MODELLING OF ADSORPTION-INDUCED MECHANICAL LOADING ON
MICROCANTILEVERS USING FULL-FIELD MEASUREMENTS
#
N. GARRAUD#∗, F. AMIOT¤, F. HILD#, J.P. ROGER∗
Laboratoire de Mécanique et Technologie, ENS Cachan, CNRS-UMR 8535, Université Paris 6
61 Avenue du Président Wilson, F-94235 Cachan, FRANCE
∗
Laboratoire d'Optique Physique, ESPCI Paris, CNRS-UPR A0005, Université Paris 6
10 Rue Vauquelin, F-75231 Paris CEDEX 05, FRANCE
¤
Department of Micro and Nanotechnology NanoDTU, Technical University of Denmark
Oeørsteds Plads, DTU Building 345 east DK-2800 Kgs. Lyngby, DENMARK
ABSTRACT
The present study deals with micro-mechanical-systems and their interactions with their environment. We focus here on the
mechanical effects induced by the adsorption of decane-thiol molecules onto a gold coated microcantilever for which a
downward bending is measured. The obtained displacement field characterizes the loading distribution acting on the beam.
Using a dedicate identification procedure, a heterogeneous shear stress field is found to be the best match for the adsorption
process studied herein.
Introduction
The displacement fields are measured in-situ and in real time during an adsorption process thanks to a Nomarski imaging
interferometer. The adsorption experiments and results are described, several possible modellings, as well as the
identification procedure used to model the resulting mechanical loading.
3
Microcantilevers are used as test structures (70 x 20 x 0.84 µm , see Figure 1). They are made of silica (thickness of 770 nm),
covered by a titanium adhesion layer (20 nm) and a gold layer (50 nm).
Figure 1: SEM picture of microcantilever (70x20x0.84 µm3)
Because of their high surface / volume ratio, the mechanical behaviour of the microcantilever is driven by surface phenomena,
such as adsorption of biological or chemical species. In this paper we focus on mechanical effects induced by the adsorption
of neutral molecules (Berger et al. [1]).
To precisely establish a mechanical loading modelling of adsorption, a system based on an interferometric measurement is
used, providing a full out-of-plane displacement field of the microcantilever under scrutiny. Therefore, it is used as a sensor,
namely, its displacement is an indicator of the reaction taking place on its interface. The aim of the identification step is to
obtain a modelling of the loading applied on the beam.
Experimental set-up
The out-of-plane displacement measurement is performed with a Nomarski shear-interferometer (Amiot and Roger [2])
described in Figure 2.
Figure 2: Schematic view of the interferometric imaging set-up
A monochromatic light beam, which is emitted by an electroluminescent diode (λ = 760 nm), is split thanks to a Wollaston
prism into two coherent beams with a small angle between each other. After focusing by an objective and reflection on the
gold surface, the two beams are recombined. The optical path difference between them, related to the object topography, is
written as a phase difference Φ, which gives an interference pattern recorded by a CCD array (256x256 pixels, 8-bit
digitisation). To improve the sensitivity of the method, the optical phase difference is modulated by a polarization modulator.
A phase integration method is then applied to obtain a phase map (Figure 3).
Figure 3: Measured phase map along the beam
The interference pattern in Figure 3 is the result of the difference of two topographic images, translated by a shear-distance dw
introduced by the Wollaston prism. Thus the measured phase is related to the topography w by taking into account the shear
distance dw (Amiot and Roger [2])
Φ ( x, y ) =
4π n
ιλ
[ w( x, y ) − w( x − d w , y )] + B [
∂w( x, y ) ∂w( x − d w , y )
−
]
∂x
∂x
(1)
with n the refractive index of the medium, λ the diode wavelength, ι a parameter depending on the numerical aperture of the
objective lens, B a coefficient related to the objective lens, obtained after a calibration step of the system. By comparing phase
maps before and after loading, full out-of-plane displacement fields along the beam surface are obtained.
The measurement noise of the system is characterized, by performing reproducibility experiments for various image
accumulation numbers N. Noise is assumed to be described by a Gaussian, zero mean, and its variable whose standard
deviation variation is photon noise driven, linked with and thus depends on the locally measured intensity value, and thus with
-0.5
the accumulation. The noise standard deviation for low accumulation numbers decreases as N until reaching a limit due to
the system thermomechanical drift. The reproducibility is optimal for 200 accumulations and reaches 50 pm for the out-ofplane displecement.
An adsorption of neutral molecules of decanethiol, whose chemical formula is CH3(CH2)9SH, on the gold surface is performed.
The experiments are carried out in a cell maintained at 22° C. The microcantilever surface is electro chemically cleaned while
-2
-1
varying its potential from 0 to 0.85 V in a KCl solution at a concentration of 10 mol.l until a stable voltammogram is obtained.
-2
-1
Then the beam is first observed in a solution composed of KCl (10 mol.l ) during 20 minutes. Last, a solution is added,
whose composition is KCl (10-2 mol.l-1) + 1 wt% of ethanol, and decanethiol molecules that are to be adsorbed (7 x10-8 mol.l-1).
Results
Figure 4 shows the average phase change field along the beam. The injection moment (t = 20 minutes) is represented by the
solid line, while the dashed one is the limit between the substrate and the cantilever.
Figure 4: Phase change during decanethiol adsorption
After injection, the measured phase changes quickly along the beam, and then levels off, while it does not change on the
substrate. One uses this phase change field to obtain a mechanical loading modelling.
Loading modelling
The information given by the phase change allows one, thanks to an identification method, to obtain the loading
characteristics. The loading model is the key ingredient, i.e., a wrong load description biases the identification. Thus, a fullfield displacement allows one to distinguish between various models the best, contrary to point measurements. One assumes
that the molecules adsorption is a distributed phenomenon on a surface, and cannot be concentrated; therefore only
distributed loadings are presented.
Uniform pressure
A distributed loading may be modelled by a pressure, as a volume force (e.g., weight). Assuming that a uniform pressure,
denoted by p, acts uniformly on an Euler-Bernoulli beam, whose length is L, width is b and uniform flexural stiffness is EI, the
corresponding displacement field reads
Yp =
pb
(6L ²x² − 4Lx3 + x4)
24EI
(2)
with x the considered abscissa along the beam.
Uniform shear stress
Figure 5: Uniform shear stress (a) and heterogeneous localised shear stress (b) modelling
During adsorption, the surface tends to increase in order to adsorb more molecules. This surface phenomenon, acting parallel
to the gold surface, may be modelled by a uniform shear stress (Figure 5-a). Assuming a uniformly distributed shear stress,
denoted by
τs acting on the upper gold surface, the displacement field reads
Ys =
τ sδb
6EI
(3L² − x3)
(3)
with δ the distance to the neutral axis.
Quadratic shear stress
Assuming a quadratic distributed shear stress along the beam, denoted by
τqua such as
τ qua(x) = A x² + B x + C
(4)
L2
Ysqua = δb [C (x²L3 − x3L² + x 4 L − x5 1 ) + D(x²
− x3 L + x 4 1 ) + E(x²L − x3 1)]
2EI 3
2
10
2
3
12
3
(5)
the displacement field reads
Three parameters (C, D, E), describe the loading distribution
τqua, and thus describe the displacement field of the beam.
Heterogeneous localised shear stress
The molecules adsorption is known to operate by growth of adsorbed small islands (Godin et al. [3]). Each adsorbed small
island leads to a localised strain on the microcantilever surface (Figure 6). The deformed shape is evaluated by assuming
that:
- there is no any interaction between adsorbed islands of characteristic length Dloc
- each of these islands is self-balanced with the piece of beam onto which it is adsorbed
- the island acts locally on the beam with a linear shear-stress field τ, whose highest value is τloc on the island boundary.
Figure 6: Localised stress induced by a single one adsorbed island
In that case, the overall displacement is obtained as an assembly of all elementary ones when eliminating all rigid body
motions. For one island, the bending moment M reads
M(X loc) = δ b ∫
Dloc / 2
X loc
2δ bτ loc
Dloc
τ(y)dy =
∫
Dloc / 2
X loc
ydy =
δ bτ loc
[Dloc ² / 4 − X loc ²]
Dloc
(6)
where Xloc is the abscissa of the considered point on the island. The abscissa along the beam x is described by the number of
considered island m, and the local abscissa on this island Xloc is
x = (m − 1 / 2)Dloc + X loc
(7)
Assuming a uniform flexural stiffness EI along the beam, and adding all contributions of the island displacements, the global
displacement field reads
Yloc(m, X loc) =
δ bτ loc Dloc 3 θ 2
12EI
(
4
−
θ4
6
+
θ(m − 1 / 2)
3
+
m(m − 1)
+ 1)
6
32
(8)
with θ = Xloc / Dloc.
If the island length Dloc is small in comparison with the length of the beam, its deformed shape is estimated as the
displacement on the island boundaries, with the condition θ = -1/2. Thus, the displacement field reads
Yloc(m) =
δ bτ loc Dloc3
12EI
(m − 1)²
and
x = (m − 1)Dloc
(9)
and last, by eliminating m, the deformed shape resulting from a heterogeneous localised loading model is
Yloc =
δ bτ loc Dloc
x²
12EI
(10)
Projection
In the previous parts, we focused on the displacement along the beam as a result of adsorption. This leads to various
displacement fields expressed by polynomials (Figure 7). Thanks to full-field displacements, a key element in the present
analysis, an identification method allows us to obtain an appropriate deformed shape.
Figure 7: Various displacement fields shapes
The four different loading modellings are inducing various deformed shapes, allowing us to pick the best model. When using
an identification method with the measured phase variation and Equation (1) relating phase and displacement, an error
indicator is written for each loading model and then compared.
Identification
The identification for finding the model parameters is performed by minimizing η2 for each time step
η² =
1
N pixels
∑
pixels
(Φ mes(x) − Φ model(x))²
= 1
N pixels
σ(x)²
∑
pixels
χ(x)²
(11)
where Φmes is the measured phase along the beam, Φmodel a test function, obtained from Equation (1), depending on the
chosen modelling and parameters, σ(x) a standard deviation for the phase (Amiot et al. [4]), and Npixels the total number of
pixels. This minimization uses displacements that are statically admissible, thus yielding least squares error indicators, which
are used to assess the modelling quality.
Figure 8-a shows the identified displacement field using a heterogeneous shear stress model (Equation (10)). Figure 8-b
shows the displacement of the cantilevers edge during the experiment. The microcantilever experiences a downward bending
induced by decanethiol adsorption. This results from a covalent bond between gold and sulphur atoms (Au-S, thiol bond),
which is thermodynamically favourable, namely, the upper gold surface tends to increase to allow for the adsorption of more
molecules. The adsorption and deformation energies of the beam are compensated at equilibrium.
In Figure 8-right, the kinetics of the reaction is followed. A strong change appears directly after decanethiol injection.
Figure 8: Identified displacement change along the beam for heterogeneous modelling (a), displacement
of the end of the beam (b)
Minimizing η2 by a least squares method using the three deformed shape Yp, Ys, Ysqua, Yloc, one obtains global error indicators
2
rp, rs , rsqua and rloc as the minimum of η averaged over several time steps after injection
•
•
•
•
rp
= 0.12
with p/EI = 7.7 x 10
rs
= 0.074
with
rsqua= 0.013
with
rloc = 0.026
with
-11
µm
-4
τs.δ /EI = 4.4 x 10-9 µm-3
τqua normalised distribution presented Figure 9
τloc.δ .Dloc/EI = 4.59 x 10-7 µm-2
Among the suggested models, quadratic shear stress matches best the displacement measured. In Figure 9 is related the
identified shear stress distribution, when normalised by its maximum: the sign of this loading changes along the beam.
Therefore, even though the error indicator is lower than others loading models, this one is not physically acceptable.
Figure 9: Identified quadratic shear-stress distribution
Among others models, heterogeneous shear matches best the measured phase field. In Figure 10, the local error indicator χ²
is mapped for this loading model (averaged on the beam width). This indicator, compared with 1, is small enough, thereby
validating the modelling quality of the heterogeneous shear stress to describe adsorption of decanethiols. A uniform pressure
does not have a physical signification for molecule adsorption, and the error indicator is the worst one. That of uniform shear
stress loading is better, but adsorption seems to occur in the form of several adsorbed islands.
Figure 10: Local error indicator χ² map for the heterogeneous shear model
Additional analyses may be performed to increase the accuracy of the loading description, for example, by taking into account
a flexural stiffness distribution, or other non-uniform loading along the beam.
Conclusion
In this paper, the importance of a good loading description is shown when full-field measurements are available. In the
present case, a Nomarski interferometer is used to analyse bending induced by molecule adsorption on the surface of a
microcantilever beam. On the basis of the understanding of the physical phenomenon, i.e. molecules adsorption, it is
concluded that a set of small islands on the beam surface is likely.
References
1.
2.
3.
4.
Berger, R., Delamarche, E., Lang, H.P., Gerber, C., Gimzewski, J.K., Meyer and E., Guntherodt, H.J., “Surface stress in
the self-assembly of alkanethiols on gold probed by a force microscopy technique “, Appl. Physics A, 66, S55-S59 (1998).
Amiot, F., Roger, J.P., “Nomarski imaging interferometry to measure the displacement field of micro-electro-mechanical
systems”, Appl. Optics, 45 (30), 7800-7810 (2006)
Godin, M., Williams, P.J., Tabard-Cossa, V., Laroche, O., Beaulieu, L.Y., Lennox, R.B., Grütter, P., “Surface stress,
kinetics, and structure of alkanethiol self-assembled monolayers”, Langmuir, 20, 7090-7096 (2004).
Amiot, F., Hild, F., Kanoufi, F., Roger, J.P., “Identification of the electro-elastic coupling from full multi-physical fields
measured at the micrometer scale ”, submitted for publication.