2-38
State of the art & theory
2.2.7
FluidFM cantilever
The rectangular FluidFM cantilever (as shown in Figure 2.26) is made of silicon nitride and
based on microfabrication technology. The channel consists of a sandwich of two 350 nm
thick silicon nitride layers, with a silicon sacrificial layer in between. It goes from the very
tip of the cantilever to the heart of the chip 1400 μm away. The two thin silicon nitride
layers are reinforced by a protective silicon layer on the critical spots and bonded to a
glass wafer for stabilization and handling. To contact the channel in the chip through the
glass, a connection is drilled by sand blasting.
20 µm
Figure 2.26 FluidFM cantilever. The pillars in the channel are for stabilization. The thick
border around the cantilever is where the two silicon nitride wall-layers bond together.
The FluidFM cantilever has a length of 200 µm, a height of 1.7 µm and a width of 37 μm.
However, 6 µm of this width are due to winglets which are only 0.7 µm high (see also
Figure 2.27). They are where the upper and the lower silicon nitride layer stick together.
The channel inside (see Figure 2.27) is 1 μm high and 30 μm wide. Two rows of pillars
increase the channel stability without increasing the cantilever spring constant
significantly, as we found through simulations in Section 2.4.3.
In the early days, the fluidic reservoir was several ml large and 30 cm away from the
FluidFM chip. This introduced a variable hydrostatic offset, as every cm of water head
equals 1 mbar of pressure.
Recently, the cantilevers are available pre-glued onto a special polymer clip, which
simplifies the mounting on the AFM as well as the connection with the pressure controller.
The clip also contains a small liquid reservoir of 15 μl, giving reserves for several hours of
operation.
2-39
5 µm
Figure 2.27 Channel of a FluidFM cantilever. In this FIB cut through a tipless cantilever, a
winglet can be seen on the left side.
2.2.8
FluidFM probes
At the far end of each FluidFM cantilever an opening is needed to access the
microchannel inside. This opening comes in sizes between 100 nm and 8 µm, and gives,
together with the tip shape, the functionality to a FluidFM cantilever. Thus, it can act as
nanosyringe or pipette, as nanolithographic tool or to grab microscopic objects.
Pyramidal FluidFM probe
Similar to a regular AFM tip, this design has a pyramid as tip. Due to anisotropic etching
the pyramid comes with an angle of 54.7° (see Figure 2.28). With a quadratic footprint of
10×10 μm2, the height is 7 μm. Depending on the fabrication procedure, these tips can
have different types of openings:
20 µm
Figure 2.28 Scanning electron micrograph of a pyramidal FluidFM tip.
2-40

State of the art & theory
Applying the process of “corner-lithography” (J. W. Berenschot H. V. Jansen, M.
C. Elwenspoek 2008; Berenschot 2012) the pyramid has 4 symmetrical holes at
each side. While potentially useful to inject into cells, the four holes make the tip
structurally weak and thus a sensitivity calibration on a hard surface cannot be
properly performed.
1 µm
Figure 2.29 Pyramidal FluidFM tip etched with corner lithography.

Using corner lithography again, the pyramid can have an opening at its very tip,
these “apex tips” were used by R. Grüter for precise lithography in liquid (Grüter
2013).
Figure 2.30 FluidFM apex tip from (Grüter 2013).

We also have cantilever available with a hollow channel but no apex opening.
Here the opening can be drilled by focused ion beam with precisely controlled
dimensions (see Figure 2.31). If the opening is drilled into one face of the pyramid,
the resulting “syringe tip” can be used for injection into cells as presented in
Chapter 4.
2-41
300 nm
Figure 2.31 Syringe tip opening, milled by focused ion beam.
Tipless probes
Instead of a pyramid, the tipless cantilevers have a circular opening at their end (Figure
2.32). They are fabricated on wafer scale by photolithography and useful to grab cells
(Section 3.4) or colloids (5.4). Being easier in fabrication and more reliable in handling,
this cantilever type is the workhorse of current FluidFM experiments. The typical opening
diameters are 2, 4 and 8 µm.
20 µm
Figure 2.32 Tipless FluidFM cantilever with 8 µm opening.
2.2.9
The AFM cantilever spring constant
The vertical spring constant k [N/m] is one of the most important properties of a cantilever.
While attached to a chip of millimeter dimensions, the cantilever itself is much smaller. To
obtain the desired spring constants, the dimensions of the cantilever are chosen
according to classical mechanics. For a single side clamped cantilever with uniform cross
section k is given by (Bhushan 2010):
2-42
State of the art & theory
k
3EI
L3
(2)
Where E is the elastic modulus [Pa] and L the length [m] of the cantilever. I is the moment
of inertia of the cantilever cross section.
For a simple rectangular cantilever, I is:
wt 3
I
12
(3)
With the cantilever thickness t [m] and the width w [m].
Thus, resulting in the simple expression (Sader 1993):
Ewt 3
k
4 L3
(4)
Typical cantilevers are single µm thick, up to 50 µm wide and 500 µm long. Commercially
available lever types have typically spring constants ranging from 0.01 to 50 [N/m].
However, for AFM cantilevers both Young‟s modulus and the thickness t cannot be known
accurately (Cleveland 1993) for two reasons:

To obtain a high force-resolution, AFM cantilevers usually have to be very thin (< 1
µm) and their production comes affected by a considerable variation in thickness.
As seen in Equation (4), the thickness is cubed for the calculation of the spring
constant. Thus, variations in thickness impact the spring constant heavily.

Formation of non-stoichiometric silicon nitride and anisotropic growth of the film
lead to difficulties defining the Young‟s modulus.
This is why experimental methods are necessary to determine the spring constant of a
cantilever. In Section 2.2.10 we show the Sader method, which was used for all
experiments in this thesis.
2.2.10
FluidFM spring constant
Due to the channel in the cantilever, the thickness of a FluidFM cantilever is increased
compared to a regular AFM cantilever. This results in a high spring constant, which was
around 0.5 to 3 N/m for all cantilever used in this thesis.
2-43
Analytical approximation
The spring constant can be estimated with the rational already presented in Equation (2).
The moment of inertia Ic of the cross section of a rectangular cantilever with a concentric
rectangular channel is:
I c  wt 3  wctc3
(5)
Where wc and tc are the width and height of the channel, respectively. The spring constant
k for a hollow cantilever is then:
kh 
E
wt 3  wctc 3 
3 
4L
(6)
The calculated spring constant for an idealized FluidFM cantilever is thus 0.94 N/m, with E
= 250 GPa, L = 200 µm, wc = 30 µm, tc = 1 µm, and a wall thickness of 350 nm.
Experimental (Sader) method
Knowing the exact cantilever spring constant is crucial for AFM force measurements. As
explained in Section 2.2.9, it cannot be determined precisely without a measurement.
The method used in this thesis is based on the well-known theory of John Sader. He
found a relationship between the cantilever spring constant k and the fundamental
resonance frequency in vacuum ωvac (Sader 1999):
2
k  M e  wtLvac
(7)
Here ρ [kg/m3] is the density of the cantilever. Me [1] is the normalized effective mass of
the cantilever, it depends on the cantilever geometry and can only be found with
numerical methods.
Me 
k
2
mvac
(8)
Where m is the total mass of the cantilever. For a long (L/w > 5), rectangular cantilever
Sader found (Sader 1995):
M e  0.243
(9)
In a follow up work Sader could then relate ρ and ωvac with the surrounding fluid and the
cantilever quality factor (Sader 1999). For a rectangular cantilever this lead to:
2-44
State of the art & theory
k  0.1906 f w2 LQ f ( ) 2
(10)
Where ρf [kg/m3] is the density and Γ [ ],the frequency dependent hydrodynamic function
of the surrounding fluid. The quality factor of the cantilever in the fluid Qf [1] should be
clearly larger than 1 for the method to be precise. The lumped constant 0.1906 still
contains the effective mass.
Both the resonant frequency and the quality factor can be precisely determined by
measuring the thermal noise spectrum of the cantilever, while L and w can be measured
with a microscope. Neither the thickness nor the density nor the elastic modulus of the
cantilever are needed, all of which could introduce great uncertainties.
Fortunately, the Sader Method for rectangular cantilever is still valid for FluidFM cantilever
for three reasons:

The method is independent of the thickness and material properties of the
cantilever (Sader 2012), as long as it has a uniform cross section. The
hydrodynamic function is the same for all long and thin rectangular cantilevers,
irrespective of the structure within the cantilever. A FluidFM cantilever fits these
requirements well.

The pillars in the channel, which could contribute to a non-uniform cross section,
have only a minute influence of 1% on k as we show in Section 2.4.3.

The effective mass Me is a geometry dependent factor and can only be determined
numerically. As it turns out in the simulations of Section 2.4.3 the hollow FluidFM
cantilever has the same effective mass as any long, rectangular cantilever.
The experimentally found values for the FluidFM cantilever used in this thesis are a
resonance frequency around 73-85 kHz, a quality factor of around 110 and a spring
constant of 0.5 to 2 N/m.
2.2.11
Force spectroscopy
During a force spectroscopy the extension of the z-piezo is scanned and the
corresponding deflection of the cantilever is monitored. It is a valuable tool to probe
adhesion (Leckband 1995), find the elasticity of the substrate (Touhami 2003) or observe
molecules unfolding (Rief 1997). An example force spectroscopy curve is shown in Figure
2.33.
2-45
Force spectroscopy principle
When the cantilever tip is moved [nm] the cantilever angle changes, which is registered by
the photo detector as voltage V [V]. To translate the detector signal into a distance, the
sensitivity S [nm/V] is obtained by performing a force spectroscopy on a hard substrate
such as glass or metal. In contact, the movement of the AFM piezo is exactly
compensated by the lever, giving the inherent voltage to nm relation. Care has to be taken
in liquid, as the buffer influences the refractive index, thus the laser path and thus the
sensitivity. To derive the force F [N], the spring constant k of the cantilever is additionally
needed [nN/nm]:
F  VSk
(11)
Conventionally repulsive forces have a positive sign, while attractive forces are depicted
as negative. When an AFM probes comes close to the target substrate, the observed
forces are dominate by interface effects. This can include electric potentials, local
chemistry and also depends of mechanical properties of both the cantilever and the
sample. Despite these variations the typical features of a force-spectroscopy curve are as
follows:
I.
If the AFM probe is far away from the surface, no force is measured. The curve is
flat.
II.
While approaching the surface a “snap-in” can be observed, it is the moment,
when the interface forces suddenly are so strong that they attract the cantilever to
the substrate.
III.
When contact is established the repulsive forces dominate, and the deflection rises
depending of mechanical properties of cantilever and substrate. Thus, information
about the substrate elasticity is contained in this phase.
IV.
When retracting the cantilever again from the surface it typically sticks a bit to the
surface. The maximum strength of this interaction is called adhesion force. The
adhesion between surfaces is governed by the deformation of the two bodies in
contact, and the surface forces acting between them. The deformation depends on
the surface forces and at the same time the surface forces will depend on the
deformed geometry of the bodies. This interdependence complicates the theory of
adhesion and is therefore still under debate (Kappl 2002). The adhesion can be
dependent of the applied load, loading rate and contact time. It can thus reveal
much about the mechanical properties of the materials involved (Kappl 2002).
2-46
State of the art & theory
Soft cantilever in air
Deflection [V]
0.13
0.08
Forward
0.03
Retract
-0.02
Snap-in
Adhesion
-0.07
3.3
3.8
4.3
4.8
Piezo position [µm]
Figure 2.33 Example of a force spectroscopy curve: The attractive forces show as
negative deflection signal. The snap in can be observed during the approach, while the
retraction exhibits adhesion. The hysteresis observed between the approach and the
retract curve is the piezo hysteresis.
The Hertz model
Analyzing the contact phase (page 2-45III) of a force spectroscopy, the elastic modulus of
the substrate can be derived. For this thesis we used the classical contact model
proposed by Heinrich Hertz in 1882 (Wikipedia 2013a) to find the elasticity of cells
(Chapter 6).
The classical solution for non-adhesive elastic contact between a spherical colloid and a
flat surface is given by:
A  Rd
(12)
with A being the radius of the contact area, R the radius of the colloid and d the depth it is
pressed into the surface (Wikipedia 2013a). Also, d is related to the applied force F:
F
4 * 0.5 1.5
ER d
3
(13)
Where E* is the effective elastic modulus:
 1  12 1  22 
E 


E2 
 E1
*
1
(14)
2-47
Where E1 and E2 are the respective elastic moduli [Pa] and ν1 and ν2 are the Poisson‟s
ratio [1] associated with each body.
The Hertz model considers elastic contact between two spherical bodies and makes strict
assumptions. Extensions have been made to account for different geometries such as a
pyramidal AFM tip (Sneddon 1965; Lin 2007a). The Hertz model was also derived for
small deformations, large deformations can thus result in a considerable error (Dintwa
2007).
JKR Model for adhesive contact
The Hertz model does not account for adhesive forces between the tip and sample and
thus extensions have been made to deal with such surface forces for more accurate
results (Lin 2007b). In our case, we used the JKR model for the contact of polystyrene
beads on glass (Chapter 6).
The JKR model, presented by Johnson, Kendall and Roberts, describes the radius of the
contact area A as follows:
1

3
 R 
A
4 
 E* 
3 

 Fa  Fl  Fa

2
3
(15)
Where Fa is the measured adhesion force and Fl is the externally applied loading force.
Relation between adhesion force and sphere radius
The Derjaguin approximation is a method to express the force acting between two bodies.
It was presented by the Russian scientist Boris Derjaguin in 1934. For a sphere of radius
R and a planar surface, it states:
F  h   2 RW  h 
(16)
Where F(h) is the interaction force at close separation distance h and W(h) is the
interaction energy per unit area (Assemi 2006). This indicates that the adhesion force of a
colloid should be linearly proportional with its contact radius; a consideration of
importance for the experiments in chapter 6.
2-48
State of the art & theory
Loading rate
The adhesion of a chemically coated cantilever is influenced by the loading rate rl [N/s] as
the individual molecules unbind more likely if they have more time. The loading rate is the
cantilever velocity v [m/s] times the spring constant k [N/m]:
rl  vk
(17)
The most likely unbinding force of a single molecule Fu is influenced by the logarithm of
the loading rate (Merkel 1999; Lee 2007):
Fu 
kBT  rl x
ln 
k k T
x
 off B



(18)
Where kb is the Boltzmann constant, T is the temperature, koff is the off rate at zero speed
and x the bond displacement.
2.2.12
Noise in the Skeleton setups
The noise of an OBD AFM system depends ideally of the thermal vibrations of the
cantilever. As harmonic resonator it will constantly vibrate at its resonance frequencies
fuelled by thermal energy. The electronic noise in the laser/detector components can also
have some influence in low end AFM systems.
Under experimental conditions other sources often can dominate, for example when
working with cells extracellular matrix debris can float through the laser path and disturb
the signal. Mechanical noise of the substrate can be considerable if the AFM table is not
mechanically isolated properly or if the lab is exposed to acoustic noise.
This cumulated noise at the detector is then translated into a force noise through the
cantilever sensitivity (explained in Section 2.2.11) and spring constant (2.2.9).
For the Skeleton setups the effective detection noise was determined with a data
acquisition card at a bandwidth of 1000 Hz under very good experimental conditions in
buffer. The Skeleton I resolution was clearly limited by the photodetector/electrical noise,
as the incoming laser signal was too weak. The Skeleton II resolution was mainly limited
by the cantilever thermal vibrations, such that drift and dirt in the solution became the
dominant source of error.
2-49
2.3 Flow in the FluidFM cantilever
It was important for us to quantitatively estimate the volumes dispensed
by FluidFM. How much would flow through the channel, what would be
the necessary pressure? In this section we present how to find such an
estimate by theoretical considerations. These are later backed up by
simulations in Section 2.4.2 and measurements in Chapter 6.
2.3.1
Basic assumptions
To study the flow in the FluidFM setup, the choice of the proper physical model was
important. The following four assumptions are common in microfluidics (Ho 2010) and
were also made for this thesis:
1. Continuum assumption: The discrete particles of the fluid can be seen as a
continuous medium. In nano channels the molecular nature of a fluid can become of
importance. For liquids, this non-continuum effect becomes visible as an anomalous
diffusion near the channel walls. Experiments with water showed that films as thin as 2
nm, or 10 molecular diameter, still behave as predicted by the continuum approximation
(Karniadakis 2005). As we always worked with aqueous buffer and geometries larger than
100 nm, the continuum assumption holds for the experiments in this thesis.
2. The fluid is Newtonian. This means that the stress in the fluid is directly proportional
to the strain. The proportionality constant is the viscosity of the fluid. The assumption is
valid for both gas and aqueous buffers under normal lab conditions (20° C, 1 atm).
3. The liquid has a constant density; it is incompressible and has constant viscosity.
This is valid as long as the fluid velocities are clearly below the speed of sound in the
medium, which applies for all experiments in this thesis.
4. No-slip boundary conditions apply at the channel walls. Thus, the fluid molecules on
the very wall do not slip along the wall; instead they are fixed to it. Depending of the
microscopic surface geometry and the hydrophobicity of the wall it is hard to predict the
slip, if any. However, the experiments in Chapter 6 show that the no-slip condition is
satisfied in our setup.
2-50
State of the art & theory
2.3.2
Laminar flow regime
The flow of a fluid can be divided into two regimes:
A) The laminar flow regime, where a fluid flows smoothly without lateral mixing of the
streamlines. Laminar flow can often be described analytically and simulated comfortably.
B) The turbulent flow regime, where the flow is curly and characterized by a cascade of
eddies of decreasing size mixing the components of the flow quickly. Turbulence is a
highly complex phenomenon. Even though a wealth of data on turbulent flow is reported in
the literature (Uriel Frisch 1995), there are only inaccurate numerical methods and
virtually no analytical tools to model practical engineering cases.
Using the Reynolds number we will see that for FluidFM laminar flow dominates.
Reynolds Number
The Reynolds number Re relates the geometry of the flow with the inertial and viscose
forces in the fluid. For Re below 2300 (Hardt 2007) the viscous forces are dominant and
flow is laminar. The Reynolds number is defined as:
Re 
u  Dh

(19)
Where u is the mean flow velocity [m/s] and ρ the density of the fluid [kg/m3] they
represent the inertial part of the flow. µ is the dynamic viscosity [Pa s], and indicates the
viscous part of the flow. Dh is the hydraulic diameter [m] and the characteristic length of
the system.
For the cylindrical holes of the tipless cantilever and the FIB drilled pyramids, the hydraulic
diameter is given by:
Dh  do
(20)
Where do is the diameter of the opening. For the case of water (ρ  1000, µ  0.001), this
allows estimating a maximal flow speed umax below which the flow is still laminar:
umax 
2300
 Dh
This also gives a maximum allowed volumetric flow Qmax [m3/s]:
(21)
2-51
Qmax 
umax do 2 2300  do 2

 do
4
d
4
(22)
Using the smallest opening available dmin = 100 nm this gives us the maximal laminar flow
rate for all FluidFM experiments in this thesis:
Qmax (dmin )  1010
(23)
This is equivalent to 100 nL/s. As the flow rates where typically below 1 nL/s, laminar flow
through the opening can be assumed for all experiments in this thesis.
in the rectangular FluidFM channel, the hydraulic diameter Dh is (Hardt 2007):
Dh 
2wc tc
wc  tc
(24)
Dh is roughly 2 µm for a FluidFM channel the maximum laminar flow in the channel Qmax_c
is:
Qmax_ c  umax Ah 
2300
w t  104
2wctc c c

wc  tc
(25)
Where Ah is the hydraulic cross section. With Qmax_c much larger than the critical flow at
the opening, the flow is bound to be laminar in the channel as well.
2.3.3
Hydrodynamic resistance and flow
Resistance
In a simple model, the flow Q through a channel can be calculated as the pressure drop
p [Pa] across the channel divided by the hydrodynamic resistance Rh [Pa s/m3]of the
channel (Hardt 2007):
Q
p
Rh
(26)
Analog to electrical resistances also hydrodynamic resistances can be added up to find
the total resistance of several systems in series. For the case of a FluidFM cantilever this
gives:
Rh _ total  Rh _ hole  Rh _ channel
(27)
2-52
State of the art & theory
According to Poiseuille-Hagen (Hardt 2007) the hydrodynamic resistance the cylindrical
hole can be calculated as follows:
Rh 
128 Lc
 Dh 4
(28)
Where Lc is the length of the channel [m] equaling the wall thickness of a FluidFM
cantilever (350 nm).
For the rectangular channel inside the cantilever the resistance can be calculated as:
Rh 
12 Lc
wctc 1  0.630 tc / wc 
3
(29)
With a channel length of 1400 µm the total hydrodynamic resistance of a FluidFM
cantilever depends only of the opening for small diameters below 500 nm as can be seen
in Figure 2.34.
The pillars in the FluidFM microchannel cannot be taken into account analytically. The
simulations shown in Section 2.4.2 allow estimating that the pillars inside the
microchannel lead to a 7% higher hydrodynamic resistance compared to a microchannel
without pillars.
For the pyramidal tips, the pyramid itself has a negligible influence on the flow resistance
as seen in the simulations in Section 2.4.2.
The resistance of the tubing to the cantilever and in the probeholder can be neglected, as
they have macroscopic hydraulic diameter, at least 100 times larger than in the cantilever.
Flow
The flow through a FluidFM channel can now be estimated. Assuming a pressure of 50
mbar and a typical opening diameter of 2 µm:
Q  9 1011
(30)
This is 9 pL/s and results in the velocity u1 in the channel and u2 at the exit:
u1  3 104
u2  3 103
(31)
Therefore flow speeds in the range of mm/s can be expected for typical FluidFM
experiments where pressures of mbar are used.
2-53
FluidFM hydrodynamic resistance
Hydr. resistance [Pa s/m3]
1E+25
1E+24
1E+23
1E+22
1E+21
1E+20
1E+19
1E+18
1E+17
0
200
400
600
800
1000
1200
Opening diameter [nm]
Figure 2.34 Calculated hydrodynamic resistance of a tipless FluidFM cantilever,
depending of the opening at the apex. The resistance of the cantilever channel dominates
for openings larger than 500 nm.
2.3.4
Surface tension
Liquid molecules always attract each other: as consequence molecules at the liquid
surface feel a net force attracting them to the inner region where a lot of other liquid
molecules are present. A liquid adapts its surface until external and internal forces are in
equilibrium, this is described by the Young-Laplace equation (Carter 1988).
It describes which pressure difference is needed to bend a fluidic surface:
 1
1 
p    
 R R 
y 
 x
(32)
Where γ is the surface tension [N/m], and Rx and Ry are the principal radii [m] of curvature
of the surface as displayed in Figure 2.35.
In case of the radial symmetry of FluidFM openings with radius RO [m]. this simplifies to:
p 
2
RO
(33)
To press out a drop from a FluidFM cantilever this surface tension has to be overcome,
which is γ = 7.28×10-2 N/m between air and water.
2-54
State of the art & theory
Figure 2.35 Surface tension from (Wikipedia 2013b). Here a small piece of the surface is
shown with the associated principal radii and the forces acting on each edge.
The pressure to eject a water droplet into air is 9.7 bar in case of a 300 nm diameter
opening, it is the same pressure needed to suck in water into an empty cantilever with a
300 nm opening. This has several consequences:

For openings of 1 µm and more, the water can still exit the cantilever in air and is
observed as a droplet. This is used to check the cantilever functionality while still in
air.

For small openings the pressure barrier is prohibitively high. Thus, liquid coming
out the cantilever cannot be observed while still in air. For this reason we often
work with a fluorescent liquid in the channel to check whether the cantilever is
already filled up.

The cantilever should be filled with the buffer solution while still in air. The
advancing liquid interface can push the air in the cantilever easily while air is also
outside the cantilever. It is much more difficult to press out the air while immersed
in buffer due to the strong air-water surface tension at the cantilever opening.

This same pressure barrier also applies when we want to suck in water into an air
filled channel. As the maximal underpressure is 1 bar, this makes it impractical to
fill up the cantilever from the front. In general a FluidFM cantilever is thus always
filled with liquid from the back.
Surface tension is also important while filling the channel in the cantilever. However, here
we have three interfaces and surface tensions: liquid-air γla, liquid-wall γlw and wall-air γwa.
All interface forces together can be described empirically as a contact angle θ of the liquid
with the wall (Hardt 2007):
 lw   wa   la cos
(34)
Angles above 90° indicate a hydrophobic surface, where high angles are desirable for
repellent coatings as in Figure 2.36.
2-55
Figure 2.36 Hydrophobic surface from http://www.laurelproducts.com.
A small contact angle θ < 0° implies that the liquid can wet the surface easily; to fill the
cantilever a low contact angle is thus favorable. The silicon nitride of FluidFM cantilever
should already have a low contact angle (Tsukruk 1997) for water in air. Probably due to
processing residues the contact angle in the channel is higher than expected.
Nevertheless, we can fill the cantilever within a minute by applying several hundred mbar
of pressure.
2.4 Investigating FluidFM with simulations
The FluidFM
cantilever has geometrical features which cannot be
covered by analytical expression, in particular the pillars in the channel
and
the
pyramidal
end.
These
are
investigated
here
by
COMSOL
simulations.
2.4.1
Introduction to Comsol Multiphysics
While theory is usually limited to rather simple cases, experiments are often too complex
and expensive in resources as well as in time. Therefore it can be a good option to
consider simulations to gain additional information or to provide a starting point for the
design of following experiments.
Simulations give solutions for a model of the physical reality. So whether the information
gained from simulations makes any sense, depends of the assumptions used for this
model. Even a good approximation might still neglect effects which would occur in the real
world. Either because the problem would become too complex, or because the effect is
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State of the art & theory
not known when the model is developed. However, if conducted with care, a simulation
can give an insight into the governing physics of a setup.
There are several factors which are important for the outcome of the simulation.

How many spatial dimensions are considered and whether there is time
dependence.

The mathematical model chosen to represent the physics.

The boundary conditions applied.

Last but not least, how the space is meshed: Is it in real space, finite element,
finite volume, difference or in a Fourier equivalent. How fine is the meshing and
can it capture all important effects?
Even though finer meshing improves quality or can even be imperative to obtain a
reasonable solution, there are practical limitations to it. Both memory consumption and
processing time increase tremendously with the size of the computed problem. Simulated
geometries are usually translated into huge matrices of linearly coupled systems and the
number of unknowns is called degrees of freedom (DOF). Modern 64 Bit personal
computers can handle up to 350 kDOF. If more DOF are necessary, heaps of memory
and processing time are needed to solve the problem.
For this work all simulations were conducted with the finite element software COMSOL
Multiphysics versions 3.5 to 4.3b. It offers a simple geometrical model editor and also
allows importing CAD data. As the name indicates, multiple physical models are included
and can be coupled with each other. Some of those are preassembled in modules to
serve the needs of common problem types. Three of them were of special interest for this
thesis: The fluid flow module in Section 2.4.2, the structural mechanics module in Section
2.4.3 and the AC/DC module in Chapter 6.
Using the FluidFM technology it is possible to dispense and manipulate tiny amounts of
liquid with precise positioning on a surface. It is therefore both desirable and important to
know, how large the outflow through the nanochannel is with respect to the applied
pressure.
It is straightforward to calculate the hydrodynamic resistance for structures with a constant
and simple perimeter analytically as seen in Section 2.3.3. However, a structure like the
pyramid at the FluidFM tip and the pillars in the channel are much harder to take into
account. For this reason several simulations have been carried out.
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2.4.2
Simulating flow through the FluidFM cantilever
Governing equation - Navier Stokes
The most complete model in fluid dynamics, at least at macroscopic dimensions, is given
by the Navier-Stokes equations. Yet, a couple of assumptions are connected to it. The
fluid should be homogeneous, isotropic and Newtonian. For an incompressible and
stationary analyzed problem with no external forces, this leads to following two equations
for the fluid velocity vector u:
 (u)u      pI M    u  uT  
(35)
 u  0
(36)
Here p is the local pressure [Pa], ρ [kg/m3] is the density of the fluid and µ [Pa s] the
dynamic viscosity, IM is the identity matrix and T indicates the transpose of a matrix.
Boundary conditions
To analyze the resistance, a pressure gradient was applied along the simulated structure
while observing the resulting flowrate.
The fluid inlet at the back was set to a higher constant pressure than the outlet at the tip.
p0 was the constant inlet pressure:
p  p0
(37)
At the outlet the flow could leave unhindered where n is the normal vector on the
boundary.
n  u  uT   0
(38)
For the walls the no slip condition was used, which means that the water molecules at the
walls cannot move in flow direction, but are kept in place by friction forces. This constraint
can be expressed as follows:
u0
(39)
Geometrical model
As the flow rate was of interest, only the fluid channel of the cantilever (see Section 2.2.7)
was modeled, neglecting the surrounding walls. It is clear from theoretical considerations
(Section 2.3.3) that the outside tubes leading to the chip were an insignificant part of the
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State of the art & theory
final hydrodynamic resistance. To model the complete channel inside of the chip would be
exact, but still too large to simulate it efficiently. The field of computational fluid dynamics
(CFD) is one of the most demanding simulation disciplines. Even simple geometries can
exceed in calculation complexity. In the case of the hollow cantilever the big scale
differences were a challenge: The nanochannel in the tip could have a diameter of 100
nm, while the microchannel in the chip had dimensions of 1400×1×30 µm3.
To find the influence of the pyramid it sufficed to look only at the pyramid itself. The
resistance of the rest of the channel could be approximated analytically and was added to
the solutions during post processing.
We were interested in the hydrodynamic resistance generated by the pyramid (see Figure
2.37) as it could not be covered analytically. The pyramid dimensions itself remained
constant (10×10×7 µm3). The physical parameters of the water were taken from the
COMSOL library and were applied at normal conditions (20°C, 1 atm).
Study & meshing
The simulations were performed three dimensional, steady state case with the COMSOL
fluid flow module. Different geometries were simulated and evaluated parametrically. The
incompressible Navier-Stokes model was used in an approach known as direct numerical
simulation. The physics controlled mesh resulted in around 300 kDOF, finding a solution
took around 15 minutes per geometry condition.
5 µm
Figure 2.37 Model of the liquid in the FluidFM pyramid. The liquid is confined between two
walls which are vertically 1 µm apart.
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Influence of the pyramid
We wanted to know the influence of the pyramid on the overall flow resistance.
Therefore the liquid flow through the pyramid was analyzed by cutting the pyramid tip at
different distances from the pinnacle as seen in Figure 2.38.
A
B
5 µm
Figure 2.38 Pyramid cut at different distances from the pinnacle. A) at 100 nm. B) at 2000
nm. In B) it is visible, that the liquid is confined between two pyramidal walls.
This allowed simulating the hydrodynamic resistance of the pyramid seen by an apex
channel of a certain diameter. This was then compared to the resistance generated by the
apex channel itself, and to the resistance of the microchannel in the cantilever. The results
are shown in Figure 2.39.
The conclusions are:

The pyramid resistance is almost always negligible

The analytic approximation in Section 2.3.3 gives a good approximation of the flow
rates even for pyramidal FluidFM tips.

The pyramid could only contribute to the flow resistance if the cantilever walls are
a few nm thin and if the opening is smaller than 200 nm.

For openings above 500 nm the resistance of the microchannel in the cantilever
dominates.
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State of the art & theory
Hydr. resistance [Pa s/m3]
Influence of pyramid resistance
1E+25
1E+24
1E+23
1E+22
1E+21
1E+20
1E+19
1E+18
1E+17
1E+16
Total resistance
Pyramid resistance
Cantilever channel
0
200
400
600
800
1000
Opening diameter [nm]
Figure 2.39 The influence of the pyramid flow resistance is negligible. The major
resistance comes from the nano-channel drilled through the 350 nm thick cantilever walls.
Influence of the pillars in the FluidFM channel
Using the same approach as in the last section we analyzed the influence of the pillars in
the channel.
The same COMSOL module and boundary conditions were used in this case and are
therefore not repeated.
The geometry of the analyzed channel section covered 4 columns of pillars. The pillars
had a diameter of 3 µm and were spaced regularly 11 µm apart.
Comparing the hydraulic resistance of the cantilever channel with and without pillars
showed that the pillars increased the resistance by 7%. This value was then used to
compare the theoretical and the measured flow in Chapter 6.
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15 µm
Figure 2.40 Simulated cantilever channel section with pillars. Here the flow speed is
indicated from slow (blue) to fast (red).
2.4.3
Mechanical simulations of the hollow cantilever
The mechanical properties of FluidFM cantilever can be estimated with theoretical
approximations in Section 2.2.10. Here these predictions are put to test, by simulating the
cantilever mechanically with COMSOL Solid Mechanics. We were interested how the
cantilever spring constant and resonance frequencies would be affected by the hollow
channel within and the simulations were then compared with measurements. Finally, the
effective mass was also calculated to validate the Sader method for FluidFM.
Governing equation
The equations used in COMSOL Solid Mechanics are based on the principle of virtual
work. All internal strain work must equal the external load work. In the linear elastic,
stationary case, used for these simulations, this is expressed in COMSOL as:
t
W  0   ( test : s  utest FV )dv   (utest Fs )ds   (utest FL )dl   (U test
Fp )
V
S
L
(40)
p
εtest is the test strain tensor, s the stress tensor, utest is the test displacement. The work
derivative δW should be 0 and equals the sum of all force × displacement work in all
domains (Volumes V, face S, Edges L and points p) minus the strain work. The F‟s
represent the respective force densities in each term.
Boundary conditions
The cantilever was fixed on one end, thus the displacement ud [m] was:
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State of the art & theory
ud  0
(41)
On the free end of the cantilever we applied an edge load FL to study the spring constant:
F  FL
(42)
To study the eigenfrequency of the cantilever the edge load was zero.
Geometrical model
The model of the cantilever can be seen in Figure 2.41. The cantilever was simulated
along its full length of 200 µm with a channel height of 1 µm and a wall thickness of 350
nm. The stabilization pillars were included as cylinders of 3 µm diameter with 11 µm
spacing. The cantilever also features winglets of 700 nm height and 3 µm width along the
full length; they are added in fabrication as additional area where the upper and lower wall
of the channel can bond together.
The material properties of the cantilever are those of the COMSOL material library for
silicon nitride.
50 µm
Figure 2.41 Mechanical model of cantilever in COMSOL. It includes the two silicon nitride
wall-layers and the pillars in the channel.
Meshing
For the computations we usually meshed the geometry “physics controlled” with the “fine”
element size.
Spring constant
The spring constant was found by dividing the edge load by the edge displacement
(Figure 2.42). The value of 1.04 N/m is close to that predicted by the mechanical theory of
0.94 N/m in Section 2.2.10.
The winglets are responsible for this 10% discrepancy, whereas the pillars only increases
the spring constant by 1 %.
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50 µm
Figure 2.42 Deflected cantilever in COMSOL. The deflection is indicated from small
values (blue) at the base of the cantilever to large values (red) at the free end.
Resonance frequency
The resonance frequency was found by studying the eigenmodes of the cantilever. They
were calculated without damping and thus should be close to the measured values in air.
The simulated value of 79 kHz is well within the range of experimentally measured values
in Section 2.2.10. This indicates that this model represents the real cantilever to a good
extent.
Simulating the cantilever without winglets resulted in a slightly higher resonance
frequency of 82 kHz. Taking away the pillars in addition increased the simulated
resonance frequency to 83.5 kHz. This makes sense, as both are an additional resonance
volume for the standing wave in the cantilever; adding them to the cantilever increases the
wavelength.
Effective mass – validation of Sader method
As explained already in Section 2.2.10 the Sader method should also be valid for FluidFM
if the effective mass Me of the FluidFM cantilever is comparable to the effective mass of a
filled, long rectangular cantilever.
Me 
k
m 2
(43)
For any filled rectangular cantilever this should theoretically be 0.243 (Sader 1995), for the
simulated FluidFM cantilever, with pillars and winglets, this was 0.243. The mass m was
found by an integration of the cantilever volume multiplied with its density. Indeed Me used
by Sader and Me found in the simulations are identical within the simulation error.
This confirms that the Sader experimental method can be applied to measure the spring
constants of FluidFM cantilever.