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The present work analyzes theoretically and verifies the advantage of utilizing rectangular microcantilevers with long-slits in microsensing applications. The deflection profile of these microcantilevers is compared with that of typical rectangular microcantilevers under the action of dynamic disturbances. Various force-loading conditions are considered. The theory of linear elasticity for thin beams is used to obtain the deflection-related quantities. The disturbance in these quantities is obtained based on wave propagation and beam vibration theories. It is found that detections of rectangular microcantilevers with long-slits based on maximum slit opening length can be more than 100 times the deflections of typical rectangular microcantilevers. Moreover, the disturbance (noise effect) in the detection quantities of the microcantilever with long-slits is found to be always smaller than that of typical microcantilevers, regardless of the wavelength, force amplitude, and the frequency of the dynamic disturbance. Eventually, the detection quantities of the microcantilever with long-slits are found to be almost unaffected by dynamic disturbances, as long as the wavelengths of these disturbances are larger than 3.5 times the microcantilever width. Finally, the present work recommends implementation of microcantilevers with long-slits as microsensors in robust applications, including real analyte environments and out of laboratory testing.

The recent advances in nanoscience and nanotechnologies have led to the development of innovative and highly sensitive microsensors. These microsensors are now becoming pivotal tools in exploring many chemical and biological phenomena. An example of these sensors is the microcantilever. The deflection of the microcantilever was first used for atomic force microscopy [

The detection capability of the microcantilever is influenced by the disturbance level in the adjacent medium. Fritz

A remarkable microcantilever assembly among the assemblies proposed in the work of Khaled

In this work, the advantage of utilizing the rectangular microcantilever with long-slit established by Khaled

The geometry of the typical rectangular microcantilever considered in this work is shown in

When the length of the microcantilever is much larger than its width, Hooke's law for small deflections can be used to relate the microcantilever deflection at a given cross-section to the effective elastic modulus

The boundary conditions for

The magnitude of microcantilever stress at bottom surface (

If a concentrated force in the direction of the

For this case, the effective elastic modulus is the same as the elastic modulus (_{oF}_{oF}

The solution of _{F}

The maximum deflection (_{F}_{max}

We define the concentrated force deflection indicator _{F}_{F}

When one side of the microcantilever is coated with a thin film of receptor, the microcantilever will bend if the analyte molecules adhere on that layer. This adhesion causes a difference in the surface stresses across the microcantilever cross-section (Δ

For this case, the effective elastic modulus varies with the elastic modulus according to the following relationship:

Δ_{Δ}_{σ}

The maximum deflection due to analyte adhesion is obtained from

_{Δ}_{σ}_{Δ}_{σ}

The one degree of freedom model that can best describe the disturbance in the tip deflection of the typical rectangular microcantilever, _{d}_{eff}_{1} is the effective mass of the microcantilever at its tip and _{eff}_{1} is the effective stiffness of the microcantilever at its tip. _{o}_{eff}_{1} and _{eff}_{1} are given by the following expressions[_{o}

The total maximum deflection of the typical rectangular microcantilever, _{t}_{do}

Define the clearness indicator of the microsensor deflection signal (_{D}_{F}_{Δ}_{σ}

The geometry of the rectangular microcatilever with long-slit is shown in _{o}

The length of the microcantilever with long-slit is considered to be much larger than its width. As such, Hooke's law for small deflections can be used to relate the microcantilever deflection at a given cross-section to the effective elastic modulus

Let the middle cross-section of the beam on the left side of the long-slit be loaded by a normal concentrated force of magnitude

Accordingly,

The boundary conditions of

The magnitude of the maximum stress occurs at (_{oF}_{oF}

The solution of _{F}

If the position of RB above the concentrated load is taken as the datum of the rectangular microcantilever with long-slit, then the maximum deflection in that microcantilever denoted by (Δ_{F}_{max}

The deflection of LB and RB in the positive _{F}_{max}_{F}_{max}

By using _{F}_{max}

As (Δ_{F}_{max}_{F}_{max}_{F}_{max}_{LB}_{F}_{F}_{max}

Define the first detection enhancement indicator of the rectangular microcantilever with log-slit due to concentrated force loading _{1,F}_{F}_{F}_{1,F}

The maximum values of (_{F}_{max}_{LB}_{F}

When one surface of LB is coated with a thin film of receptor, it will bend if analyte molecules adhere on that layer due to induced tension/compression surface stress. This adhesion causes a difference in the surface stresses across the microcantilever section (Δ

The effective elastic modulus for this case is shown in

The boundary conditions of

The solution of _{Δ}_{σ}_{max}

The deflection of LB and RB in the positive _{F}_{max}_{F}_{max}

By using _{Δ}_{σ}_{max}

As (Δ_{Δ}_{σ}_{max}_{Δ}_{σ}_{max}_{Δσ}_{max}_{LB}_{Δ}_{σ}_{Δ}_{σ}_{max}

Define the first detection enhancement indicator of the rectangular microcantilever with long-slit due to prescribed differential surface stress loading _{Δ}_{σ}_{Δ}_{σ}_{Δ}_{σ}_{Δ}_{σ}

The maximum values of (_{Δ}_{σ}_{max}_{LB}/d_{Δ}_{σ}

The one degree of freedom model that can best describe the disturbance in the deflections at the midsections of LB and EB of the rectangular microcantilever of the long-slit, _{d}_{eff}_{2} is the effective mass of the LB or RB at their midsections, _{eff}_{2} is the effective stiffness of the LB or RB at their midsections and _{o}_{eff}_{2} and _{eff}_{2} can be shown to be equal to the following [

The particular solution of the differential equation given by _{s}

Thus, the total maximum deflection of LB and RB denoted by _{tL}_{tR}_{d1}

Since the position of the midsection of RB is taken as the datum of the rectangular microcantilever with long-slit, the maximum total deflection in that microcantilever denoted by (Δ_{t,F}_{max}_{t,Δσ}_{max}

By inspection of

The clearness indicator of the deflection signal of the present microsensor (_{D}_{F}_{Δ}_{σ}

To compare between the clearness indicator of the rectangular maicrocantilever with long-slit and that of typical microcantilever, the second detection enhancement indicator (_{2,D}_{D}_{1} = _{o}ω_{o}^{2}/_{2} = _{o}ω_{o}^{2}(_{o}W

The present analytical methods for the rectangular microcantilever with long-slit were tested against an accurate numerical solution using finite element methods and accounting for all mechanical constraints induced by the geometry. Among these constraints is the torsion effect of the concentrated force and restraining the wrapping of the side beams due to the end portions of the microcantilever. The deflection contours for the present microcantilever with _{o}^{−9} N is shown in ^{−2} and a poisons ratio of _{F}_{max}_{F}_{max}

_{1,F}_{1}_{Δ}_{σ}_{F}_{max}_{LB}/d_{1,F}_{1}_{Δ}_{σ}_{F}_{max}_{LB}/d_{1,F}_{F}_{max}_{LB}/d_{1}_{Δ}_{σ}_{Δσ}_{max}_{LB}/d_{1}_{Δ}_{σ}

_{o}_{F}_{Δ}_{σ}_{F}_{Δ}_{σ}_{o}_{o}_{o}_{Δ}_{σ}

_{2,F}_{2}_{Δ}_{σ}_{o}_{2,F}_{2}_{Δ}_{σ}_{o}_{2,F}_{2}_{Δ}_{σ}_{o}_{2}_{Δ}_{σ}_{2} value.

_{o}_{2,F}_{2,F}_{2,F}_{2,F}_{o}_{o}_{2,F}_{o}_{2,F}_{2}_{Δ}_{σ}_{o}

An investigation verifying the advantage of using rectangular microcantilevers with long-slits in microsensing applications was performed in this work, based on analytical solutions. The detection capabilities of these microcantilevers were compared against that of typical rectangular microcantilevers under the action of dynamic disturbances. Concentrated force loadings and prescribed surface stress loadings were considered as the sensing driving forces. The theory of linear elasticity for thin beam deflections was used to obtain the detection quantities. The disturbance in these quantities was obtained using the wave propagation and beam vibration theories. The defection profile of the rectangular microcantilever with long-slit was validated against an accurate numerical solution utilizing finite element method with a maximum deviation less than 11 percent.

It was found that the detection of rectangular microcantilevers with long-slits based on their maximum slit opening length can be more than 100 times the maximum deflection of the typical rectangular microcantilever. Furthermore, the disturbance (noise) in the deflection of the microcantilever with long-slit was found to be always smaller than that of the typical microcantilevers, regardless of the wavelength, force amplitude, and the frequency of the dynamic disturbance. Moreover, good mixing the analyte solution was found to produce better detection capability and smaller disturbance in the detection of the microcantilever with long-slit than weakly-mixed ones. Eventually, detections of the microcantilevers with long-slit were found to be practically unaffected by dynamic disturbances as long as the wavelengths of these disturbances are larger than 3.5 times the width of the microcantilever. Finally, the present work strongly suggests implementation of microcantilevers with long-slit as microsensors in real analyte environments and out of the laboratory testing.

clearance length for rectangular microcantilever with long-slit (μm)

microcantilever thickness (μm)

elastic modulus (N μm^{−2})

concentrated force (N)

area moment of inertia (μm^{4})

stiffness (N/μm)

typical rectangular microcantilever or slit length (μm)

_{o}

length of rectangular microcatilever with long-slit (μm)

moment (N μm)

mass (kg)

surface stress model index

_{o}

dynamic disturbance force per square of frequency of disturbance (N s^{−2})

time variable (s)

total microcantilever width (μm)

axis of the extension dimension (μm)

effective elastic modulus (N μm^{−2})

first deflection indicator

deflection (μm)

_{d}

Amplitude of disturbance in deflection (μm)

detection clearness indicator

slit thickness (μm)

_{1}

the first detection enhancement indicator

_{2}

the second detection enhancement indicator

wavelength of the dynamic disturbance (μm)

Poisson's ratio

surface stress (N μm^{−1})

dynamic disturbance frequency (s^{−1})

_{o}

first natural frequency of typical rectangular microcantilever (s^{−1})

_{s}

first natural frequency of rectangular microcantilever with long-slit (s^{−1})

disturbance

concentrated force condition

prescribed surface stress condition

effective value

left beam of the rectangular microcantilever with long-slit

right beam of the rectangular microcantilever with long-slit

Schematic diagrams and the corresponding coordinate system for typical rectangular microcantliever (MC): (

Schematic diagrams and the corresponding coordinate system for the rectangular microcantliever with long-slit: (

Deflection profile for rectangular microcantilever with long-slit with _{o}^{−2}, ^{−9} N.

Effects of maximum rectangular microcantilever with long-slit side beams dimensionless deflection {(_{F}_{max}_{LB}/d_{1,F}

Effects of maximum rectangular microcantilever with long-slit side beams dimensionless deflection {(_{Δσ}_{max}_{LB}/d_{1}_{Δ}_{σ}

Effects of the dimensionless frequency of dynamic disturbance (_{o}_{o}ω_{o}^{2}/_{F}

Effects of the dimensionless frequency of dynamic disturbance (_{o}_{o}ω_{o}^{2}(_{o}W_{Δ}_{σ}

Effects of the dimensionless dynamic disturbance wavelength (_{o}ω_{o}^{2}/_{2,F}

Effects of the dimensionless dynamic disturbance wavelength (_{o}ω_{o}^{2}(_{o}W_{2}_{Δ}_{σ}

Effects of the dimensionless frequency of dynamic disturbance (_{o}_{o}ω_{o}^{2}/_{2,F}

Effects of the dimensionless frequency of dynamic disturbance (_{o}_{o}ω_{o}^{2}(_{o}W_{2}_{Δ}_{σ}

Maximum value of rectangular microcantilever with long-slit side beams deflection that produces detection enhancement indicator due to concentrated force loading larger than unity.

( |
10^{−2} |
10^{−3} |
10^{−4} |
10^{−5} |
---|---|---|---|---|

[(_{F}_{max}_{LB}/d_{γF}_{=1} |
23.928 | 246.81 | 2,490.1 | 24,969 |

[(_{F}_{max}_{LB}/d_{γF}_{=1} = 0.23467 ×exp(−1.00593× |

Maximum value of rectangular microcantilever with long-slit side beams deflection that produces detection enhancement indicator due to prescribed differential surface stress loading larger than unity.

( |
[(_{Δσ}_{max}_{LB}/d_{γΔσ}_{=1} |
( |
[(_{Δσ}_{max}_{LB}/d_{γΔσ}_{=1} | ||
---|---|---|---|---|---|

10^{−2} |
21.157 | 10^{−2} |
7.4114 | ||

10^{−3} |
238.55 | 10^{−3} |
103.90 | ||

10^{−4} |
2,464.4 | 10^{−4} |
1155.5 | ||

10^{−5} |
24,888 | 10^{−5} |
12067 | ||

10^{−2} |
12.854 | 10^{−2} |
3.9373 | ||

10^{−3} |
159.14 | 10^{−3} |
66.535 | ||

10^{−4} |
1,699.3 | 10^{−4} |
775.55 | ||

10^{−5} |
17,408 | 10^{−5} |
8288.2 |