^{1}

^{2}

^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The present work analyzes theoretically and verifies the advantage of utilizing ɛ-microcantilever assemblies in microsensing applications. The deflection profile of these innovative ɛ-assembly microcantilevers is compared with that of the rectangular microcantilever and modified triangular microcantlever. Various force-loading conditions are considered. The theorem of linear elasticity for thin beams is used to obtain the deflections. The obtained defections are validated against an accurate numerical solution utilizing finite element method with maximum deviation less than 10 percent. It is found that the ɛ-assembly produces larger deflections than the rectangular microcantilever under the same base surface stress and same extension length. In addition, the ɛ-microcantilever assembly is found to produce larger deflection than the modified triangular microcantilever. This deflection enhancement is found to increase as the ɛ-assembly’s free length decreases for various types of force loading conditions. Consequently, the ɛ-microcantilever is shown to be superior in microsensing applications as it provides favorable high detection capability with a reduced susceptibility to external noises. Finally, this work paves a way for experimentally testing the ɛ-assembly to show whether detective potential of microsensors can be increased.

The rapid growth of nanotechnology has led to the development of new sensing devices of micrometer size coined as microsensors. These devices can be used to detect, measure, analyze, and economically monitor low concentrations of chemical and biological agents. The monitoring of a specific substance is pivotal in many applications, especially for clinical purposes in order to screen a patient for the presence of a disease at an early stage [

The magnitude of microcantilever deflection is of the order of nanomenters and it is usually measured using optical methods. The performance of the microcantilever as a sensing device is affected by the noise level in the surrounding environment. For example, Fritz

In this work, the advantage of utilizing microcantilever assemblies including the ɛ-assembly established by Khaled

The geometry of the rectangular microcantilever considered in this section is shown in

The boundary conditions for

For a concentrated force exerted on the rectangular microcantilever tip (_{aF}

The above result is based on a realistic linearly increasing bending moment from the base prescribed by:

For thin cross-sections, the surface stress, σ, can be calculated from the following equation:

The surface stress at _{aFo}

The maximum deflection which occurs at the microcantilever tip (

For a bending moment _{aM}

The surface stress at the base section which is denoted by _{aMo}

The maximum deflection which is the deflection at the microcantilever tip is equal to:

When the microcantilever is coated on one side with a thin film of receptor, it is usually bent due to analyte adhesion on that layer. This adhesion causes a differential in the surface stress across the microcantilever section yielding a bending moment at each section. The bending moment _{aΔσ}

This is because the effective elastic modulus for this case is given by

The geometry of the microcantilever assembly (b) is shown in

Note that ^{3}/12. Note that

The boundary conditions for

The solution of _{bF}

Using _{bFo}

The maximum deflection occurs at the tip (

For a bending moment

The solution of

As such, the maximum deflection is expected to be equal to:

Using _{cMo}

When a receptor layer is coated on one side of assembly (b)-side beams (SB),

The solution of

The maximum deflection due to analyte adhesion is then equal to:

Define the first deflection indicator _{pU}_{o}_{bF}_{bM}_{bΔσo} are equal to:

The geometry of the microcantilever assembly (c) is shown in

The solution of _{cF}_{1} is equal to:

The surface stress at the base section _{cFo}

Define the second deflection indicator _{cU}_{cIBU}_{cU}_{o}) loading. The last two types of force loadings will be described later on. As such, _{cF}

Now, let a bending moment

The boundary conditions are given by _{2} is equal to:

The surface stress at _{cMo}

The second deflection indicator for assembly (c) for the current moments loading _{cM}

If the top surfaces of the side beams of assembly (c) are coated with a receptor while the receptor coating on the intermediate beam is on its bottom surface, then the deflection equations of assembly (c) changes to:

The solution for

The deflection indicator for assembly (c) due to the alternating analyte adhesion on the surfaces _{cΔσ}

The deflection indicators _{cF}_{cM}_{cΔσo} can be shown to be equal to the following:

The present analytical methods were tested against an accurate numerical solution using finite element methods and accounting for all mechanical constraints induced by the assemblies. Among these constraints is restraining the wrapping of the side beams due to the presence of the small connecting beam at ^{−12} Nμm is shown in ^{−2} and a poisons ratio of _{cM}_{cIBM}

_{bF}_{cF}_{bF}_{cF}_{bM}_{cM}_{bΔσ}_{cΔσ}

_{cF}_{cF}_{cF}_{cM}_{cΔσ}_{cΔσ}

A theoretical investigation on improving deflections of microcantilevers sensors is presented in this work based on analytical solutions. Three different mcirocantilevers were analyzed. These are: (a) the rectangular microcantilever, (b) the modified triangular microcantilever assembly, and (c) the ɛ-microcantilever assembly. The deflection theory of thin beams is utilized to obtain the deflection profile for each microcantilever. Different force loadings were considered including concentrated force, concentrated moment and constant surface stress. Different deflection indicators were defined and computed. It was found that both the modified triangular microcantilever assembly and the ɛ-microcantilever assembly produce larger deflections than the rectangular microcantilever under the same base surface stress and same extension length. The deflection of the former microcantilevers can be 280% and 425% above that of the rectangular microcantilever for concentrated moment and constant surface stress cases, respectively. In addition, the ɛ-microcantilever assembly was found to produce larger deflection than the triangular microcantilever assembly. The deflection of the ɛ-microcantilever intermediate free end may reach 200% above that of the triangular microcantilever assembly. It was found that deflection enhancement due to ɛ-microcantilever increases as the assembly free length decreases. The cited conclusions were found to be valid for the different force loading conditions. The analytical results were validated against an accurate numerical solution utilizing a finite element method. The analytical and numerical solutions were found to be in good agreement. Based on our analysis, the ɛ-microcantilever assembly was found to provide a superior and the best favorable high detection capability with the least susceptibility to external noise in microsensing applications. As such, it is recommended to experimentally test it to show whether detective potential of microsensors can be increased.

Base length of the microcantilever assembly

Elastic modulus (N μm^{−2})

concentrated force (N)

Area moment of inertia (μm^{4})

microcantilever or assembly extension length (μm)

moment (N μm)

surface stress model index

microcantilever thickness (μm)

microcantilever width (μm)

axis of the extension dimension (μm)

effective elastic modulus (N μm^{−2})

deflection (μm)

first deflection indicator

second deflection indicator

Poisson’s ratio

surface stress

concentrated force condition

moment condition

constant differential surface stress condition

the intermediate beam of assembly (c)

the side beams of assembly (c)

Schematic diagrams and the corresponding coordinate system for microcantlievers (MC) assemblies:

Deflection profile for assembly (c) using numerical solutions with ^{−12} Nμm, ^{−2} and ν = 0.33, deflections (U) are in μm.

Effects of the relative dimensions of the microcantilevers assemblies (b) and (c) on the first performance indicators _{bF}_{cF}

Effects of the relative dimensions of the microcantilevers assemblies (b) and (c) on the first performance indicators _{bM}_{cM}

Effects of the relative dimensions of the microcantilevers assemblies (b) and (c) on the first performance indicators _{bΔσ}_{cΔσ}

Effects of the relative dimensions of the microcantilevers assemblies (b) and (c) on the second performance indicators _{cF}

Effects of the relative dimensions of the microcantilevers assemblies (b) and (c) on the second performance indicators _{cM}

Effects of the relative dimensions of the microcantilevers assemblies (b) and (c) on the second performance indicators _{cΔσ}