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Modeling flow and transport in nanofluidic devices
Brian Storey (Olin College)
Collaborators: Jess Sustarich (Graduate student, UCSB)
Sumita Pennathur (UCSB)
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First…. the 30,000 foot view
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Microfluidics – Lab on a chip ca. 1990
• Microfluidics deals with the behavior, precise control and manipulation of fluids that are geometrically constrained to a small, typically sub-millimeter, scale. (Wikipedia)
Stephen Quake, Stanford Thorsen et al, Science, 2002
Micronit
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Dolomite Dolomite Prakash & Gershenfeld, Science, 2007
Seth Fraden, BrandeisAgresti et al, PNAS 2010
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Nagrath et al, Nature 2007
Circulating tumor cells, MGH
H1N1 Detection, Klapperich BU
Neutrophil Genomics, MGH
Kotz et al, Nature Med. 2010
CD4 cell count, Daktari Diagnostics
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“Hype cycle”
Gartner Inc.
Microfluidics?
Nanofluidics?
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Nanofluidics• Nanofluidics is the study of the behavior, manipulation, and
control of fluids that are confined to structures of nanometer (typically 1-100 nm) characteristic dimensions. Fluids confined in these structures exhibit physical behaviors not observed in larger structures, such as those of micrometer dimensions and above, because the characteristic physical scaling lengths of the fluid, (e.g. Debye length, hydrodynamic radius) very closely coincide with the dimensions of the nanostructure itself. (Wikipedia)
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Nanofluidics is interesting because…
• Faster, cheaper, better– analogy to microelectronics.
• “the study of nanofluidics may ultimately become more a branch of surface science than an extension of microfluidics.” George Whitesides
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Some background.
Flow in a channel.
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Pressure driven flow is difficult at the nanoscale
High pressure Low pressure
Pressure driven flow of a Newtonian fluid between parallel plates has a parabolic velocity profile. The fluid velocity is zero at the walls and is maximum along the centerline.
𝑈𝑚𝑎𝑥=Δ𝑃𝐿
𝐻2
3𝜇
H
About 100 atmospheres of pressure neededto drive reasonable flow in typical channels
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The electric double layer
--------
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-
-
-
+
0 1 2 3 4 50
0.5
1
1.5
2
2.5
3
X
C
counter-ions
co-ions
-
-
-
-
-
-
-Glass + water
0HSiOSiOH 3
Glass Salt water
Debye length is the scale where concentrations of positive and negative ions are equal.
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Electroosmosis (200th anniversary)
Electric field
- - - - - - - -
+
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+
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-
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- - - - - - -
+ ++
++
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-
-
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+ -
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+ -
--
-
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-
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Double layers are typically small ~10 nmVelocity profile in a 10 micron channel
0 0.2 0.4 0.6 0.8 1 1.2-1
-0.998
-0.996
-0.994
-0.992
-0.99
-0.988
-0.986
-0.984
-0.982
-0.98
Velocity
y
0 0.2 0.4 0.6 0.8 1 1.2-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Velocity
y
EU slip Helmholtz-Smolochowski
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Pressure-driven Electrokinetic
Molho and Santiago, 2002
Electroosmosis-experiments
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The specific problem – Detection.
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FASS in microchannels
Low cond. fluid High cond. fluidHigh cond. fluid
V
+
Chien & Burgi, A. Chem 1992
σ=10 σ=10σ=1
E=1
E=10
E Electric fieldσ Electrical conductivity
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FASS in microchannels
--
-
-
--
-
-
-
Low cond. fluid High cond. fluidHigh cond. fluid
Sample ion
V
+
Chien & Burgi, A. Chem 1992
-
σ=10 σ=10σ=1
E=1 n=1
E=10
E Electric fieldσ Electrical conductivityn Sample concentration
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FASS in microchannelsV
+
Chien & Burgi, A. Chem 1992
--
-
-
--
-
-
-
Low cond. fluid High cond. fluidHigh cond. fluid
Sample ion -
E=1 n=1
n=10
σ=10 σ=10σ=1
E=10
E Electric fieldσ Electrical conductivityn Sample concentration
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FASS in microchannels
---
--
-
---
Low cond. fluid High cond. fluidHigh cond. fluid
Sample ion
V
+
Chien & Burgi, A. Chem 1992
-
Maximum enhancement in sample concentration is equal to conductivity ratio
E=10
E=1
n=10
σ=10 σ=10σ=1
E Electric fieldσ Electrical conductivityn Sample concentration
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FASS in microchannels
Low cond. fluid High cond. fluidHigh cond. fluid
V
E
+
Chien & Burgi, A. Chem 1992
dP/dx
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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FASS in microchannels
0 5 10 15 20 25 300
1
2
3
4
5
6
X
time
Low conducti
vity fl
uid
Sample io
ns
Simply calculate mean fluid velocity, and electrophoretic velocity.Diffusion/dispersion limits the peak enhancement.
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FASS in nanochannels
• Same idea, just a smaller channel.• Differences between micro and nano are quite
significant.
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Experimental setup2 Channels: 250 nm x7 microns
1x9 microns
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Raw data 10:1 conductivity ratio
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Micro/nano comparison
10
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Model
• Poisson-Nernst-Planck + Navier-Stokes• Use extreme aspect ratio to get simple
equations (strip of standard paper 1/8 inch wide, 40 feet long)
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Full formulation 100+ years old
Concentration of positive salt ions,
Concentration of negative salt ions,
Concentration of sample ions,
G
Navier-Stokes for the fluid velocity vector,
Conservation of mass
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Analysis procedure
• Make dimensionless, with separate scales for channel height, H, and length, L.
• Define • Throw out (carefully) terms with any power of in front
of them.• Solve the zeroth order problem.• Go back to equations and throw out terms with or
higher. • State the first order problem. • Integrate (or average) across the depth of the channel.
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Zeroth order electrochemical equilibrium
Relative concentration at centerline, Conc. of positive salt ions = negative
Debye length/channel height. Constant ~ 0.1
Once potential is solved for, concentration of salt ions, conductivity, and charge density are known.
Integrate w/ B.C.
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Proceeding to next order in
0
0
0
0
Enbunxt
n
Ebuxt
Ebux
xu
s
Flow is constant down the channel
Current is constant down the channel.
Conservation of electrical conductivity.
Conservation of sample species.
u is velocityρ is charge density E is electric fieldb is mobility (constant)
σ is electrical conductivity n is concentration of sampleBar denotes average taken across channel height
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Assume distinct regions yields jump conditions
0 5 10 15 20 25 30-1
0
1
x
y
Velocity
-1
0
1
y
Sample ions
-1
0
1
y
Potential
High cond.
21
121
121122
0
)(
0
EbuEbudtdL
EbuEbuLLdtd
dxEbuxt
L
x=0 x=L
L1 L2High cond.Region 1
Low cond.Region 2
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Total pressure & voltage drop
dx L1
0u
0 5 10 15 20 25 30-1
0
1
x
y
Velocity
-1
0
1
y
Sample ions
-1
0
1
y
Potential
High cond.L1 L2
High cond.Region 1
Low cond.Region 2
𝑢=−𝐸 𝜁 (1− 𝜙𝜁 )+ 𝑑𝑃𝑑𝑥 13
Zeroth order velocity field
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Characteristics
0 5 10 15 20 25 300
1
2
3
4
5
6
X
time
1 micron
Enhancement =13 Enhancement =125
Low co
nductivit
y
0 5 10 15 20 25 300
1
2
3
4
5
6
Xtim
e
250 nm
Low co
nduc
tivity
Sample
ionsSa
mple ions
10:1 Conductivity ratio, 1:10mM concentration
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Why is nanoscale different?
0 5 10 15 20 25 30-1
0
1
x
y
Velocity
-1
0
1
y
Sample ions
-1
0
1
y
Potential
High cond.
High cond.
High cond. High cond.
High cond.
High cond.Low cond.
Low cond.
Low cond.
X (mm)
y/H
y/H
y/H
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Focusing of sample ions
- -
Low cond. buffer High cond. bufferHigh cond. bufferUσ
Us,lowUs,high
Debye length/Channel Height
Us,high
Uσ
Us,low
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Simple model to experiment
Simple model – 1D, single channel, no PDE, no free parameters
Debye length/Channel Height
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Focusing of conductivity characteristicsfinite interface
0 5 10 15 20 25 300
1
2
3
4
5
6
X
time
Low co
nductivit
y
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Shocks in background concentration
Mani, Zangle, and Santiago. Langmuir, 2009
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Towards quantitative agreement
•Add diffusive effects (solve a 1D PDE)•All four channels and sequence of voltages is critical in setting the initial contents of channel, and time dependent electric field in measurement channel.
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Model vs. experiment (16 kV/m)
Model
Exp.
250 nm 1 micron
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Model vs. experiment (32 kV/m)
Model
Exp.
250 nm 1 micron
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Conclusions
• Model is very simple, yet predicts all the key trends with no fit parameters.
• Future work– What is the upper limit?– Can it be useful?– More detailed model – better quantitative
agreement.
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Untested predictions
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Characteristics – 4 channels1 micron channel 250 nmchannel
Red – location of sampleBlue – location of low conductivity fluid
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