Transcript Bulk electroconvective instability at high Peclet numbers
Bulk electroconvective instability at high Peclet numbers Brian D. Storey (Olin College) Boris Zaltzman & Isaak Rubinstein (Ben Gurion University of the Negev)
Physical setup
y x Binary electrolyte (C+,C-) Equations •Poisson-Nernst-Planck •Incompressible Navier-Stokes •Fixed potential •Fixed concentration of C+ •No flux of C Solid surfaces are charge selective (electrode or ion exchange membrane).
Steady state (no flow) V=1
1.5
1 0.5
0 0 4 3.5
3 2.5
2 Double layer, Debye =0.01 0.2
E, flux of C+ Bulk is electro-neutral, linear conc. profile 0.4
x V=1 0.6
0.8
1 Double layer, Debye =0.01 Typical dimensionless Debye =0.0001 or less
Current-voltage relationship
2 1 0 0 6 5 4 3 Resistor at low voltage 5 10 Voltage 15 Observed 1D Solution 20
Different views on bulk stability
Conflicting reports of bulk instability in present geometry Microfluidic observations of bulk instability with imposed concentration gradients •Bulk instability . Grigin (1985, 1992) •Bulk instability , but not sufficient for mixing. Bruinsma & Alexander (1990) •Bulk instability . Rubinstein, Zaltzman, & Zaltzman (1995).
•No bulk instability . Buchanan & Saville (1999) •No bulk instability . Highlighted problems with all earlier works reporting instability. Limited parameter space. Lerman, Zaltzman, Rubinstein (2005) Lin, Storey, Oddy, Chen & Santiago (2004) El Mochtar, Aubry, Batton (2003)
Bulk electroconvective (BE) model
c
t
v
c
1
Pe
2
c
Convection/Diffusion of concentration i
D D
1 1
c
c
0 Current continuity Re
v
t
v
v
P
2
v
2 Navier-Stokes
v
0 Incompressibility First 2 equations are derived from Poisson-Nernst-Planck, assuming electro-neutrality.
Parameters
Pe
k b T e
2
d
4
D
0 Re Pe
D
0
V
V applied k b T
/
e D
D
D
0 Peclet, approx. 1 for KCl in water Reynolds, approx .001 (so we disregard) Ratio of applied voltage to thermal voltage (25 mv) Ratio of diffusivity of ions
Hoburg-Melcher (HM) limit D=1, Pe= ∞, low V analysis
c
t
v
D D
1 1 2
c
0
c
1
Pe
0 2
c
c
0 Purely imaginary spectrum
Modified Hoburg-Melcher (MHM) Pe= ∞, low V analysis
c
t D D
v
1 1 2
c c
1
Pe
2
c
0
c
0
Summary
•D>1, Real, S 2 <0, Stable •D<1, Real, S 2 >0, Unstable •D=1, Imag, Oscillations ~
S
2
S
2 1
D
1
D
;
D
D
D
Finite voltage, Pe=∞
MHM model (Pe=∞), low V limit MHM model (Pe= ∞)
Unstable Stable
Bulk electroconvection (BE) model low V analysis
unstable
L=-68 k=4.74
L
I
Pe 4 2
D D
1 1 ;
D
D
D
Current, I max =4
Summary
•D>1, Real, Stable •D<1, Real, Unstable (threshold) •D=1, Stable
Pe=9.9
BE at finite voltage, D=0.1
Unstable
BE at finite voltage D>1
Unstable
MHM model (Pe=∞)
BE model, Pe=10000, V=4
Real Imag
Conclusions
• • • • • Bulk instability can exist, in theory. New bulk instability mechanism found when D+ < D-, that can occur at low V.
Many previous studies only considered D+=D-, Pe ~ 1.
Whether D+ > D- or vice versa can lead to different behaviors. Unresolved questions: – Are there cases where this instability could be experimentally observed?
– How does bulk instability relate to instability in extended space charge region? (Zaltzman and Rubinstein, 2006). – Does asymmetry in electrolyte matter in microfluidic applications? (Oddy and Santiago, 2005).
– Does this instability matter in concentration polarization flows observed in nanochannel applications?
Kim, Wang, Lee, Jang, Han (2007)
Steady state (no flow) V=20
4 Double layer, Debye =0.01 Double layer, Debye =0.01 3.5
E, flux of C+ 3 2.5
2 1.5
1 0.5
0 0 0.2
V=20 0.4
Bulk is electro-neutral, linear conc. profile 0.6
0.8
x 1
Finite voltage, Pe=10000
Unstable Unstable
BE
Stable
BE, low V MHM model (Pe= ∞)
Finite voltage, Pe=10000
Unstable
V=4
Unstable
BE, full
Stable
MHM model (Pe= ∞)
Bulk electroconvection (BE) model low V, D=1 HM
Low voltage limit, Pe=10000
Stable
BE, low V limit