Download Supplimentary material of the *Current controlled dielectric breakdown

Liquid glass electrodes for nanofluidics
Sanghyun Lee1, Ran An, and Alan J. Hunt2
1
E-mail:[email protected]
2
To whom correspondence should be addressed; Tel: 734-615-0331, Fax: 734-936-2116, E-mail:
[email protected]
1. Breakdown current control by in-series resistance.
The breakdown current is limited to nanoamperes by a high in-series resistance. The
serial circuit composed of the nanochannel and the NLGE tip is similar in its operation to
high impedance mode current control circuits. In a high impedance mode, the resistor (the
nanochannel) has much higher impedance than the components (the tip), thereby stabilizing
the current regardless of the non-linear characteristics of the target components. The
difference between conventional high impedance mode circuits and the NGLE is that the
impedance of the nanochannel is not significantly larger than that of the tip. Consequently
the in-series resistance of the NLGE acts more to limit the breakdown current than to hold
it constant.
I NGLE 
VT
V
 T  I max
Rch  Rtip Rch
INLGE : the current in the NLGE [nA], Imax : the maximum current [nA], VT : the total
electric potential [V], Rch : the resistance of the nanochannel [MΩ], and Rtip : the resistance
of the tip [MΩ]. With configurations of the NLGE used here, the resistances of the tip
(~350 Mohm) and the nanochannel (~190 Mohm) become similar, and the breakdown
current is stabilized to about 20 nA.
2. Nano-glass wall thickness measurements.
The thickness of the nano-glass wall is one of the major parameters determining the
dielectric breakdown threshold. Measuring the wall thickness is thus important, but
challenging due to the resolution limits of light microscopy, and the subsurface location of
the NLGE tip. As shown in Fig. S1a, the microchip is composed of a PDMS header and
the glass substrate (microscope cover slip, 25×25mm2, CORNING 0211, zinc titania glass),
wherein the NLGE is inscribed in three-dimensions.
As shown in Fig. S1b and c, the optical microscope image is processed using bicubic
interpolation to estimate the intensity between adjacent pixels. In Fig. S1c, along the red
horizontal dot-line, the dark area between the bright straight channels includes the nanoglass wall. The resulting smoothed intensity contour is then used to locate the critical
intensity corresponding to the interface between glass and fluid.
The intensities along the nanochannel (yellow vertical dot-line in Fig. S1c) are plotted
in Fig. S1d. As shown in Fig. S1d, since the diameter of the nanochannels has been
measured to be 650nm by SEMs1, the critical intensity can be determined as 16%. Applying
this critical intensity to determine the ends of the channels give the wall thickness as 180
nm as shown in Fig. S1e.
a
PDMS header
Corning0211
NLGE
b
c
1µm
1µm
d
Intensity across the channel (%)
e
Intensity across the tip (%)
96
96
80
80
64
64
48
48
32
32
650nm
16
16
0
0
0
400
800
Length (nm)
1200
1600
180nm
0
350
700
1050
1400
Length (nm)
Figure S1 | Optical microscope images (100X, N.A. 1.3 oil immersion objective lens). a, The
NLGE microchip is composed of the PDMS header and the glass substrate containing nanochannels
and the tip. b, Raw image of the tip. c, After bicubic interpolation. d, Intensity changes across
channel diameter, where the reference intensity can be determined based on the known channel
diameter. e, By matching the reference intensity with intensity changes across the tip, the thickness
of the tip can be determined.
The maximum surface roughness of the fs-laser machined nanochannel is about 50nm
as shown in Fig. S2. This level of roughness cannot be resolved by light microscopy, and
since the wall contains two interfaces, the uncertainties are added in quadrature to estimate
the thickness of the nano-glass wall as 180±70 nm. The thickness of the nano-glass wall in
the nano-injector was determined in the same manner.
Figure S2 | SEM showing surface roughness of the fs-laser machined nanochannel (ref.
s1).
3. Electric characterization of the NLGE and electroosmotic pressures
3.1. Electric characterization
To estimate the resistance and potential across the NLGE tip, we calculate the resistance of
the channel portion of the NLGE by examining the factors contributing to the increased
electric current when the tip is removed. This could substantially change the electrokinetics
of fluid flows, so it is important to evaluate the electrical contribution of diffusion and
convection of ions – the second and the third terms of the equation (s1)s2.

i    F  zi Di ci  Fu  zi ci
(s1)
σ is electric conductivity [S/m]; Φ is electric potential [V]; F is Faraday constant [C/mol]; z
is charge number; D is diffusivity [m2/s]; c is concentration [mol/m3]; u is bulk velocity
[m/s]. Because R110 molecules are electrically neutral in the pH=7.0 buffer, and the buffer
(100 mM phosphate) maintains electroneutrality in submicron-scale channels, (the
thickness of electric double layer is about 1 nms2), the contribution of convection to the
electric current can be assumed to be negligible. In addition, the contribution of the
diffusion of charges is also negligible since concentration gradients, if any, quickly
dissipate across micrometer-scale dimensions. Thus, the conduction is approximately ohmic,
following the first term of equation (s1). From the current after elimination of the nanoglass wall (Io=53.4 nA), the channel conductivity, ~ =1.01 S/m is calculated:

I o  ~
Ax
x
where Ax is cross-sectional area of the nanochannel [m2].
(s2)
3.2. Electroosmotic pressure
In addition to conductivity, the electric currents along the NLGE also depend on pressure
driven streaming currents. When the nano-glass wall is present, EOF produces a pressure
gradient along the nanochannels, which induces pressure driven flow along the core of the
channel, satisfying flow continuity (net flow is zero at any cross-sections in the presence of
the wall). This is illustrated in Fig. S3. The combined effects of the Ohmic and streaming
currents along the NLGE (Ie) are described by:

32eo2  

P 

 Ax
I e    ~   eo
 Ax    ~ 
x
x 
d 2  x


(s3)
where η is dynamic viscosity [Ns/m2], μeo is electroosmotic mobility [m2/Vs], and d is
diameter [m]. Using the experimentally determined electroosmotic mobility μeo (=2.25×10-8
[m2/Vs]) and a channel conductivity ~ (=1 [S/m]), the electric resistance of the nano-glass
wall can be calculated as 356 MΩ. This and other physical properties of the NLGE are
summarized in TABLE I in the main text.
Though dielectric breakdown is usually associated with material damage, the nano-glass
wall remains remarkably robust, and is able to block fluid flow even in the presence of
substantial pressures. In a closed end channel the pressure driven laminar flows have the
same flow rate as the electroosmotic flows (direction is opposite), and the pressure gradient
along the nanochannel can be calculated from:
P
 32 eo
 
2
x
 d
 

 x
(s4)
The pressure acting on the nano-glass wall of the NLGE (=Pmax-Pmin, Fig. S3) is evaluated
as 5.88 kPa.
Pmax
Patm
Pmin
Figure S3 | Channel circulation along the nanochannel inducing electroosmotic
pressure at the NLGE tip surface. Green dots represent the R110 molecules and the blue
arrows indicate the breakdown currents.
4. Depletion of the ionic current.
When we assume the number of atoms in 1cm3 volume as 1023, there are approximately
0.47×109 numbers of atoms at the NLGE tip (π×1002×150 nm3). When the impurities are
about 30%, the total numbers of the ions will be 1.4×108, which are supposed to move as a
current, when the electric field strength approaches the breakdown threshold. We also
assume that these ions are singly charged and there are no additional supplements from the
quasi-infinite surroundings, where the electric field strength is much weaker than the
breakdown threshold. Although we do not completely exclude the additional
supplementation by ions in the buffer solution, it seems unlikely that direct migration of
ions in the buffer into the solid glass structure is a dominant for the NLGE operation; it is
difficult to envision how ions could move this freely through the glass while fluid is
excluded. The electric current of 20nA requires 1.25×1011 atoms per second, which can be
sustained for only about 1.2ms if the ions are not replenished; this is at odds with the
currents we observe, which can be sustained indefinitely.
5. Numerical simulation of the NLGE
5.1. Theoretical estimation of the temperature rise
The resistive heat generation or Joule heating due to the breakdown currents increases the
temperature of the NLGE tip. To theoretically estimate the temperature rise at the tip, the
NLGE tip geometry is modeled as a hemisphere, whose thermal resistance can be explicitly
expressed. With additional simplification of an infinite homogenous substrate with constant
thermal conductivity, the steady state temperature at the spherical NLGE tip can be
calculated, providing a good initial estimate of the order of the temperature rise.
Tsurface  Q / 4kr
(s5)
Where Q is the resistive Joule heating [W], k is the thermal conductivity [W/mK], and r is
the radius of the spherical NLGE tip [m]. From equation (s5) under the experimental
conditions we estimate a 0.023K temperature rise at r=300nm with 10V external potential,
where the measured current of 20nA with the potential of 6V across the tip provides
120nW Joule heating at the tip. However, because the equation (s5) actually calculates the
average surface temperature of the spherical tip, the temperature rise at the center of the tip
should be higher than 0.023K. An analytical estimation of the center temperature is,
however, complicated due to the non-uniform heat generation inside the tip. The nonuniform heat generation inside the tip is governed by the breakdown phenomenon, posing
difficulties in analytically finding the center temperature, and thus requires the numerical
approach.
Furthermore, the numerical simulation can include more realistic conditions including the
three-dimensional structure of the NLGE tip and boundary conditions, which were all
simplified in the other theoretical approaches. Based on the numerical calculation, it is
found that the temperature of the spherical tip surface 300nm from the tip center rise about
0.03K in average, and the maximum temperature rise at the tip center is about 0.15K.
Simulation were performed using COMSOL multiphysics (version 3.5).
5.2. Governing equations for the electric fields, conductivity transition, and the heat
transfer.
The electric fields over the NLGE are first calculated based on the continuity equation (s6);
   (V  J e )  Q j
(s6)
σ is conductivity [S/m], V is electric potential [V], Je is the external current density [A/m2],
and Qj is the current source [A/m3]. Je and Qj are set to zero for the NLGE simulation.
In general, the conductivity of glass is set to a constant of 10-12 [S/m]. However, when the
field strength exceeds the breakdown threshold, the conductivity transition from an
insulator to a conductor must be considered. Generally conductivity transitions
accompanying breakdown are expressed using the empirical multiplication factor equation,
the M-factor, (s7) which was developed to describe Zener diodess3.
M 
1
 V
1  
 Vbk



n
(s7)
V is the potential [V], Vbk is the breakdown potential [V], and the exponent n determines
the slop of the transition: n is typically in the range of 3-6.
The specific breakdown phenomena of the NLGE such as the slope of the conductivity
transition are different from those in semiconductors, and we find the transition slope of the
NLGE is shallower than the M-factor. Thus we use a modified version of the multiplication
factor to better follow the experimentally measured conductivity transition of the NLGE:
M
1

V
1 
 Vbk



n

1

V /d 
1 

 Vbk / d 
n

1

E
1 
 E bk



n
(s8)
d is the thickness of the tip [m], E is the electric field [V/m] and Ebk is the critical electric
field for the breakdown to make the conductivity infinite [V/m]. Also, the modified Mfactor is expressed by the electric field strength.
Thus, the transition of the conductivity of glass is modeled as the equation (s9);



E 
1 

 Ebk 
n
(s9)
σo is the low field conductivity (=10-14) [S/m].
The analyzed electric fields and the resultant current density over the NLGE provide the
basis for the heat transfer analysis. In the heat transfer analysis, only conductive heat
transfer in glass is considered, although convective heat transfer along the nanochannels
will additionally cool down the tip. This may overestimate the heating somewhat, but
allows a simple numerical application of the heat transfer equation:
   (kT )  Q
(s10)
k is the thermal conductivity [W/mK], T is temperature [K], and Q is the heat generation
[W], which is set equal to the resistive heating by the breakdown current at the NLGE tip.
5.3. Relation between I-V characteristics and conductivity transition.
For a planar tip the conductivity (σ) can be simply approximated by dividing conductance
by the area (Ao) and the electric field (E or V/Lo) of the tip.
I
Ao
Lo
V
 
Lo I
I

Ao V Ao E
(s11)
σo is the low field conductivity.
When considering the stabilization resistance of the nanochannel:
I
V
Rtip  Rch
 
Lo
Ao (V / I  Rch )
(s12)
Thus, the conductivity transition of a planar tip can be directly calculated from the
measured I-V characteristics.
However, if the tip thickness, L(x,y), is not constant like the NLGE, the conductivity is not
a constant either, but it has a distribution, σ(x,y), because the electric field strength, E(x,y),
over the tip area varies.
I    ( x, y) E ( x, y )dxdy    ( x, y)
V
dxdy
L ( x, y )
(s13)
When considering the resistance of the nanochannel;
I  
 ( x, y )V
dxdy
L( x, y )   ( x, y )( Lch /  ch )
(s14)
Lch and σch are the length [m] and the conductivity [S/m] of the nanochannel, respectively.
When the tip is spherical shape, the tip thickness, L(r), is a function of radius.
I 
R
I 
2r (r )V
dr
L(r )   (r )( Lch /  ch )
2r (r )V
2
2
0 t o  2 R  2 R  r   ( r )( Lch /  ch )
(s15)
dr
(s16)
to and R are the minimum thickness of the tip at the center [m] and the radius of the
nanochannel [m], respectively.
Thus, the measured I-V characteristics of the NLGE cannot explicitly give the conductivity
transition of the spherical tip. In this case, the conductivity transition based on the modified
M-factor assumption can be solved through the numerical simulation, which iteratively tries
different values for the exponent of the modified M-factor until the simulation best matches
the measured I-V.
5.4. Three dimensional structure of the NLGE and mesh generation
The effects of the three-dimensional geometry of the NLGE tip on reversible dielectric
breakdown are modeled by numerical analysis. As shown in Fig. S4, the geometry and the
meshes in the COMSOL simulation are three-dimensional. Although the actual outside
geometry of the substrate is finite as a rectangle, the outmost geometry of the substrate can
be treated as infinite, because the scale of the NLGE is million times smaller than that of
the substrate.
Figure S4 | Three-dimensional structures of the NLGE used in the COMSOL
simulation. When assuming the outer boundary is infinite, the three-dimensional domain
can be modeled as axially symmetric, which can be solved using the two-dimensional
axisymmetry module in the COMSOL, wherein users define the problem in the twodimensional mesh domain representing the half cross-section of the axisymmetric volume,
as shown in Fig. S5; thereby, the simulation speed and accuracies can be improved with
less computational power.
Figure S5 | Axisymmetric mesh domain around the NLGE tip. This simplifies the
dimensions and increases the accuracy owing to the finer and the more meshes.
Because the numerical simulation is not allowed to cover the infinite boundaries, we
identify a boundary size large enough to approach the infinite limit by repeating the
simulation with increasing boundaries, as shown in Fig. S6. As shown in Fig. S6, both the
temperature rise (ΔT) and the electric fields (E) at the center of the NLGE tip are 99%
converged when the size of the boundary becomes bigger than 3μm. Based on the results
and the computation power available, an outer boundary of 20μm is selected.
Normalized results at the center
1
0.99
ΔT [K]
0.98
E-fields
0.97
0
5
10
15
20
Boundary size [um]
Figure S6 | The plateau of the simulation results with increasing boundary sizes
identifies the appropriate boundary size for numerical approximation of the infinite
boundary assumption.
5.5. Simulation results
Figure S7 | Three-dimensional simulation result of the electric field distribution of the
NLGE, showing the electric field confinement at the tip center.
a
b
c
d
Figure S8 | Axisymmetric simulation results. a, The electric fields. b, Conductivity. c, Current
density. d, Resistive Joule heating.
References
s1. Ke, K., Hasselbrink, E. F. & Hunt, A. J. Rapidly prototyped three-dimensional
nanofluidic channel networks in glass substrates. Anal. Chem. 77, 5083-5088 (2005).
s2. Probstein, R. F. Physicochemical hydrodynamics. 2nd edition, Wiley-Interscience, ISBN
0-471-45830-9 (2003).
s3. Bart Van Zeghbroeck, Principles of semiconductor devices. web book edition, Univ. of
Colorado, http://ecee.colorado.edu/~bart/book/book/title.htm.
s4. John R. Reitz, Frederick J. Milford, and Robert W. Christy, Foundations of
Electromagnetic Theory. 4th edition, ADDISON WESLEY, ISBN 0-201-52624-7 (1993).