Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Mathematical analysis of electromigration

dispersion fronts

Ivan C. Christovhttp://christov.tmnt-lab.org/

School of Mechanical EngineeringPurdue University

Microscale Flows: Electrokinetics (G24.00001)

69th Annual Meeting of the APS DFD

Portland, Oregon

November 22, 2016

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 1 / 8

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Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Introduction: Taylor–Aris (shear) dispersion

peak conc. convects withcross-sectionally averagedvelocity, U

(a) initial con�guration t = 0

(b) at large times, t >> Ο(R2/D),

Gaussian pro�le fortemp. with peak as

x = Ut ; dispersion

De� = D (R2U2/D2)

plug of heated �uid

x = 0

x = 0 x = Ut x = 2Ut

“The transport process that leads to the spread of

this cross-sectionally averaged temperature pulse

turns out to resemble a pure axial conduction (or

diffusion) process and is therefore called Taylor

dispersion.” (Leal, Advanced Transport Phenomena, 2007, §3-H-2)

Key physics:shear enhances diffusion.

ApplicationsI measuring molecular

diffusivity of solutes(Taylor, Proc. R. Soc. A 1954)

I chromatography,separations(Golay, Gas Chromatography 1958)

I but, limits throughput/resolution in microfluidics(Bae et al., Lab Chip 2009)

What about migration ofions in electro-osmoticflows?

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 2 / 8

Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Theory of Taylor–Aris dispersion

Diffusive passive tracer advected by a flow in 2D obeys

∂c

∂t+ vx(y)

∂c

∂x=

∂

∂x

(D∂c

∂x

)+

∂

∂y

(D∂c

∂y

).

Let c(x , z , t) = c̄(x , t) + c ′(x , y , t) and vx(y) = vx + v ′x(y).

For L/h� vxh/D0 and |c ′|/c̄ � 1, can separate the evolution of themean c̄ from fluctuations c ′ to obtain a macrotransport equation:

∂c̄

∂t+ vx

∂c̄

∂x≈ ∂

∂x

(D̄∂c̄

∂x

)− v ′x

∂c ′

∂x,

∂

∂y

(D∂c ′

∂y

)≈ v ′x

∂c̄

∂x.

NB: ‘dispersion’ in the sense of ‘dispersal’ (not ω(k)).(G.I. Taylor, Proc. R. Soc. A 1953; Aris, Proc. R. Soc. A 1956; Brenner & Edwards, Macrotransport Processes, 1993,

see also Griffiths & Stone, EPL, 2012; Christov & Stone, Granular Matter, 2014)

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 3 / 8

Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Electromigration dispersion due to electro-osmotic flow

Assuming infinitely thin Debye layers and simplest three-ion model(Ghosal & Chen, J. Fluid Mech. 2012; also APS DFD 2010 & 2011):

Φt +∇ ·[(V + E )Φ

]= Pe−1∇2Φ, n · ∇Φ = 0 (x ∈ Sw ),

−∇P +∇2V = 0, ∇ · V = 0, V1 = u∗E1 (x ∈ Sw ),

∇ · [(1− Φ)E ] = 0, ∇× E = 0, n · E = 0 (x ∈ Sw ).

The electromigration dispersion equation can derived by standardmethods (e.g., Pagitas, Nadim & Brenner, Physica A, 1986), is

∂Φ

∂T+

∂

∂X

(Φ

1− Φ

)=

1

Pe

∂

∂X

{[1 + ku2

∗Pe2

(Φ

1− Φ

)2]∂Φ

∂X

},

where 0 < Φ < 1, Pe = v0h/D(> 0) and u∗ = ueo/v0(> 0).

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 4 / 8

Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Reduction to an ODE under the traveling wave ansatz

For right-traveling waves Φ(X ,T ) = F (Z ), Z = (X − cT )Pe, c > 0,letting κ := ku2

∗Pe2(> 0), we obtain after an integration:

F ′ = [1− c(1− F )](1− F )F/[(1− F )2 + κF 2

]︸ ︷︷ ︸=G(F )

,

which is a form of Darboux’s ODE (Murphy, ODEs and their solutions, 1960).[Note that (1− F )2 + κF 2 cannot be 0, and 0 < F < 1!]

Equilibria (G = 0) are F0 = 0, F1 = 1− 1/c , F2 = 1;their stability is determined by G′(Fi ).

General solution exact solution can be found implicitly:[(c − 1)2κ+ 1

]ln[1− c(1−F )]−

[(c − 1)κ ln(1−F ) + lnF

]c = (Z −Z0)(c − 1)c .

Goal: determine structure of allowed front-type (“kink”) solutionsbased on this expression.

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 5 / 8

Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Step 1: Study stability diagram of equilibria

◑

Figure: G(F ): RHS of the ODE; κ = 1 without loss of generality; equilibriumpoints are F0,1,2 (open for unstable, filled for stable, half-filled for neutral).

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 6 / 8

Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Step 2: Construct kinks connecting stable equilibria

Figure: Solid: exact kink solutions illustrating all possible scenarios;Dashed: Taylor shock (G.I. Taylor, Proc. R. Soc. Lond. A, 1910) approximation:F (Z) ≈ 1

2

[F (+∞) + F (−∞)

]+ 1

2

[F (+∞)− F (−∞)

]tanh(2Z/`), ` =

∣∣∣∣ F (+∞)− F (−∞)F ′(0)

∣∣∣∣.I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 7 / 8

Introduction Electromigration dispersion Traveling wave reduction Dispersion fronts Summary

Summary

Obtained and classified the exact front solutions to the equation ofelectromigration dispersion, via a reduction to Darboux’s ODE.

I Permanent waveforms unlike classical Taylor–Aris dispersion.

Showed that for (dimensionless) front speeds > 1, bistability allowstwo co-existing front solutions (one increasing, one decreasing).

Open problems:

I Are the kink solutions “global attractors,” emerging from any IC?

I Can bistatiblity be exploited in applications to chromatography andseparations? Traveling wave electrophoresis? (Edwards et al., Phys. Rev. Lett., 2009)

Ref.: I.C. Christov, “Nonlinear waves in electromigration dispersion ina capillary,” Wave Motion (2017) to appear; arXiv:1603.08277.

Thank you for your attention!

I. C. Christov (Purdue) Electromigration dispersion fronts APS DFD 2016 8 / 8

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THE SIGNATUREAlways use the correct artwork for the schools of Engineering signatures. The left side is always the university name as independent usage or the name of the division level for lockup usage with Frutiger Black font. The right side can be changed to the name of the College or divisions with Frutiger Light font.

VISUAL IDENTITY

LOGO USAGELOGO WITHTHE SCHOOL NAME