cycle (%)0 10 20 30 40 50

0

20

40

60

80(c)

2.5 µm 3.0 µm 3.5 µm 4.0 µm 4.5 µm

Curr

ent (

µA)

Duty Cycle (%)

0 10 20 30 40 500.00

0.04

0.08

0.12

0.16

0.20

0.24 2.5 µm 3.0 µm 3.5 µm 4.0 µm 4.5 µm

(d)

Effic

iency

(%)

Duty Cycle (%) 2.5 3.0 3.5 4.0 4.50.00

0.05

0.10

0.15

0.20(e)

Effic

iency

(%)

Diameter (µm)0.10 0.12 0.14 0.16 0.18 0.20

0.00

0.05

0.10

0.15

0.20(f)

Effic

iency

(%)

λD~

Silicon membrane EO pumps (a) Flow rate; (b) Maximum back pressure; (c) Current; (d) Maximum efficiency; (e) Efficiency vs. diameter; (f) Efficiency vs. dimensionless Debye length.

Measured Efficiency of the Electroosmotic Pump (I)

23

0 10 20 30 40 500

50

100

150

200 D=200nm D=100nm D=20nm

Flow

Rat

e (µL

/min

/cm2 )

Duty Cycle (%)

(a)

0 10 20 30 40 500.0

0.5

1.0

1.5

2.0

2.5

3.0 D=200nm D=100nm D=20nm

(b)

Pres

sure

(kPa

)

Duty Cycle (%)0 10 20 30 40 50

0

5

10

15

20 D=200nm D=100nm D=20nm

(c)

Curr

ent (

µA)

Duty Cycle (%)

0 10 20 30 40 500.00

0.05

0.10

0.15 D=200nm D=100nm D=20nm

(d)

Effic

iency

(%)

Duty Cycle (%)0 50 100 150 200

0.00

0.02

0.04

0.06

0.08

0.10(e)

Effic

iency

(%)

Diameter (nm)0 5 10 15 20 25 30

0.00

0.02

0.04

0.06

0.08

0.10

(f)

Effic

iency

(%)

λD~

Measured Efficiency of the Electroosmotic Pump (II) Silica-coated AAO EO pumps (a) Flow rate; (b) Maximum back pressure; (c) Current; (d) Maximum efficiency; (e) Efficiency vs. diameter; (f) Efficiency vs. dimensionless Debye length.

24

A Simplified Model of the EO Pump

Theoretical Modeling (I)

25

Poisson equation:

2

0

,e

r

ρψε ε

∇ = − where ( ).e ze n nρ + −= −

,B DDze k T aψ ψ λλ= / = /

Debye length:

Dimensionless Poisson-Boltzmann equation:

22

1 sinh .D

ψ ψλ

=∇

Boundary conditions:

0 10 .Br r

d ze k Tdrψ ψ ζ ζ

= =| = , | = ≡ /

1 20

2 22r B

Dk T

z e nε ελ

/

∞

=

Electric Potential Distribution Assume the equilibrium Boltzmann distribution equation to be applicable,

Dimensionless:

exp( ), exp( ).B B

z e z en n n nk T k T

ψ ψ∞ ∞+ −+ −= − =

The electrical potential distribution for ζ = 50 mV.

0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0

~

ψ

( r )

/ ζ

~

~

~

~

~λD=0.4λD=0.01

λD=10

λD=1.0

λD=0.1

~r

Theoretical Modeling (II)

a: radius of channel

26

Velocity Profile

0

1

0,

.

r

r

dd rψ

ψ ζ

=

=

| =

| =

2 0z z ePV Ez

η ρ∂⋅∇ − + =

∂

0 0

0.

zr

z r a

dVdrV

=

=

| = ,

| =

2 2

0 ( ) ( )4

r B zz

k TE a r dPVze dz

ε ε ψ ζη η

−= − + ⋅ −

Poisson-Boltzmann Equation: Navier-Stokes Equation:

Boundary conditions:

22

1 sinhD

ψ ψλ

=∇

No-slip B.C.

Boundary conditions:

Theoretical Modeling (III)

27

( )2

( )8z eo DA

Aa dPQ V dA V A fdz

λη

= ⋅ = + −∫

( )

1

02 (1 )Df rd rψλ

ζ= −∫

where

Correction factor plotted as a function of the dimensionless Debye length.

( )Df λ

0r zeo

EV ε ε ζη

= − (Smoluchowski velocity)

(Correction factor)

Volume Flow Rate

21fJ L V= − ∇

( )021

//

f rD

J Q AL fV V L

ε ε ζ λη

= = = −−∇ ∆

Theoretical Modeling (IV)

28

The efficiency of EO pumps when coupled to an external load is defined as the ratio of mechanical hydraulic power output, as seen by the external load, to electrical power input:

Q PI V

χ | | ⋅∆=

⋅

,

,

Qmax

∆Pmax ∆ Pressure

Volume Flow Rate

Theoretical calculation of maximum efficiency

Theoretical Modeling (V)

29

The maximum efficiency is given by

0.01 0.1 1 100.0

0.4

0.8

1.2

1.6 ζ = 100 mV ζ = 68.8 mV ζ = 50 mV

Max

imum

Effi

cienc

y χ m

ax (%

)

λD~

(0.40, 1.26)

(0.36, 0.65)

(0.38, 0.94)

1/20

2 2 ; /2

r BD D D

k T az e n

ε ελ λ λ∞

= =

( ) ( )max maxmax

14

D DP QV I

λ λχ

∆ ⋅≈

⋅

Theoretical Modeling (VI)

Changing the dimensionless Debye length to optimize the maximum efficiency

30

Normalized maximum efficiencies vs. Dimensionless Debye length

0.01 0.1 1 10 1000.0

0.5

1.0

1.5 Theoretical efficiency for silicon Experimental efficiency for silicon Theoretical efficiency for AAO+SiO2

Experimental efficiency for AAO+SiO2

Norm

alize

d M

axim

um E

fficie

ncy

λD~

Theory-Experiment Comparison

1/20

2 2 ; /2

r BD D D

k T az e n

ε ελ λ λ∞

= =

• Green: Silicon-based EO pumps (ζ ~ -68.8 mV) • Black: Silica-coated AAO based EO pumps (ζ ~ -42.3 mV)

PRE 83, 066303 (2011).

31

To obtain the optimal maximum efficiency of EO pumps, the size of the micro/nanochannel should be ~5 times the Debye length.

Maximum efficiency, for a zeta potential of 100 mV, is about 1%.

Verified through agreement between the theory prediction and the experiments for the trend of maximum efficiency varying with channel diameters.

Key to break the efficiency upper bound: Achieve super-slip at the fluid-solid interface.

Advantages of membrane-type EO micropumps • Very thin (~ 50 µm or less) and no moving parts

• Low operation voltages (by about tens voltages)

• Easily-controlled accurate flow rate

• Low cost of fabrication

• Suitable for both small and large area applications

Summary

Maximum Efficiency of the Electroosmotic Pump

32

Publications Maximum efficiency of the electro-osmotic pump Zuli Xu, J. Miao, N. Wang, W. Wen, and Ping Sheng, Phys. Rev. E 83, 066303 (2011). Digital flow control of electroosmotic pump: Onsager coefficients and interfacial parameters determination Zuli Xu, J. Miao, N. Wang, W. Wen, and Ping Sheng, Solid State Communications 151, 440 (2011).

Micropumps based on the enhanced electroosmotic effect of aluminum oxide membranes J. Miao, Zuli Xu, X. Zhang, N. Wang, Z. Yang, and Ping Sheng, Advanced Materials 19, 4234-4237 (2007).

33