All the above provides a general foundation of the theoretical model and more specific,
practical details follow. The modelled domain, naturally, embraces only a fraction of the
real pipette, typically it is 40 – 60 μm of the pipette tip and liquid meniscus (Figure 2.3).
As mentioned above, SECCM pipettes do not have axial symmetry and one has to build a
full 3D modelled domain. Owing to symmetry perpendicularly to the septum (Figure 2.3b),
only half of the pipette makes up the modelling domain eventually. Geometrically the
domain is characterized by a number of parameters: tapered angle θ, pipette diameter d, thickness of the septum s, pipette asymmetry d1/d2, meniscus height or tip-to-substrate
distance mh, width of meniscus bottom mw. θ, d, s and d1/d2 can be obtained from SEM
image of the pipette. Depending on the wetting properties of the substrate, meniscus may
have to be presented as a tapered cone rather than a cylinder. This can be easily
incorporated in the model, but requires an independent measurement of mw. Since the
pipette is “cut” at some length, the voltage at the top the modelled domain, Eeff, is
obviously a fraction of EC, which is, along with mh is not an experimentally measured
51
a b
Figure 2.3. 2D projections of the modelled domain of SECCM tip. a) Side projection of the domain depicts the walls of glass pipette and the septum running through the middle and electrolyte solution (blue). The actual modelled domain is within this blue area. b) Bottom projection of the lower part of modelled domain. Outer dashed circle depicts meniscus bottom circumference that generally is larger than the tip diameter. Light blue colour is to show mainly the “meniscus part” of the domain.
Although theta-pipettes typically possess elliptical rather than circular cross-section
(perpendicularly to the longitudinal pipette axes), and this has been included in the
SECCM model,35 it seems fairer not to account for this subtlety for two reasons. First, one
has to introduce second tapered angle as the tapered cone of this type is characterized by
two tapered angles.* The precision of SEM measurements does not allow to distinguish the two angles. Second, practical precision of the measurements based on imaging is also not
as high (drift in iC and iS, fluctuations in iAC, variation in the meniscus shape during
scanning and the very way the model is “tuned” to the experiment).
* ratio of tangents of these angles depends on the ratio of major and minor semi-axis of bottom (or top)
52 The solution returned by the model is for a fixed geometry or, to put it differently, fixed
meniscus height and fixed Eeff. Real pipette oscillates with a certain frequency and
generally changes its average position above the substrate. The approach to finding two
unknown model parameters – Eeff and mh – and linking experimental data with theoretical
model was essentially laid out in ref35 and is presented here with some (small but valuable)
additions.
Thus, one performs a series of computations covering a range of Eeff and mh, computing
the ionic current for each pair (mh, Eeff) and effectively generating a 3D surface (eq 2.19).
C,st st( h, eff)
i f m E (2.19)
This is the so-called stationary tip conductance current. Owing to the tip oscillation, an
alternating conductance current is generated with average, denoted iC,osc (experimentally
measured as direct component of conductance current), generally differing from iC,st as mh
decreases. If A is amplitude of oscillations (not peak-to-peak distance but the factual amplitude of the sinusoid) then the direct current can be calculated from iC,st according to
formula (2.20):
C,osc 1/ 2( (st h , eff) st( h , eff))
i f m A E f m A E (2.20)
Figure 2.4 gives a graphical presentation of the above said, sketching how the stationary
approach curve would differ from non-stationary/oscillatory one.
The alternating component of conductance current, iAC is obtained through expression
(2.21). The factor 1/ 2 2 relates peak-to-peak amplitude (expressed by the difference
under the modulus sign) to the amplitude of alternating current as measured by the
equipment in use, which is RMS.
AC st h eff st h eff
1
( , ) ( , )
2 2
53 Both currents – iC,osc and iAC – have their experimental counterparts and are schematically
plotted in Figure 2.5a and b. Intersection of experimental and computed surfaces occurs
along a 3D curve but one needs only its projection on (mh, Eeff)-plane, which is referred to
as common line for the given type of current. If direct and alternating currents are consistent with the model than the common lines for each of them intersect at one point
that is the sought pair of the two model parameters – (mh, Eeff).
Figure 2.4. Schematic presentation of conductance current during approach plotted vs average meniscus height: stationary approach curve iC,st (black) as is and with sinusoidal oscillations superimposed (gray), direct
54
a b
c
Figure 2.5. Schematic presentation of determination of Eeff and mh from the theoretical model and
experimental values of iC,osc and iAC. a) Computed iC,osc surface (orange) and plane (gray) corresponding to
experimental value of this quantity. b) The same as in a but for iAC. c) The common lines (1 for iAC, 2 for
iC,osc) along which experimental and computed surfaces intersect. The common point of the common lines
corresponds to a singular pair (mh, Eeff) consistent for both direct and alternating current components.