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Microcantilevers were first introduced as imaging probes in Atomic Force Microscopy (AFM) due to their extremely high sensitivity in measuring surface forces. The versatility of these probes, however, allows the sensing and measurement of a host of mechanical properties of various materials. Sensor parameters such as resonance frequency, quality factor, amplitude of vibration and bending due to a differential stress can all be simultaneously determined for a cantilever. When measuring the mechanical properties of materials, identifying and discerning the most influential parameters responsible for the observed changes in the cantilever response are important. We will, therefore, discuss the effects of various force fields such as those induced by mass loading, residual stress, internal friction of the material, and other changes in the mechanical properties of the microcantilevers. Methods to measure variations in temperature, pressure, or molecular adsorption of water molecules are also discussed. Often these effects occur simultaneously, increasing the number of parameters that need to be concurrently measured to ensure the reliability of the sensors. We therefore systematically investigate the geometric and environmental effects on cantilever measurements including the chemical nature of the underlying interactions. To address the geometric effects we have considered cantilevers with a rectangular or circular cross section. The chemical nature is addressed by using cantilevers fabricated with metals and/or dielectrics. Selective chemical etching, swelling or changes in Young's modulus of the surface were investigated by means of polymeric and inorganic coatings. Finally to address the effect of the environment in which the cantilever operates, the Knudsen number was determined to characterize the moleculecantilever collisions. Also bimaterial cantilevers with high thermal sensitivity were used to discern the effect of temperature variations. When appropriate, we use continuum mechanics, which is justified according to the ratio between the cantilever thickness and the grain size of the materials. We will also address other potential applications such as the ageing process of nuclear materials, building materials, and optical fibers, which can be investigated by monitoring their mechanical changes with time. In summary, by virtue of the dynamic response of a miniaturized cantilever shaped material, we present useful measurements of the associated elastic properties.
Microcantilevers were first designed and fabricated for use as force sensors. Possessing an extremely high force sensitivity, in the piconewton (pN) range, the cantilevers have made Atomic Force Microscopy (AFM) [
A cantilever sensor can be operated in two different modes: the static mode, where the cantilever deflection is monitored, and the dynamic mode, where the cantilever resonance is monitored. The deflection of a cantilever can be due to number of processes such as molecular adsorption, thermal effects, electric and magnetic fields, and fluid flow. Adsorptioninduced deflections are attributed to changes in the surface free energy and are observed only when a differential adsorption occurs between the cantilever surfaces. Depending on the mode of operation, several methods for reading the movement of a cantilever have been developed. These readout techniques can be applied to a single cantilever or to arrays of cantilevers.
One of the first applications of the microcantilever was in sensitive mass balance measurements, where it served as a micro resonator. As a result, a mass resolution in the picogram range was achieved [
Based upon the simultaneous measurements of bending and resonance frequency, a miniature magnetic force balance was developed by Finot
Applications of microcantilevers as label free biological and chemical sensors have been demonstrated by many groups. Good overviews of the early work have been written by Raiteri and Thundat [
The ability of microcantilevers to function as a bioassay was demonstrated by the detection of prostate cancer disease [
Microcantilevers were also found to be particularly suitable for the chemical sensing of vapours and gases [
Another application, where the use of microcantilevers has not been fully exploited, is the study of mechanical properties of materials. Microcantilevers have proven to be powerful tools for the investigation of the mechanical properties of microsystems that is otherwise unattainable, or not easily achievable by other more macroscopic approaches [
The determination of physical parameters is of significant interest for optimizing the design of mechanical structures. The accurate measurement of mechanical properties is contingent upon a rigorous understanding of the length scale dependence. The natural length scale will depend on such structural features of the material as the average grain size, and the dislocation length. To further classify the material properties, one may distinguish the “thin” microstructure, with length scales below a grain size, from a “thick” microstructure or a macrostructure, with length scales encompassing many grains.
In order to optimize the performance of the meso and macroscale devices, material engineers have recognized the need for a better understanding of the processes in the microdomain. Such optimization efforts require similar investigations of the mechanical properties at the nanoscale. Five parameters are usually used to characterize the mechanical response of the material:
The
The
The
The
The
For thin mechanical structures, the elastic (E, ν) or inelastic response (s) play a major role. The viscoelastic properties of silicon microcantilevers can usually be avoided. However, amorphous solids such as glass, polymers can be characterized with a viscosity η (10^{1821} Pa.s for glass) in the plastic regime. The viscosity is defined mathematically as the ratio of the shearing stress to the velocity gradient in the material, i.e., the material's resistance to flow.
Continued miniaturization of mechanical structures will lead to increased influence of surface stress [
While, many mechanical properties of macrostructures may exist in handbooks, as shown in
Unlike other mechanical oscillators, an advantage of the cantilever may be that it is not limited to only one type of material. For example, fabrication of SAW devices is restricted to piezoelectric substrates. Silicon was initially the material of choice in the microfabricated devices because of its favorable electrical and mechanical properties, enabling inexpensive, batchfabricated, highperformance sensors and transducers that could be easily interfaced with advanced microelectronics [
Recently polymers have been applied in the fabrication of microdevices because of their desirable properties (e.g. biocompatibility and cost) [
Metallic microcantilever beams of various thicknesses and lengths have been fabricated by bulk micromachining [
Other materials used in cantilever and micromechanical oscillator fabrication include, glass [
Microfabrication techniques for cantilevers [
Here we review our recent research on the measurement of the mechanical properties of cantilever shaped materials. First, we present the advantage of the small size of the cantilever realtime, insitu measurement of their mechanical properties. Then, we will introduce the necessary theoretical background of the elastic and inelastic parameters for discerning the bulk and surface mechanical properties. We also discuss the limitations of the continuum mechanics as well as the effect of the environment such as the pressure and the temperature. Finally, we will discuss applications in various areas such as the ageing process in nuclear plant, the setting of cement or the etching or swelling of coatings.
One of the underutilized capabilities of a microcantilever is its ability to carry out
For static gas measurements, a prior vacuum is often required. A special cell holding the cantilever can be designed easily in the laboratory to withstand pressures ranging from a secondary vacuum to almost 15 bar [
For dynamic gas measurements, a control of the gas flow is required (
In liquids, the injection is usually performed using a cell of small volume (around 1 mL) using a syringe (
The cantilever response can be read out by several methods depending on the static or dynamic mode of operation but also on the spring constant of the cantilever. The cantilever deflection can be easily measured using a position sensitive detector (PSD) monitoring the reflection of a laser beam from the cantilever. The cantilever curvature can also be obtained optically with subnanometer resolution with a processing speed of about ten cantilevers per second [
For cantilevers with low spring constants, the dynamic response,
For cantilevers with higher spring constants, when the deflection is not easily detectable, the cantilever can be excited mechanically (using a piezoelectric transducer [
The elastic parameters, Young's modulus E and Poisson's ratio?
In the static mode,
We therefore have:
The uncertainty in the evaluation of the Young modulus increases with the complexity of the system. Errors of 35% have been found for cylindrical cantilevers, whereas using a simplified model to describe the Vshaped cantilevers, results in 25% uncertainty [
Bilayer cantilevers,
The resultant spring constant k_{bilayer}, can then be shown to be given by the more complex expression:
The second method used to determine the Young modulus is the mechanical tuning method. The resonance frequencies of a micromachined cantilever has been extensively employed in the determination of the elastic modulus 103 of thin films [
The equation governing the cantilever motion, that is, the time dependent deflection
The corresponding eigenfrequencies are given by:
As a first approach, the problem may be approximated using a simple mechanical harmonic oscillator with the frequency:
From a measurement of the fundamental mode and depending on the cantilever geometry, the effective Young modulus can be determined. Both changes in the spring constant
Determination of resonance frequencies of small cantilevers seems to be the most suitable way for estimating the Young modulus. This is particularly useful since, signals measured in frequency domain often display sharp peaks allowing for very small frequency shifts to be measured. But as the cantilever dynamics is very sensitive to the environmental conditions, the measurements are preferably performed in a vacuum chamber. The accuracy of the measurements may further be studied by the use of eigenmodes of higher frequencies.
The experimental procedure is based on the periodic excitation of the fixed end of the cantilever and the detection of its natural resonance frequency
Although the vibration energy of the cantilever is less for the second mode than for the first mode, the bending angle is larger at the cantilever end and therefore more easily detectable by an optical method. hod operating in the MHz range.
If we consider a bilayer rectangular beam, the effective Young modulus of the film E_{f} can be obtained from the eigenfrequencies of the uncoated (
The effective mass density of the bilayer cantilever is given by:
Changes in the cantilever mass as a function of the coating thickness are shown for various materials in
The frequencies of the coated cantilever (
As shown in
The Poisson ratio ν of a thin material is a very important quantity for stress analysis and structural dynamics. However, thus far, the determination of this parameter remains difficult.
The Poisson ratio is a function of the Young modulus
The shear modulus
The Poisson ratio must be considered for the flexural frequencies of relatively large cantilevers (compared to the cantilever length).
The flexural frequency is in direct proportion to the elastic parameters following the equation
Note that in the major cases where L ≫ b, D remains close to 1.
Continuum mechanics is applicable if the resonance frequency
As the effect of geometrical parameters on the results is of great importance, only results obtained in the linear behavior in
This section is devoted to the measurement of the inelastic parameters of materials using cantilevers. We will first consider the effect of residual stress in simple or bimaterial cantilevers. When the cantilever becomes relatively thin compared to the cantilever length and when the cantilever coating is a monolayer, the surface stress must be considered. Anelastic behavior and fracture strength are also discussed.
Residual stress is a tension or compression, which exists in the bulk of a material without application of any external loads. The residual stress can vary from 500 MPa to 500 MPa.
The main factor that causes this stress is the grain boundary rather than the grain size. Two kinds of residual stress are therefore usually defined: the macro stress corresponds to the behavior of few grains whereas the micro stress deals with submicroscopic areas, within a grain.
Residual stress may be created during the manufacturing process of a material, or it may accumulate in a structure over many years in operation. In either case, this stress can have a serious negative effect on a product's quality, durability and lifetime. Accurate detection of residual stress is an important element of the quality control process and helps predict the service lifetime of the product [
To illustrate, we note that the residual stress depends strongly on the film thickness: the highest compressive stress is created in the first 200 nm of a deposited film and the stress is relaxed significantly if the film gets thicker than 350 nm. A structure with many crystal defects can also generate a stress, which can be minimized by annealing. At high temperatures, the atoms can rearrange themselves, thus the number of crystal defects decreases, thereby reducing the stress. Generally, compressive residual stress is benefic for the fatigue life since it delays crack initiation and propagation. Tensile stress on the contrary reduces the mechanical performance of materials. Such phenomena can be at the origin of the observed asymmetric oscillations of coated cantilevers.
The residual stress can be represented by a uniform stress s _{R} and a gradient stress. The uniform residual stress is relieved though the free end of a single material cantilever. This component can be measured using a bilayer cantilever, in which the residual strains in the two materials are different leading to a bending moment. The bilayer cantilever is usually bent by several micrometers, corresponding to a radius of curvature R_{s} given by:
Assuming σ_{f} to be uniform, the radius of curvature
A more realistic approach consists in considering the uniform stress for both the bare cantilever s _{R} and the coated one s_{Rf}. The cantilever will bend but it is still difficult to isolate s _{R} and s _{Rf}. The radius of curvature is given by:
The gradual residual stress causes the single material cantilevers to bend. The detection of the bending of single material cantilevers provides then a convenient method to measure the internal stress gradient:
The residual stress can be also obtained from the resonance frequency. For a bridge, that is, a beam supported at both ends we have:
Cantilevers have been used to determine the yield strength. The limit of validity of the elastic regime for thin films is thickness dependent, especially due to the changes in the microstructures related to the fabrication process. As the thickness is reduced, the yield strength increases with usually a decrease in the grain size. The yield strength appears to be constant for a film thicker than 1 μm (
Surface stiffness can be viewed, in a top down perspective, as a residual stress near the surface when the thickness of the film tends to zero. This stiffness can arise, for instance, from the surface roughness of the cantilever or some localized mechanical defects [
On the other hand, from a bottom up point of view, the surface stress finds its origin at the molecular scale. It [
As previously noted, mass effects due to molecular adsorption do not contribute to a significant shift in the resonance frequency. The mass of a monolayer will result in a shift of a few Hz. However, shifts of around 100 Hz for a monolayer have been measured. Such results suggest that molecular interactions somehow affect the resonance frequency. To describe this effect, simultaneous measurements of the resonance frequency and the adsorptioninduced cantilever bending have been used to determine the variation in the spring constant. Plotting the change in surface stress as a function of the chemical concentration, the surface excess of adsorbed molecules and, therefore, the mass adsorbed can be determined [
The surface stress can be viewed as the sum of two contributions: one is an axial force per unit length and the second is a moment (N.m) per unit cross section. A variation in the moment induces a cantilever bending but not a frequency shift. If the central part of the beam is under compression, the surface must be under tension, and the forces are balanced. No shear stress exists between the bulk and the surface layer, except at the very end.
In a first order approximation, Stoney's equation may be used to estimate the differential stress Δs from the cantilever deflection
Typical values of surface stress encountered are around 30 mN/m in the case of gas adsorption, 50 mN/m in the case of thiol binding, and 10 mN/m in the case of protein binding. However, inadequate modeling can lead to significant error in the estimation of the surface stress: around 10% in microscopic experiments, whereas for macroscopic cantilevers, the surface stress could be overestimated by a factor of 5 if the mass effect is neglected [
The effect of surface morphology on the surface stress such as the surface roughness is also controversial. Unlike prior reports that suggest the surface roughness enhances adsorptioninduced stress, we observe that nanometersize roughness may slightly decrease the adsorption kinetics and the associated surface stress [
The changes in the axial force can be used to determine the surface stress [
The changes in the surface tension are thus of the order of the spring constant for the more flexible cantilevers. A frequency shift of 50 Hz for a cantilever with k=0.5 N/m leads to a surface tension of 1 mN/m. In the dynamic method, the adsorptioninduced surface stress appears to be less than that obtained by the static approach of Stoney.
When the cantilever is excited, it reaches a new mechanical state, and this equilibrium does not appear instantaneously, consistent with the observation that the corresponding relaxation time is not zero. The time lag between the response of the cantilever and the periodic excitation gives rise to a hysteresis loop. Since the hysteresis is accompanied by energy dissipation, the “compliance” or “rigidity”, relating the stress and the strain, may be defined via a generalized complex elastic modulus:
The behavior of the cantilever is characterized by the response function E(ω), where the real part E′(ω) describes the energy stored by the cantilever and the imaginary part E″(ω) describes the energy dissipated by the cantilever.
The phase angle
by fitting the vibration amplitude in frequency domain using the following formula:
by the direct use of the resonance peak by measuring its width Δ
by the decrement logarithmic curve
The following equation may be considered when distinguishing
Cantilevers have been used to determine the dependence of the internal friction on temperature in the range between 50 to 150°C. A slight decrease in damping
When residual stresses are sufficiently large, they can lead to fracture or delamination either after processing or during the application of subcritical loads. For instance, fracture strength of polysilicon in uniaxial tension could vary between 2.2 and 4.3 GPa, depending upon the details of the fabrication process.
Flexural elements such as cantilevers are naturally concerned with the effects of cyclic loading on material failure [
The reliability of the mechanical measurements will depend on both the accuracy of the measurements and the validity of the continuum mechanics. Analysis of the literature shows controversial measurements; for instance, measurements of
The frequency
The model for calculating Young's modulus presupposes an isotropic material with a constant thickness, no surface roughness, and no texture effects. These effects may cause a mean variation in the measured signals. Laser acoustic methods are insensitive to microscopic methods such as nanoindentation or microcantilevers.
A large difference in the measurement of the Young modulus for sprayed coatings was found for the cases of tension and compression, which was explained in terms of microcracks [
It was necessary to experimentally confirm the influence of the dimensions of the cantilevers on their fundamental frequency of resonance
To apply the mechanics of the continuous media require that the smallest dimension of the sample, in fact the thickness of the cantilevers, is at least 20fold larger than the larger characteristic dimension of material, that is to say the grain size for a polycrystalline solid. Variation of the reference resonance frequency remained lower than 2% as long as
The nonlinear response of a cantilever at large deflections is sometimes also overlooked. A general study of cantilever beam nonlinearity under a variety of loading conditions was performed with analytical and finite element analyses. The cantilever nonlinearity was found to increase with increasing cantilever deflection. The linear analysis was found to underestimate the applied load by up to 15% [
Variations in the environmental parameters of the cantilever, namely the change in temperature and the pressure surrounding the cantilever can strongly affect the cantilever response compared to its behavior in high ultra vacuum.
The doublelayer microcantilevers are very sensitive to the variations in temperature because of the “bimetallic” effect in connection with the difference between the thermal dilation coefficients of various materials. The cantilever deflection becomes then extremely sensitive to temperature changes. Two temperature modes were distinguished:
For small temperature changes Δ
For higher temperatures, the sensitivity in temperature is attenuated due to non linearity of the cantilever response. In addition to the sensitivity in deflection, the sensitivity
The temperature sensitivity is thus at the origin of new types of sensors [
Microcantilevers with quantum wells were also fabricated for manipulating, in realtime, the energy states, thus providing photon wavelength tunability. Applications were then found in an effective and rapid change in electron energy levels for photon detection devices, such as InSb microcantilevers and small arrays of GaAs/GaAlAs microcantilever. Uncooled Infrared (IR) radiation detector were then designed at room temperature [
The pressure effect on microcantilever is clearly visible in the dynamic mode but also in the static mode for both gas and liquid environments.
Various gases, such as helium and nitrogen with pressures between 10^{2} and 10^{5} Pa were used to investigate specific molecular properties (
Among the various flow characteristics, the Knudsen number
Where λ is the mean free path of the gas molecules, b the cantilever width, dg the density of the gas (air=1)?σ_{c} the cross section of collision of the molecules.
Three regimes are distinguished: the molecular regime (
In the molecular mode, the properties of the gas, considered as rarefied, are difficult to reach by macroscopic parameters like the temperature. If no variation in frequency and temperature is detected, the
In the viscous regime, the intermolecular collisions control mainly the gas properties. The cantilever acceleration is the paramount parameter determining the frequency dependence of the resonance; the increase in the pressure induces the uptake of effective mass of the cantilever. In this mode, the frequency of resonance and the quality factor vary in accordance with:
Q decreases in an identical way in the viscous and molecular regimes. The deflection being stable in this regime, the temperature can be considered as constant in this zone.
The transitional regime, suitable for the passage from the molecular mode to the viscous mode, is explained by the equilibrium between the effects of the speed and the inertia.
In liquids, the cantilever motion is damped by a viscous term. The damping can be used to determine the viscosity?? and the density ? of very small volumes of fluids. The Reynold number is used to account for the geometry of the vibrating cantilever as well as in the description of the viscous properties of the liquid.
The general equation of motion of the cantilever in a medium can be written as:
The damping due to the viscosity of the fluid is given by:
The solution of the equation of motion is a complex quantity. The angular resonance frequency is given by:
Therefore,
Initially the cantilever can be calibrated in vacuum or in a fluid with known properties for determining its intrinsic resonance properties. Later the cantilever can be resonated in unknown fluids.
Cantilevers provide the opportunity to develop a new method for the identification of material damage and/or for the experimental verification appropriate for the evolution of the damage laws [
Palladium cantilevers were used to investigate the issue of the storage of tritium, a radioactive isotope of hydrogen [
Young's modulus
When the palladium is completely hydrided in β phase (PdH_{0.6}), a 10% swelling in material volume occurs;
Cantilevers with rectangular and circular cross sections were hydrogenated. The circular cross section was preferred for dynamic analysis since rectangular sections induce irreversible bending by gradients of residual stress generated by the hydrogen insertion differing from one cantilever face to the other.
An isotopic effect (
Let us consider the radioactive decay of PdT_{x}, helium3 atoms are produced following the equation
One of the limitations in the communications by optical fibers is the mechanical resistance in the long run of fibers in aggressive and varied environments such as underwater or underground spaces in the subway. The ageing of optical fibers with respect to moisture or temperature is not completely understood. Hydrogen H_{2} at the origin of the growth of defects (SiOSi + H_{2}0 → SiOH, SiOH) seems to induce the most dramatic degradations. Several mechanical models based on the finite elements, discretizing the fiber by elements of 1 mm length, were developed to analyze the problems of fracture. The resonance frequency was used to determine the unknown Young modulus of these fiber elements [
The influence of the radius of the fiber core was studied using 2 monomode fibers (
The effective Young modulus
The mechanical setting [
The determination of the role played by such interfaces remain however difficult by conventional methods (rheology, techniques involving an inflection of the beam, or ultrasonic) requiring samples of centimeter size. Since hydration is highly exothermic, the miniaturization of the samples on a scale lower than the millimeter is recommended, especially for better modeling of the inclusion/matrix interface. Mortar based cantilevers have been fabricated with millimeter lengths and micrometric thicknesses.
Resonance frequencies of the mortar cantilevers cover an interesting range between 1 and 100 kHz corresponding to a region of the spectrum that has not been explored by either rheology (Hz) or ultrasonics (MHz). We therefore sought to investigate the various contributions to the elastic properties of the composite cantilevers to determine the influence of the parameters such as the hydration time, the porosity of the cement paste and the inclusion concentration within the matrix. The formalism of the continuous media could still be applied to the pure cement cantilevers thicker than 1 μm. For mortar cantilevers, namely including glass balls of 40 μm in diameter, the minimal thickness was fixed at 300 μm.
The viscoelastic limit was determined by requiring that the deflection of 5 mm long cantilevers must not exceed 75 μm to satisfy the deformation criterion of 0.02%. Results were analyzed in terms of acoustic speed, measured very precisely starting from the variations of the resonance frequency
Variations in surface stress can be generated reversibly by water adsorption [
Silica is known in microelectronics to be very sensitive to hydrofluoric (HF) acid. Microcantilevers can be used then as alternative sensors for heavy postanalysis methods. Both the deflection and the resonance frequency of the microcantilevers were analyzed according to the acid flow and concentration as shown in
The stoichiometry and the roughness of the sensitive layers play a paramount role in the surface reactivity. For the lowest concentrations (< 10 ppm), the cantilever deflection provides the most sensitive signal. In the case of Si_{3}N_{4} coatings, a linear and small variation is induced in the surface tension compared to the case with SiO_{2} coatings. Frequency shift was explained in terms of mass loss at high concentrations. The nonlinearity of the deflection observed for the SiO_{2} levers arises from the etching, which initially commences on the sides, and continues in the transverse direction.
Absorption of organic vapors such as benzene and hexane in thin film sensors can lead to changes in the film stress. PECVD membranes deposited on microcantilevers are advantageous on many points: they provide continuous films without porosity, are chemically inert, and possess physically stable defects. In addition to a strong capacity for gas absorption, their high selectivity makes them very competitive in the field of polymeric membranes. For example, the selectivity of butane/methane is 4 for polymers PDMS (PolyDiMethylSiloxane) and 15 for plasma polymers. A 1 μm thick polymeric membrane (aSiOC:H) [
The QCM (
The cantilever bending was used to measure the stress variation
The tension
Concerning the selectivity of the polymeric films, the hexane induces clearly the most important bending (
The study reproduced in the dynamic mode confirms the selectivity with the vapors already noticed in static mode (
In summary, we have shown that cantilevers provide genuine tools for the investigation of the mechanical properties of small volumes of materials and their temporal evolution under gaseous environments.
As temperature gauges, doublelayered microcantilevers operating in deflection mode, can reach extreme thermal sensitivities.
As pressure sensors, using properties such as the quality factor, microcantilevers operating under dynamical resonance mode, can detect various molecular modes defined by the Knudsen number..
As mass sensors, cantilevers exhibit sensitivities on the order of pg/Hz. Converted to the frequency shift per unit area; this sensitivity is 10 times higher than that generally obtained by other types of piezoelectric sensors (quartz microbalances, surface acoustic waves).
It can be concluded that even if the mass change must be considered, the high sensitivity of microcantilevers to molecular adsorption comes from the change in the mechanical properties. A rigorous analysis as a function of the size and the dynamic and the static behavior of the cantilever enables one to discern between:
The bulk properties: such as the change in the Young modulus, which can be measured by analyzing the resonance frequency response. This was illustrated by the studies of the ageing process of three materials: metal tritides, optical fibers, and composite materials such as cement.
Film and interface properties such as the residual stress, and the surface stress were studied by the etching or the absorption process using polymeric or inorganic thin films.
Future trends will consider smaller cantilevers to investigate the mechanical properties of nanosized cantilevers in the MHz regime. Silicon and carbon cantilevers are promising for further exploration. The use of other materials having electro or photomechanical abilities will bring the subject one step further. Simultaneous analysis of the mechanical response of the cantilevers with local electrical and/or optical properties is also an interesting challenge.
Schematics of the setup for insitu measurements of the mechanical properties in gaseous environment in static (AB), under flow (CD) and in liquid (E).
Geometrical parameters of cylindrical, rectangular and V shaped cantilever.
Changes in the effective spring constant of a bilayer cantilever Kd compared to a bare cantilever Ko as a function of the ratio between the Young's modulus of the coating Ed and that of the cantilever Es for various thicknesses of silicon cantilever a) h_{f}/h = 3.10^{3}, b) h_{f}/h = 3.10^{2}, c) h_{f}/h = 1.10^{1}, d) h_{f}/h = 3.10^{1}. Calculations were performed for a typical silicon cantilever having a length L = 200 μm, a width b = 40 μm and a thickness of h = 300 nm. This corresponds approximately to a spring constant of 5mN/m and a mass of 4.8 ng.
Frequency spectrum of a rectangular beam under vacuum
Changes in the cantilever mass M as a function of the ratio between the coating h_{d} and the cantilever h_{s} thicknesses in the case of a) bio (
Fundamental resonance frequencies
Changes in yield strength with the film thickness and film grain size.
Deviation of the measured frequency from the theoretical reference frequency as a function of the thickness of the cantilever.
The pressure dependence of the deflection, the
Changes in the Young modulus of palladium cantilever as a function of the stoichiometry
Resonance frequency peaks of a cylindrical palladium cantilever under vacuum, hydrogen, deuterium and tritium.
Temporal evolution of the frequency spectrum of a monomode optical fiber under a hydrogen bar.
Evolution of the resonance frequency f of the cement cantilever standardized by the thickness h according to the reverse of the square length
Response to HF exposure of cantilevers covered with Si_{3}N_{4} films (dashed line) and SiO_{2} films (solid line) (A) in deflection and (B) in resonance frequency.
Temporal variation measured by QCM of the number of moles of vapor absorbed by the aSiOC:H film standardized by the saturated vapor pressure for each gas and the volume of film.
A) Absorption kinetics of hexane vapor at various pressures monitored by the bending of the cantilever coated with aSiOC :H polymer. B) Cantilever selectivity in deflection of the aSiOC :H polymer to hydrocarbon vapors (P=220 mbar).C) Cantilever deflection as a function of the vapor pressure standardized by the saturated vapor pressure for each gas P_{S}. D) Frequency response of the cantilever as a function of the vapor pressure standardized by the saturated vapor pressure for each gas P_{S}.
Some mechanical parameters of soft and hard materials.
E [ 
ν  s _{Y} (MPa)  s _{F}(MPa)  

LB film  anisotropic 
anisotropic 
1535  
Rubber  0.01  0.1  0.5  412  25 
Polystyrene  2  0.35  30  30 
Aluminum  70  0.33  50  7101000 
Silicon  150  0.17  300  700 
Single Carbon nanotube  1000  0.17  not reached  not reached 
Ratio of the n^{th} frequency over the fundamental frequency. Lines 2 and 3 correspond to rectangular cantilevers and lines 4 and 5 to cylindrical ones.
 

b/L=0.2  6.85  19.1  36.4  62.4 
b/L=0.1  6.44  18  35.4  57 
d/L=0.03  6.05  16.9  
d/L=0.03  6.19  17.3  
Theory  6.27  17.6  34.4  56.9 
Sound velocities of some metals.



 
Cantilever  6.85  19.1  36.4  62.4 
Ultrasonic  6.44  18  35.4  57 
Frequency shift of a microcantilever with a Young's modulus
99.4%  71.4%  24.8%  
99.3%  70,2%  23.9%  
98.9%  58.5%  19,5% 