1.
CHAPTER-1
INTRODUCTION
Arrays of micro cantilevers are increasingly being used as physical, biological, and
chemical sensors in various MEMS applications. In this work, the overall sensitivity
of the micro cantilevers that are used in MEMS applications by increasing their
deflection and frequency characteristics due to shape are changed are studied. To
improve the sensitivity investigated the basic and modified design C-section micro
cantilevers are studied.
A micro cantilever is a device that can act as a physical, chemical or biological
sensor by detecting changes in cantilever bending or vibrational frequency. It is the
miniaturized counterpart of a diving board that moves up and down at a regular
interval. This movement changes when a specific mass of analyte is specifically
adsorbed on its surface similar to the change when a person steps onto the diving
board. But microcantilevers are a million times smaller than the diving board having
dimensions in microns.
The ability of label-free detection, scalability to allow massive parallelization, and
sensitivity of the detection range applicable to, in vivo problems are some of the
important requirements for a future generation of biosensors. Although generally used
in the topological investigations of surfaces such as in atomic force microscopy,
arrays of micro cantilevers are attracting much interest as sensors in a wide variety of
applications. Micro cantilever sensors have emerged as a universal, very powerful and
highly sensitive tool to study various physical, chemical, and biological phenomena.
As biosensors, they are found to display label-free, real-time and rapid assaying
features.
This project proposes the stoney equation first used to calculate the surface stress
induced moment and then applied to the microcantilever free end to produce
deflection. The stress analysis of the proposed design, We found that the stoney
equation is predicting cantilever deflections, and that for increasing the sensitivity of
a microcantilevers biosensors increasing the cantilever thickness is more practical.
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The cantilevers transduce changes in temperature, mass, or surface stress into a
(nano) mechanical response. Antibody immobilization, followed by specific antigen
binding or DNA hybridization, induces surface stresses in the cantilever; due to
Vander Waal, electrostatic, or steric interactions. These surface stresses bend the
cantilever. Bending of the cantilever can be detected optically. A piezoresistor strain
gauge embedded in the cantilever experiences a change in strain due to the bending
and responds with a change in resistance (electrical detection).
Surface stresses produced due to DNA hybridization, antibody immobilization are
small (5 N/m to 0.5 N/m). These stresses should be effectively translated to strain in
the piezoresistive layer so as to cause a detectable change in resistance, i.e., for the
electrical detection of surface stresses observed in bio sensing applications, composite
piezoresistive cantilever structures are used as the sensing elements. In such a
cantilever structure, it is necessary to understand how the variations in material type,
thickness and piezo layer placement with in the cantilever stack.
Micro cantilever biosensors generally use optical deflection readout technique to
measure the absorption at induced deflections, and hence in assaying the unknown
species present in a media. In micro cantilever biosensors, the accuracy of
measurement strongly depends on the accurate determination of the surface stress
induced deflections. The deflections usually range a few tens to a few hundred of a
micrometer. Measuring deflections of this order requires extremely sophisticated
readout arrangements. Therefore, increasing the sensitivity of a micro cantilever
without increasing the complexity in the deflection detection is a major challenge.
The overall sensitivity of a micro cantilever biosensor depends on the design
sensitivity of the cantilever and the measurement sensitivity of the deflection
measurement system. A sensitive cantilever design should efficiently convert the
bimolecular stimulus into a large cantilever deflection, whereas the measurement
sensitivity should ensure that the deflections measured are only induced because of
the bimolecular stimulus and not due to some ambient disturbance source. The design
sensitivity of the cantilever can be improved by changing the cantilever design in such
a way that for a given surface stress larger deflections can occur. This scheme can be
realized by reducing the bending stiffness of the cantilever or by using softer
cantilever materials. To improve the measurement sensitivity, the fundamental
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resonant frequency of the cantilevers should be made as large as possible, because the
accuracy in deflection measurement depends not only on the deflection occurred, but
also on the signal-to-noise ratio.
Most of the noise in the deflection signal can be attributed to flow and thermal
induced excitations. To improve the signal-to-noise ratio, and hence the measurement
sensitivity, the resonant frequency of the cantilever should be made as high as
possible. Thus, to increase the overall cantilever sensitivity, one should select a design
that exhibits both higher deflection and higher resonant frequency.
To improve the design sensitivity of cantilevers various designs and schemes have
been reported. Silicon micro cantilevers are commonly used in biosensors. However,
due to high elastic modulus, silicon cantilevers exhibit extremely low deflections for a
given surface stress change. Therefore to increase the deflections, polymer cantilevers
can be used. Since the elastic modulus of polymers is generally much lower than
silicon, the deflections induced are magnified many folds. Polymer cantilevers,
however, have a major drawback in being very temperature sensitive, because of the
thermal bimetallic effects. Thermal induced deflections exceeding the surface-stress
induced deflections are not uncommon. Hence, polymer cantilevers require a fine
control of the surrounding. The other way to improve design sensitivity is to increase
the cantilever deflection by changing the shape of the cantilever. By reducing the
moment of inertia of a cantilever its bending stiffness can be reduced, which results in
higher deflection.
For calculation of deflection analytically the Stoney’s equation is being used. The
stoney equation is the main basis for the cantilever deflection for thin films. The
deflection of cantilever main depends on the material to be used, surface stress,
bending moment etc., The C-section micro cantilever is being used for the analysis of
deflection. For various surface stress values the deflection is being used and the
graphs are being drawn.
With the objective of increasing the deflection and resonant frequency at the same
time, the project investigates the deflection and stress analysis of C-Section profile
micro cantilevers. The surface-stress induced deflection in the micro cantilever is
modeled by an equivalent in-plane tensile force acting on the top surface of the
cantilever, in the length direction. Commercial finite element method (FEM) software
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ANSYS Multi-physics is used in this analysis. All the cantilevers are investigated for
deflection and stress by using ANSYS.
1.1 Mass Sensitive Detection by Microcantilevers
A micro-cantilever is a device that can act as a physical, chemical or biological
sensor by detecting changes in cantilever bending or vibrational frequency. It is the
miniaturized counterpart of a diving board that moves up and down at a regular
interval. This movement changes when a specific mass of analyte is specifically
adsorbed on its surface similar to the change when a person steps onto the diving
board. But microcantilevers are a million times smaller than the diving board having
dimensions in microns and different shapes as shown in figure 1.
Fig.1: Meshed cantilever showing fix and load ends.
Molecules adsorbed on a microcantilever cause vibrational frequency changes and
deflection of the microcantilever. Viscosity, density, and flow rate can be measured
by detecting changes in the vibrational frequency. Another way of detecting
molecular adsorption is by measuring deflection of the cantilever due to adsorption
stress on just one side of the cantilever. Depending on the nature of chemical bonding
of the molecule, the deflection may be up or down. Biochips with mechanical
detection systems commonly use microcantilever bi-material (e.g. Au–Si) beams as
sensing elements.
The Au side is usually coated with a certain receptor. Upon the binding of the
analyte (e.g. biological molecules, such as proteins or biological agents) with the
receptor, the receptor surface is either tensioned or relieved. This causes the
microcantilever to deflect, usually in nanometers, which can be measured using
optical techniques. The deflection is proportional to the analyte concentration. The
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concept has been employed in screening certain diseases such as cancer and detecting
specific chemical and biological warfare agents.
1.2 Microcantilever Deflection Detection Methods
1.2.1Deflection in Biosensors:
Biosensors are electronic devices that convert bio molecular interactions into
measurable signal. The purpose of a bio-sensor is to detect and analyses the unknown
biological elements present in a medium. Biosensors have two main elements, a bio-
receptor and transducer. Bio-receptors are target specific and known bio-molecules
that combined with the target analytic molecules, and generate a unique signal during
the reaction for sensing purpose one surface of the bio sensor is functionalized by
depositing a sensing layer of known bio receptor molecules on to it. This bio-sensitive
layer either contains the bio receptors or the bio receptors are covalently bonded to it.
The most common types of bio-receptors used in bio-sensing are based on proteins,
antibody/antigen or nucleic acid interactions. The traducer element of the biosensor
converts the bio-molecular reactions between the target and bio-receptor molecules
into a measurable signal. The signals can be measured using appropriate detection
technics like electro chemical, optical or mechanical. In bio-sensing applications
sample preparation and molecular labeling of the target analytic is a basic requirement
therefore, label-free detection technique is critical in developing rapid, economic and
user-friendly biosensors and bio analytical kits.
Figure 2.Working principle of micro cantilever bio sensor. Functionalization of bio sensor by
depositing bio receptors (left). Surface stress induces deflection (right). Symbols ◊ and Y
represent target analytes and bio receptor molecules.
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With the ability of label-free detection and scalability to allow massive
parallelization already realized by microcantilevers biosensors, the next challenge in
cantilever biosensor development lies is achieving the sensitivity in detection range
applicable to in vivo analysis. The sensitivity of a cantilever biosensor strongly
depends on it ability to convert biochemical interaction into micromechanical motion
of the cantilever. The deflections of a cantilever biosensor are usually of the order of
few tens to few hundreds of a nanometer. Such extremely low deflections necessitate
use of advanced instruments for accurately measuring the deflections. As a
consequence, most of the applications of cantilever biosensors are done in laboratories
equipped with sophisticated deflection detection and readout techniques. The authors
believe that if the deflections of a cantilever biosensor be increased, its advantages
will be two fold. First, if the deflections are high a less sensitive readout technique
can be used to accurately measure the deflection, which will help in reducing the cost
of a cantilever-based biosensor kit. Second, it will help us in detecting analytes in in
vivo solution concentrations range. The concentrations of some clinically important
analytes vary between 10-4 to 10-15 mol/L. The detection of analytes in such large
dynamic range requires an extremely sensitive cantilever. This study proposes and
analyses a new high sensitive cantilever design that can assay analytes in extremely
low concentrations. Commercial finite element analysis software ANSYS is used to
analyze and compare the conventional and the proposed microcantilevers designs.
1.2.2 The Piezoresistive Deflection Detection Method
The piezoresistive method involves the embedding of a piezoresistive material
near the top surface of the cantilever to record the stress change occurring at the
surface of the cantilever. As the microcantilever deflects, it undergoes a stress change
that will apply strain to the piezoresistor, thereby causing a change in resistance that
can be measured by electronic means. The advantage of the piezoresistive method is
that the readout system can be integrated on the chip. The disadvantage is that the
deflection resolution for the piezoresistive readout system is only one nanometer
compared with one Angstrom by optical detection method. Another disadvantage with
the method is that a piezoresistor has to be embedded in the cantilever. The
fabrication of such a cantilever with a composite structure is more complicated.
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The piezoresistor material in the beam must be localized as close to one surface of
the cantilever as possible for maximum sensitivity. The type of doping being used for
fabrication of the piezoresistive material is an important factor. The piezoresistive
coefficient of N-type silicon is greater than that for P-type. The resistance of a
piezoresistive material changes when strain is applied to it. The relative change in
resistance as function of applied strain.
The sensitivity of a piezoresistor varies proportionally to the thickness t and the
radius of curvature. The Gage Factor is proportional to Young’s Modulus, E, which is
the intrinsic characteristic of material. The Gage Factor can also be calculated directly
by straining the cantilevers and measuring the resistance change.
1.3Bending Behavior of Cantilever Beams
Uniform surface stress acting on an isotropic material increases (in the case of
compressive stress) or decreases (in case of tensile stress) the surface area. If this
stress is not compensated at the opposite side of a thin plate or beam, the whole
structure will bend. Between the areas of compressive stress and tensile stress, there is
a neutral plane which is not deformed. Due to bending, a force F is acting at a
distance of x in the neutral plane results in a bending moment M=F.x. Therefore, the
radius of curvature R is given by:
(1)
Where E is the apparent Young’s modulus and I is the moment of inertia given by
the following equation for rectangular beams
(2)
The change in the surface stress at one side of the beam will cause static bending,
and the bending moment can be calculated as:
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1.4 Objective of project
• To calculate the deflection, stress concentration of C- section micro cantilever
beams
• Calculating the deflection of microcantilevers using the stoney equation.
• To analyze the deflection and stress concentration of same beam in the finite
element software ansys
• The mathematical and simulated values have been found and compared and errors
are determined in this project
1.5 PREFACE:
CHAPTER 1: This chapter deals with the introduction of the project about micro
cantilevers their deflection, surface stress and the stoney equation which is used for
the theoretical calculation and about ansys.
CHAPTER 2: In this chapter the literature review and the data collection required
for the project is being written
CHAPTER 3: This chapter deals with description and derivation of the stoney
equation required for the calculation of deflection and about ansys which is used for
the simulation is being described in this chapter
CHAPTER 4: In this chapter the results for deflection of the cantilever are being
found analytically for different values of surface stress and same as in simulation
CHAPTER 5: The errors have been determined and tabulated for different surface
stress and graphs are being drawn between surface stress vs deflection and pressure vs
deflection
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CHAPTER-2
LITERATURE REVIEW & DATA COLLECTION
2.1 CANTILEVERS
Beams are mechanical structures deeply studied in Mechanical Engineering. One
of the reasons for the study under certain approximations, e.g. small deformations,
differential equations that determine their deformation are one-dimensional, what
usually makes their behavior can be very simply explained and often obtain high
accuracy in analytical results. In addition, when performing Finite Element Modelling
(FEM) very complex structures can be simplified as consisting of many assembled
beams, what increments the importance of these structures.
There exists several kind of beams. Depending on the boundary conditions of both
edges, the single-clamped or double-clamped beams can be found. In addition,
depending on the shape of the beam and/or the cross section, infinite of types can be
found.
Fig 3: Cantilevers
2.2 MEMS (MICRO ELECTRO MECHANICAL SYSTEMS)
Silicon is the most used material for the fabrication of microelectronic circuits. The
development of microelectronics fabrication techniques has been allowing the
definition of smaller devices. Since the beginning of the 80s, numerous groups have
been working in the use of such fabrication techniques to define tri-dimensional
structures and hence what is named Micro Electro Mechanical Systems (MEMS). The
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main characteristic of these systems is the presence of a mechanical part that is
essential in the working principle of the system. The mechanical properties of the
named part are of biggest importance inside the characteristics of the whole system.
The materials used to compound the part are, for example: crystalline Silicon,
polycrystalline Silicon, Silicon nitride, Silicon dioxide, aluminum, etc. But the
preferred one is crystalline Silicon, due to its outstanding mechanical properties.
(Electronic characteristics of Si are good, but worse than those of other
semiconductors, as Gas.
In most of the cases, MEMS are used as sensors where the mechanical part
corresponds to the transducer element of the sensor. The fabrication of the transducer
element by means of silicon processing technologies allow a reduction in size of the
whole sensor and, with the reduction in size of the mechanical transducer, the
sensitivity is also improved. Even more, if the transduction principle was electro-
mechanical, the variations in the mechanical properties of the transducer (changes in
deflection, stresses, etc.) would provide an electronic signal which could be
electronically processed by some circuitry located nearby the transducer. And this was
the origin of Micro Electro Mechanical Systems (MEMS).
Fig.4 A simple micro cantilever
Talking about MEMS, it refers to devices whose mechanical parts have
dimensions ranging from 1mm to 1m, that combine both electric and mechanic
components and are fabricated using integrated circuits processing technologies
Inside the MEMS there are several types of sensors and they can be arranged in
different ways depending on the mechanical structure of the transducer (cantilever,
membrane, etc.), on the transduction principle that is being used on the circuitry to
process the signal.
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A particular type of beam is the cantilever beam. A cantilever is a beam anchored
at one end and other projecting into space. It is a well known mechanical structure
that has been widely used in constructions during the last two centuries, mainly for
bridges and balconies. In addition, they have also been used as mechanical
transducers in some sensors, e.g. with strain gauges for force or thermal gradient
measurements, as fundamental part of some devices like phonograph, etc.
2.3 MICRO CANTILEVERS IN MEMS
On the other hand, the development of microelectronics fabrication techniques has
been allowing the definition of smaller devices. In addition to transistors, diodes and
other circuit elements, since the beginning of the 80s, numerous groups have been
working in the use of such fabrication techniques to accomplish what is named Micro
Electro Mechanical Systems (MEMS), what implies the fabrication of micrometric
mechanical structures. Examples of MEMS devices are accelerometers (used for
example in airbag control systems), pressure sensors, biochemical sensors (used for
medical applications, environment analysis, etc.), etc.
The mechanical part of these MEMS devices can be made of any materials that are
used in microelectronics fabrication, e.g. aluminum, silicon dioxide, silicon nitride,
polycrystalline silicon and crystalline silicon. The latter is the most used because it is
a crystalline material, hence its mechanical properties are well determined, and also
because of it outstanding mechanical properties. This fact is one of the main
advantages of silicon-based MEMS. In addition, as it has been advanced, when
decreasing dimensions, sensitivity increases (this happens not only in cantilevers but
in almost every mechanical device). Hence, if dimensions are reduced until the
micrometer range, a much deformations in cantilever profile (static mode) can be
produced by acceleration, mechanical surface stress and punctual forces, while
changes in resonant frequency can be produced by mass addition and punctual forces.
All these physical magnitudes are what directly affect the cantilever, but they can be
originated by several different phenomena, that will be commented below. In
addition, transducer sensitivity is greater with decrease in their dimensions. Hence,
the smaller the cantilever, the more sensitive it is (to any of the possible applied loads
commented before).
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Fig 5: Schematic diagram of a typical dynamic-mode operation. A functionalized cantilever
is oscillating at its resonant frequency (a). When biomolecules bind to the surface,
mass of the cantilever increases, causing the resonant frequency to decrease (b).
On the other hand, the development of microelectronics fabrication techniques has
been allowing the definition of smaller devices. In addition to transistors, diodes and
other circuit elements, since the beginning of the 80s, numerous groups have been
working in the use of such fabrication techniques to accomplish what is named Micro
Electro Mechanical Systems (MEMS), what implies the fabrication of micrometric
mechanical structures. Examples of MEMS devices are accelerometers (used for
example in airbag control systems), pressure sensors, biochemical sensors (used for
medical applications, environment analysis, etc.), etc.
The mechanical part of these MEMS devices can be made of any of the materials
that are used in microelectronics fabrication, e.g. aluminum, silicon dioxide, silicon
nitride, polycrystalline silicon and crystalline silicon. The latter is the most used
because it is a crystalline material, hence its mechanical properties are well
determined, and also because of its outstanding mechanical properties. This fact is one
of the main advantages of silicon-based MEMS. In addition, as it has been advanced,
when decreasing dimensions, sensitivity increases (this happens not only in
cantilevers but in almost every mechanical device). Hence, if dimensions are reduced
until the micrometer range, a much bigger sensitivity is achieved compared to that of
macro devices.
Cantilevers are one of the most used mechanical structures in MEMS. One of the
main reasons is that, their shape can easily defined and they can be fabricated on a
wide variety of materials by using different fabrication processes. In addition, micron
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and sub-micron cantilevers, phenomena originating beam deflection or changes in
resonant frequencies can be determined.
• Surface stress: temperature changes, DNA hybridization, Prostate Specific
Antigen (PSA) concentration, etc.
• Mass change: particles flux, PSA detection, etc.
• Force at the apex: properties of bimolecular, DNA strand separation, Vander
Waals forces, etc. The detection of this plethora of magnitudes is often allowed
because of the use of smart and specialized fabrication and post-processing of the
cantilevers. For example, to measure temperature differences, a composite
cantilever (fabricated using at least two materials) has to be used. Cantilever
coated with different polymers has been proved satisfactory in order to detect
different odorants, or in order to detect pH changes. On the other hand, a careful
choice of the beam dimensions has to be made in order to fabricate devices with
the required resolution and sensitivity for each individual application.
2.4 APPLICATIONS OF MICRO-CANTILEVERS
BIOSENSORS
However, it is as biosensing tools that micro and sub-micro cantilevers have been
undergoing the furthest development in recent years and where they have been proved
to be one of the best alternatives. As pointed out, a biosensor should allow specific
and quantitative detection of analytics, label-free detection of the biological
interaction, massive parallelization by the scalability of the sensors and high-enough
sensitivity for in vivo applications. Three types of instruments are being developed to
meet those requirements (Surface Plasmon Resonance, Quartz Crystal Microbalances
and cantilever-based sensors) and the latter is thought to be the one which fits better
in of them.
First, size of mechanical part allows high sensitivity, short response times (high
resonant frequencies), access to small volume samples and parallel integration. In
addition, by means of functionalization of cantilevers surface(s) label -free and
specific detection is achieved. This is based on the fact that some bimolecular can
only bind to one or a few different molecules (see for example antigen-antibody
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binding). This can be understood as an ability to recognize those molecules and can
be used in order to detect one of them, i.e. if a surface is functionalized with an
antigen, only its own specific antibody will bind, and hence specific detection of that
compound will be performed.
Using these specific bindings, detection of molecules is allowed by means of
forces at the apex, mass change or surface stress-induced bending, although the most
extended technique is the latter. The intermolecular forces arising from adsorption of
small molecules to a surface is known to induce surface stress and this transduction is
used to cause cantilever bending which is using any deflection measurement, finally
transduces bimolecular detection into an electronic signal.
Hence, cantilever-based sensors offer a wide range of applications that are only
limited by surface functionalization techniques. Many different compounds have been
sensed up-to-date, as for example: Prostate Specific Antigen (PSA) , biotin-avidin,
Acute Myocardial Infarction (AMI) markers, DNA (with single-base mismatch
resolution) etc.
Temperature Sensors / Heat Sensors
Changes in temperature and heat bend a cantilever composed of materials with
different thermal expansion coefficients by the bimetallic effect. Microcantilever
based sensors can measure changes in temperature as small as 10-5
K and can be used
for photo thermal measurement. They can be used as micro calorimeters to study the
heat evolution in catalytic chemical reactions and enthalpy changes at phase
transitions. Bimetallic microcantilevers can perform photo thermal spectroscopy with
a sensitivity of 150 fJ and a sub-millisecond time resolution. They can detect heat
changes with Joule sensitivity.
Viscosity Sensors
Changes in the medium viscoelasticity shift the cantilever resonance frequency. A
highly viscous medium surrounding the cantilever as well as an added mass will damp
the cantilever oscillation lowering its fundamental resonance frequency. Cantilevers
can therefore be vibrated by piezoelectric actuators to resonate and used as viscosity
meters.
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Micro-Cantilever-Based Sensors
Cantilevers can be used as mechanical transducers. They are very commonly used
because of their versatility, given that loads of different signals can affect their
configuration, that is, can be sensed by means of a cantilever beam. Cantilevers can
be used in two different modes of operation, i.e. static and dynamic. In the static mode
of operation, cantilever deflection is monitored continuously in order to detect
deformations produced by external measurands. On the other hand, in dynamic mode
changes in the value of the resonant frequency are measured.
Fig.6: Static-mode operation. Two cantilevers, one of them with a gold and functionalized
layer on top and the other working as a reference cantilever (a), are exposed to the
flux of some biomolecules, which will bind with the functionalization of measuring
cantilever causing it to deflect (b).
Many of the research groups are working with the cantilever sensors. The below
said gives a detailed report on the cantilever sensors.
1) Christiane Ziegler, (2004) explains about the cantilevers based bio-sensors. He
explains briefly about the theory of cantilevers biosensors, transduction principles in
various materials. Further explains on liquid –phase sensing with particular emphasis
on biological applications. The most common method to measure the deflection of a
cantilever is the optical lever technique. Another optical detection method is based on
the interference effect
2) T. Chu Duc et al, explains about the Piezoresistive Cantilever for Mechanical
Force sensors. They describe about the piezoresistive cantilever force sensors that are
used to evaluate the impact force between microhandling tools and micro particles in
the nano-Newton range. This cantilever is based on conventional silicon wafers, the
applied force on this sensor is parallel to wafer surface. The proposed structure and its
electronic circuit suppress the effect of the vertical force with 19 dB in practice.
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3) Michel Godin et al, (2003), reported on combined in situ micromechanical
cantilever-based sensing and ellipsometry. Two cantilever-based chemical sensors are
used. First, a differential cantilever-based chemical sensor, while the second is
passivated cantilever is used as a reference. Both sensors, operated in static mode,
achieve a cantilever deflection measurement sensitivity of 0.2 nm and a surface stress
resolution of 5x10-5
N/m. Simultaneous surface stress and monolayer thickness
measurements were performed during the formation of dodecanethiol SAMs on gold.
4) Mo Yang et al, (2003) describes about the High sensitivity piezoresistive
cantilever design and optimization for analyte-receptor binding. The mechanical
design and optimization of piezoresistive cantilevers for biosensing applications is
studied using finite element analysis. The change of relative resistivity of
piezoresistive microcantilevers is analyzed in the presence of the chemical reaction at
the receptor surface under the condition of oscillating flow. Finally, the optimum SCR
modified ‘C’ piezocantilever system for biosensing is designed and the optimal
parameters are set for high sensitivity. They have developed a finite element
computational model for simulating the chemo-mechanical binding of analytes to
specific binding molecules on functionalized surfaces
5 Mohd. Zahid Ansari,(2009), reported on Comparison between deflection and
vibration characteristics of rectangular and trapezoidal profile Microcantilevers. They
concluded that Arrays of microcantilevers are increasingly being used as physical,
biological, and chemical sensors in various applications. To improve the sensitivity of
microcantilever sensors, this study analyses and compares the deflection and vibration
characteristics of rectangular and trapezoidal profile microcantilevers. Three models
of each profile are investigated. The cantilevers are analyzed for maximum deflection,
fundamental resonant frequency and maximum stress. The surface stress is modelled
as in-plane tensile force applied on the top edge of the microcantilevers.
6) Nitin S. Kale, (2009), explained about the Fabrication and Characterization of a
Polymeric Micro cantilever With an Encapsulated Hotwire CVD Polysilicon
Piezoresistor. In this they explained about the process to fabricate and characterize a
novel polymeric cantilever with an embedded piezoresistor. This device exploits the
low Young’s modulus of organic polymers and the high gauge factor of polysilicon.
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7) Daniel Ramos, (2007), reported on the Study of the Origin of Bending Induced
by Bimetallic Effect on Microcantilever. An analytical model is used for the
deflection and force of a biomaterial cantilever. Axial load is implemented upon
temperature changes.by using the stoney equation the bending in the cantilevers is
being found. For different temperature changes the stoney model is used for stress
calculation
8) Sumio Hosaka et al, (2006), reported on the possibility of a femtogram mass
biosensor using a self-sensing cantilever. In the experiments, measured the change in
the mass of adsorbed water molecules and the reaction between an antigen and an
antibody on the cantilever from the resonance frequency shift.
9) Eric Finot et al, (2008), explained about the Measurement of Mechanical
Properties of Cantilever Shaped Materials. Microcantilevers were first introduced as
imaging probes in Atomic Force Microscopy (AFM) due to their extremely high
sensitivity in measuring surface forces. As temperature gauges, double-layered
microcantilevers operating in deflection mode, can reach extreme thermal
sensitivities.
10) Ricardo C. Teixeira et al, reported on the Stress Analysis of Ultra Thin Ground
Wafers. The influence of the backside induced stress in Si wafers thinned down to
~20μm by means of an IR time-of-flight like technique. Such aggressive thinning is a
requirement for high density via interconnect, stacked die packaging and flexible
electronics.
11) Ricardo C. Teixeira et al, reported on Stress Analysis on Ultra-Thin Ground
Wafers. For the stress analysis the ground wafers of the size 100µm is used. The
stress is being found by the help of stoney equation. They concluded that even
applying several corrections for Stoney’s formula average stress on the thinned wafer
is still below 10 MPa
From the above literature survey many of the research groups are using stoney
equation for finding the deflection of the cantilever; hence we also implement stoney
equation for finding the deflection analytically.
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CHAPTER-3
Materials and Methods
3.1 Mathematical Theory of Cantilever Deflection:
The stoney equation is a fundamental expression relating the residual surface stress
(∆σ) per unit length in a film to the curvature (k) of a substrate the film is deposited
onto. The curvature does not depend on the material or the geometric properties of the
film. This equation is commonly used in determining the residual surface stress in thin
films. In its original form, the equation is given as:
(1)
Where E and t are the elastic modulus and the thickness of the substrate. Since the
cantilever plate is long and wide, in general practice E is replaced by the biaxial
modulus E/ (1-v) to accommodate he poissions ratio(v) coupling. Surface stresses in
solids are assumed analogous to the surface tension in liquids. The unit surface stress
measurement is different from that of bulk stress. For the bulk stress it is N/m2
, where
as for surface stress it is N/m. For modeling purpose the surface stress induced
deflection in a substrate is often compared to a concentrated moment induced
1deflection in a thin plate. Figure shows the schematic for a cantilever plate subjected
to a concentrated moment on its free end while the other end is fully constrained.
Modeling the surface stress induced curvature in a micro cantilevers by equating to
a concentrated moment induced curvature. The concentrated moment and the surface
stress are related as M0=∆σt/2, where t is thickness of the microcantilevers.
18
19.
Fig 7: cantilever subjected to bendining moment
Applying stoney equation, assumption that the surface stress bends the plate with
uniform curvature, into the concentrated moment induced plate bending the following
curvature relation for plate bending can be given as:
(2)
Where M0 is the applied concentrated moment, E is the elastic modulus of the
plate, I is the moment of inertia of a beam. For beam of rectangular cross-section the
moment of inertia is given as I=bt3
/12, where b and t are the thickness of the beam,
respectively. Comparing the curvature relations the following relation between the
surface stress and the moment per unit length can be established:
(3)
This relation shows that the moment is directly proportional to the surface stress
and the geometric properties of the plate. Moreover, it does not depend on the
19
20.
material properties of the plate. The governing differential equation for an isotropic,
tin plates expressing the bending and twisting moments in terms of the curvature and
deflection is given as;
(4)
(5)
(6)
Where D=Et3
/12(1-v2
) is the flexural rigidity of the plate. In these equations, the
moment are expressed in moment per unit length. Assuming Mx=My=M0 and
neglecting the shear component Mxy, the above equation can be solved to give:
(7)
If the cantilever is clamped in such a way that its y direction is restricted, the above
equation can be further simplified as:
(8)
This is well known form of stoney equation commonly used in predicting the
residual surface stress in thin films by measuring the induced deflection.
(9)
20
21.
This is the modified form of stoney’s equation. Where k is a constant dependent on
the material and geometric properties of the cantilever. For a cantilever of l/b <, 5 and
ν ≤ 0.25, the value of K lies between 0.3 and 1.05.
The above equation is used to calculate the deflection of the cantilever and the
simulation is done with Ansys. A simple java program is also written to make the
calculations very easy.
INTRODUCTION TO ANSYS
ANSYS is general-purpose finite element analysis (FEA) software package. Finite
Element Analysis is a numerical method of deconstructing a complex system into
very small pieces (of user-designated size) called elements. The software implements
equations that govern the behavior of these elements and solves them all; creating a
comprehensive explanation of how the system acts as a whole. These results can be
presented in tabulated or graphical forms. This type of analysis is typically used for
the design and optimization of a system for too complex to analyze by hand. Systems
that may fit into this category are too complex due to their geometry, scale, or
governing equations.
ANSYS provides a cost-effective way to explore the performance of products or
processes in a virtual environment. This type of product development is termed virtual
prototyping.
With virtual prototyping techniques, users can iterate various scenarios to optimize
the product long before the manufacturing is started. This enables a reduction in the
level of risk, and in the cost of ineffective designs. The multifaceted nature of ANSYS
also provides a means to ensure that users are able to see the effect of a design on the
whole behavior of the product, be it electromagnetic, thermal, mechanical etc.
The ANSYS computer program is a large-scale multi purpose Finite Element
program that may be used for solving several classes of engineering analysis. The
analysis capabilities of ANSYS include the ability to solve static and dynamic
structural analysis, steady state and transient heat transfer problems, more frequency
and buckling Eigen value problems, static or time varying magnetic analysis, and
various types of field and coupled field applications.
21
22.
The programming contains many special features which allow nonlinearities or
secondary effects to be included in the solution, such as plasticity, large strain, hyper
elasticity, creep, swelling, large deflections, contact, stress stiffening, temperature
dependency, material anisotropy, and radiation.
3.2 Structural
Modules in ANSYS
Structural analysis is probably the most common application of the finite element
method. The term structural (or structure) implies not only civil engineering structures
such as bridges and buildings, but also naval, aeronautical, and mechanical structures
such as ship hulls, aircraft bodies, and machine housings, as well as mechanical
components such as pistons, machine parts, and tools.
3.2.1 Types of Structural Analysis
The seven types of structural analyses available in the ANSYS family of products
are explained below. The primary unknowns (nodal degrees of freedom) calculated in
a structural analysis are displacements. Other quantities, such as strains, stresses, and
reaction forces, are then derived from the nodal displacements.
You can perform the following types of structural analysis. Each of these analysis
types are discussed in detail in this manual.
a. Static Analysis: Used to determine displacements, stresses, etc. under static
loading conditions. Both linear and nonlinear static analysis. Nonlinearities
can include plasticity, stress stiffening, large deflection, large strain, hyper
elasticity, contact surfaces, and creep.
b. Transient Dynamic Analysis: Used to determine the response of a structure
to arbitrarily time-varying loads. All nonlinearities mentioned under Static
Analysis above are allowed
22
23.
c. Buckling Analysis: Used to calculate the buckling loads and determine the
buckling mode shape. Both linear (Eigen value) buckling and nonlinear
buckling analysis are possible.
3.2.2 Thermal
ANSYS is capable of both steady state and transient analysis of any solid with
thermal boundary conditions Steady-state thermal analysis calculate the effects of
steady thermal loads on a system or component. Users often perform a steady-state
analysis before doing a transient thermal analysis, to help establish initial conditions.
A steady-state analysis also can be the last step of a transient thermal analysis;
performed after all transient effects have diminished. ANSYS can be used to
determine temperatures, thermal gradients, heat flow rates, and heat fluxes in an
object that are caused by thermal loads that do not vary over time. Such loads include
the following:
• Convection
• Radiation
• Heat flow rates
• Heat fluxes (heat flow per unit area)
• Heat generation rates (heat flow per unit volume) Constant temperature
boundaries.
A steady-state thermal analysis may be either linear, with constant material
properties; or nonlinear, with material properties that depend on temperature. The
thermal properties of most material vary with temperature. This temperature
dependency being appreciable, the analysis becomes nonlinear. Radiation boundary
conditions also make the analysis nonlinear. Transient calculations are time dependent
and ANSYS can both solve distributions as well as create video for time incremental
displays of models.
3.2.3 Fluid flow
23
24.
The ANSYS/FLOTRAN CFD (Computational Fluid Dynamics) offers
comprehensive tools for 'analyzing two-dimensional and three-dimensional fluid flow
fields. ANSYS is capable of modeling a vast range of analysis types such as: airfoils
for pressure analysis of airplane wings (lift and drag), flow in supersonic nozzles, and
complex, three-dimensional flow patterns in a pipe bend. In addition,
ANSYS/FLOTRAN could be used to perform tasks including:
Calculating the gas pressure and temperature distributions in an engine exhaust
manifold.
Studying the thermal stratification and breakup in piping systems.
Using flow mixing studies to evaluate potential for thermal shock.
Doing natural convection analysis to evaluate the thermal performance of
chips in electronic enclosures.
Conducting heat exchanger studies involving different fluids separated by
solid regions.
FLOTRAN analysis provides an accurate way to calculate the effects of fluid
flows in complex solids without having to use the typical heat transfer analogy of
heat flux as fluid flow.
Types of FLOTRAN analysis that ANSYS is able to perform include:
• Laminar or Turbulent Flows
• Thermal Fluid Analysis
• Adiabatic Conditions
• Free surface Flow
• Compressible or incompressible Flows
3.2.4 Magnetic
Magnetic analyses, available in the ANSYS/Multiphysics and ANSYS/ Emag
programs, calculate the magnetic field in devices such as:
24
25.
• Power generators
• Magnetic tape/disk drive &Transformers
• Electric motors
• Video display device sensors.
Magnetic fields may exist as a result of an electric current, a permanent magnet, or
an applied external field
3.2.5 Acoustics / Vibration
ANSYS is capable of modeling and analyzing vibrating systems. Acoustics is the
study of the generation, propagation, absorption, and reflection of pressure waves in a
fluid medium. Coupled acoustic analysis takes the fluid-structure interaction into
account. An uncoupled acoustic analysis models only the fluid and ignores any fluid-
structure interaction.
The ANSYS program assumes that the fluid is compressible, but allows only
relatively small pressure changes with respect to the mean pressure. Also, the fluid is
assumed to be non-flowing and in viscid (that is, viscosity causes no dissipative
effects).
3.2.6 Coupled Fields
A coupled-field analysis is an analysis that takes into account the interaction
(coupling) between two or more disciplines (fields) of engineering. A piezoelectric
analysis for example, handles the interaction between the structural and electric fields:
it Ives for the voltage distribution due to applied displacements, or vice versa. Other
examples of coupled-field analysis are thermal-stress analysis, thermal-electric
analysis, and fluid-structure analysis.
Some of the applications in which coupled-field analysis may be required are
pressure vessels (thermal-stress analysis), fluid flow constrictions (fluid-structure
analysis), induction heating (magnetic-thermal analysis), ultrasonic transducers
25
26.
(piezoelectric analysis), magnetic forming (magneto-structural analysis), and micro
electro mechanical systems (MEMS).
3.3 Communicating with ANSYS Program
There are two methods to use ANSYS.
The first is by means of the graphical user interface or GUI. This method follows
the conventions of popular Windows and X-Windows based programs.
The second is by means of command files. The command file approach has a
steeper learning curve for many, but it has the advantage that an entire analysis can be
described in a small text file, typically in less than 50 lines of commands. This
approach enables easy model modifications and minimal file space requirements.
3.3.1 Communicating via GUI
The easiest way to communicate with ANSYS is by using the ANSYS menu
system called the Graphical User Interface (GUI). GUI consists of windows, menus,
dialog boxes and other components that allow one to enter input data and execute the
ANSYS functions by simply picking buttons with mouse or typing in response to
prompts.
3.3.2 Communicating through Prompts
Commands are the instructions that direct the ANSYS program. ANSYS has
hundreds of commands, each designed for a specific function. Most commands are
associated with specific (one or more) processors and to work only with that
processor or those processors. Typing it in command window can access a command.
Every ANSYS command has a specific format.
3.4 The ANSYS Environment 6 windows
3.4.1 Input Window
Below the Utility Menu and to the left, their was an Input Window. The Input
window shows program prompt messages and operator to type in commands directly.
3.4.2 Main Menu
26
27.
Directly below Input Window you will find a vertical list of menu items referred as
the ANSYS Main Menu.
The Main Menu contains the primary ANSYS functions, organized by
preprocessor, solution, general postprocessor, design optimizer. It is from this menu
that the vast majority of modeling commands are issued.
3.4.3 Output window
The Output Window shows text output from the program, such as listing of data
etc. It is usually positioned behind the other windows and can be put to the front if
necessary.
3.4.4 Utility Menu
The window at the top of the screen, displaying a long horizontal list of menu
items, is referred to as the Utility menu. Utility Menu contains functions that are
available throughout the ANSYS session, such as file controls, selections, graphic
controls and parameters.
3.4.5 Toolbar
To the right of the Input Window, there is a Toolbar. Contains abbreviations:
Short-cuts to commonly used commands and functions. A few predefined
abbreviations are available. Requires knowledge of ANSYS commands. A powerful
feature which can use to create own "button menu" system.
3.4.6 Graphics window
Displays graphics created in ANSYS or imported into ANSYS.
3.5. Performing a typical ANSYS Analysis
A typical ANSYS analysis has three distinct steps:
i. Build the model.
ii. Apply loads and obtain the solution.
iii. Review the results.
27
28.
3.5.1 Building a Model
Building a finite element model requires more of an ANSYS user's time than any
other part of the analysis. First, by specify a job name and analysis title. Then, use the
PREP7 preprocessor to define the element types, element real constants, material
properties, and the model geometry.
The basic steps to build the model for ANSYS analysis are:
• Specify the job name and title.
• Enter ANSYS preprocessor.
• Specify the element type.
• Specify the element real constants.
• Specify the material properties.
• Define the model geometry.
• Exit preprocessor.
3.5.1.1 Specifying the job name and Analysis title
The job name is the name that identifies the ANSYS job. When the job is first
fined, it becomes the first part of all the files that are created by the software during
the analysis. By using a job name for analysis it can be ensured that no files are
overwritten. This task is not required for an analysis, but is recommended.
3.5.1.2 Defining Units
The ANSYS program does not assume a system of units for your analysis. Except
in magnetic field analyses, It may be use any system of units so long as you make sure
that you use that system for all the data you enter. (Units must be consistent for all
input data.)
Using the /UNITS command, you can set a marker in the ANSYS database
indicating the system of units that you are using. This command does not convert data
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29.
from one system of units to another; it simply serves as a record for subsequent
reviews if the analysis.
3.5.1.3 Defining Element Types
The ANSYS element library contains more than 100 different element types. Each
element type has a unique number and a prefix that identifies the element category:
BEAM4, PLANE77, SOLID96, etc.
The element type determines, among other things:
1. The degree-of-freedom set (which in turn implies the discipline-structural,
thermal, magnetic, electric, quadrilateral, brick, etc.)
2. Whether the element lies in two-dimensional or three-dimensional space.
BEAM4, for example, has six structural degrees of freedom (UX, UY, U7-,
ROTX. ROTY, ROTZ) is a line element, and can be modeled in 3-D space. PLANE
has a thermal degree of freedom (TEMP), is an eight-node quadrilateral element, and
can be modeled only in 2-D Space.
3.5.1.4 Defining Real Element Constants
Element real constants are properties that depend on the element type, such as
cross-sectional properties of a beam element. For example, real constants for BEAM3,
the 2-D beam element, are area (AREA), moment of inertia (1ZZ), height (HEIGHT),
Shear deflection constant (SHEARZ), initial strain (ISTRN), and added mass per unit
length (ADDMAS). Not all element types require real constants, and different
elements E the same type may have different real constant values.
3.5.1.5 Defining Material Properties
Most element types require material properties. Depending on the application,
material properties may be:
• Linear or nonlinear
• Isotropic, orthotropic. or anisotropic
• Constant temperature or temperature-dependent.
29
30.
3.5.1.6 Creating the model geometry
Once you have defined material properties, the next step in an analysis is
generating a finite element model-nodes and elements-that adequately describes the
model geometry. The graphic below shows some sample finite element model.
There are two methods to create the finite element model: Solid modeling and
direct generation. With solid modeling, you described the geometric shape of your
model, and then instruct the ANSYS program to automatically mesh the geometry
with nodes and elements. You can control the size and shape of the elements that the
program creates. With direct generation, you “manually’ define the location of each
node and the connectivity of each element. Several convenience operations, such as
copying patterns of existing nodes and elements, symmetry reflection, etc. are
available.
3.6 Apply loads and obtain the solution
Loading Overview
The main goal of a finite element analysis is to examine how a structure or rent
responds to certain loading conditions.
Loads: The word loads in ANSYS terminology includes boundary conditions and
externally or internally applied forcing functions, Examples of different disciplines
are;
Structural: displacements, forces, pressures, temperatures (for thermal strain),
gravity.
Thermal: temperatures, heat flow rates, convections, internal heat generation, infinite
surface.
Magnetic: magnetic potentials, magnetic flux, magnetic current segments, source
density, infinite surface.
Electric: electric potentials (voltage), electric current, electric charges, charge vs.
infinite surface.
Loads are divided into six categories:
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32.
CHAPTER-4
4.1 Mathematical calculation of C-section micro cantilevers:
The following figure is considered for our project. The cantilever is made up of
SU-8 polymer. The properties of the material are being used. The deflection is found
using stoney equation. For different values of pressure the deflection is being found.
Case-1:
For pressure 0.01 N/mm2
Surface stress (�) = 0.8 N/m
Elastic Modulus (E) = 5Gpa
Poisson ratio (v) = 0.22
Length (l) = 200µm
Width (w) = 120µm
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33.
Thickness (t) = 2.8µm
Area (a) = 17600µm
From stoney equation the formulae for deflection is
By substituting the values the equation is
= 0.37µm
Deflection (Z) =0.37µm
Case 2:
For pressure: 0.012 N/mm2
Surface stress (�) =1.05 N/m
From stoney equation the formulae for deflection is
= 0.56µm
Deflection (Z) =0.56µm
Case 3:
For pressure: 0.013 N/mm2
Surface stress (�) =1.14 N/m
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34.
From stoney equation the formulae for deflection is
= 0.54µm
Deflection (Z) =0.54µm
Case 4:
For pressure: 0.014 N/mm2
Surface stress (�) =1.2 N/m
From stoney equation the formulae for deflection is
= 0.57µm
Deflection (Z) =0.57µm
Case 5:
For pressure: 0.015 N/mm2
Surface stress (�) =1.3 N/m
From stoney equation the formulae for deflection is
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35.
= 0.61µm
Deflection (Z) =0.61µm
Simple java program to find the deflection in the micro cantilevers:
/*
* To change this template, choose Tools | Templates
* and open the template in the editor.
*/
package deflectiondetermine;
import java.io.BufferedReader;
import java.io.InputStreamReader;
/**
public class Main {
/**
* @param args the command line arguments
*/
public static void main(String[] args)throws Exception {
BufferedReader br=new BufferedReader (new InputStreamReader(System.in));
35
36.
float pasionRatio,surfaceStress,thickness;
int area;
double elasticLimit,deflection;
System.out.println("Enter the value pasionRatio t");
pasionRatio=Float.parseFloat(br.readLine());
System.out.println("Enter the value of surface Stress t");
surfaceStress=Float.parseFloat(br.readLine());
System.out.println("Enter the value of thickness t");
thickness=Float.parseFloat(br.readLine());
System.out.println("Entere the value of elasticLimit t");
elasticLimit= Double.parseDouble(br.readLine());
System.out.println("Enter the value of area t");
area=Integer.parseInt(br.readLine());
deflection=((3*(1-pasionRatio)*surfaceStress)*area)/
(elasticLimit*thickness*thickness);
System.out.println("The value of Deflection"+deflection);
}
}
36
37.
4.2 MODELLING AND SIMULATION
The surface-stress induced deflection in a microcantilever can be modelled by
applying a lengthwise in-plane tensile force at the free end of the top surface of the
cantilever. The simulations assume that the cantilevers are made of silicon, and have
an Young’s modulus of 5GPa and a Poisson’s ratio of 0.22, respectively. The
cantilevers are subject to a surface pressure (Δσ) of 0.8 N/m on their top surfaces.
Since surface stress is expressed in unit of force per unit width, multiplying the
surface stress by the cantilever width will give the total tensile force acting on the top
surface.
ANSYS PROCEDURE
Data of Microcantilevers
Sections : C-Section
Required material : Polymer (su-8)
Required element : Brick 8 node 45(isotropic, elastic)
Poisson’s ratio : 0.22
Young’s modulus : 5 Gpa
Pressure : 0.01N/mm2
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38.
Edge length : 0.005
Element type : Static (steady state)
Extrusion : 2.8e5
Software : Ansys10.0
PROCEDURE:
1. Define material
Step1: Define element type
1. Main menu>preprocessor>element type>add/edit/delete
2. add
3. structural solid (left column)
4. Brick8nodesolid (right column)
5. [o.k.]
Step 2: Define material properties
1. Main menu>preprocessor>material props>material modals
2. (double click “structural “ then “linear”, then “elastic”, then “isotropic”
3. “EX”= 5e9
4. “prxy”=0.22
5. [o.k.]
6. [close]
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39.
Step3: nodes
1. Main menu>preprocessor>modeling>create >nodes>in active cs>creat nodes in
active coordinate system.
2. Main menu>preprocessor>modeling>create >key points>on nodes> pick nodes on
current plane.
3.mainmenu>preprocessor>modeling>create>Lines>lines>straight line>pick the key
points on work plane.
[ok]
Fig.8: Creating key points
3. Main menu>create>line> st. line>pick plotted key points on working plane
4. Main menu>create>areas>arbitrary>by lines. A Plane is formed with plotted key
points.
39
40.
Fig.9: Creating plane
2. Extrude the area
Step4:
1. Main menu>modeling>operate>extrude>element opts>no. of .element divisions
=10
2. Operate>extrude>areas>along normal
Pick area
Length of Extrusion=2.8e5
Given area is extruded
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41.
Fig 10: Extrusion
3. Mesh the area
Step5:
1. main menu>preprocessor>meshing>mesh tool
2. “size control GLOBAL”=[set]
3. Element edge length=0.05
4. [o.k.]
5. [mesh]
6. [Pick all]
7. [close]
41
42.
Fig. 11: Meshing of C-Section
4. Apply loads
Step6:
1. Main menu>preprocessor>loads>applyloads>structural>displacements>onlines.
Select the lines which are to be fixed. D.O.F . O.K.
2. Main menu>preprocessor>load>define load>apply>pressure>on areas
Pick the area
Pressure applied =0.01 N/mm2
[o.k.]
42
43.
Fig 12: Applying loads
5. Obtain solution
step7: Specify static analysis type
1. Main menu>solution> analysis type> new analysis
2. Static
3. [o.k.]
Step8: solve
1. Main menu>solution>solve>current LS
2. Review the information in the status window then choose file>close.
3. [o.k.]
4. [Close] to acknowledge that the solution is done.
Step9: Results
1. Main menu>general post processor>plot results>contour plot> nodal solutions.
2. Close after observing the listing.
Step10: animate the deflection
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44.
1. Utility menu>plot ctrls>animate>deformed shape.
2. [O.K.]
3. Step11: exit the ANSYS Programme.
Fig 13: Deflection in z direction
For pressure=0.013N/mm2
Fig 14:Deflection in Z direction
44
45.
For pressure=0.014N/mm2
Fig 15: Deflection in z direction
For pressure =0.015N/mm2
Fig.16: Deflection in z direction
45
47.
CHAPTER - 5
RESULTS AND DISCUSSIONS
Basically the deflection of the cantilever beam should be high, when the sensing
element senses the material to be detected. Similarly stress of the cantilever beam
should be low so that the beam will be more resistant when any high load is applied
on the surface of the beam.
Since von mises stress is less than stress intensity the model is in safe zone.
The deflection is calculated by using stoney equation and then it is analyzed in
ansys. The errors are being determined below and are tabulated.
COMPARISIONS OF THE C-SECTION MICROCANTILEVERS:
Pressure
(p)N/mm2
Surface
Stress(∆σ)
N/m
Simulation
deflection(µm)
Analytical
Deflection(µm)
Error (µm)
0.01 0.8 0.42 0.37 0.05
0.012 1.05 0.57 0.5 0.07
0.013 1.14 0.62 0.54 0.08
0.014 1.2 0.67 0.59 0.08
0.015 1.3 0.70 0.61 0.09
Graph between surface stress vs Deflection:
47
49.
CONCLUSION
Arrays of micro-cantilevers are increasingly being used as physical, biological, and
chemical sensors in various applications. In this work, the overall sensitivity of the
micro-cantilevers that are used in mems for increasing their deflection and stress
characteristics of the cantilevers are investigated. To improve the sensitivity the basic
and modified design of c- section profile micro-cantilevers are studied. The overall
sensitivity of micro-cantilever depends on both the deflection and the stress of the
cantilever. The deflection is found analytically by the stoney equation for various
surface stress values. The same is simulated in fea software Ansys. The simulation
results and the calculated values are compared. The results shows that the analysis
with mathematical and simulated are within a minimum error of .1 (10%) at an
approval. The surface stress was successfully modeled by an in-plane tensile force
applied to the top surface of the cantilevers and the deflections ware studied.
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Study of the Origin of Bending Induced by Bimetallic Effect on
Microcantilever
15. Ricardo C. Teixeira1,2, Koen De Munck2, Piet De Moor2, Kris Baert2, Bart
Swinnen, Chris Van Hoof1,2 and Alexsander Knüettel3 on Stress Analysis on
Ultra-Thin Ground Wafers
51