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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. A commercial finite element analysis software ANSYS is used to analyze the designs. Results show paddled trapezoidal profile microcantilevers have better sensitivity.

Although generally used in topological investigations of surfaces such as in atomic force microscopy, arrays of microcantilevers are attracting much interest as sensors in a variety of applications. Microcantilever sensors have emerged as a very powerful and highly sensitive tool to study various physical, chemical, and biological phenomena. The physical phenomena can be calorimetric [_{2} sensors [

Surface stresses, in general, are generated either by the redistribution of the electronic charge at the surface due to the change in the equilibrium positions of the atoms near the surface, or by the adsorbtion of foreign atoms onto its surface to saturate the dangling bonds [

Microcantilever biosensors commonly use optical lever readout technique to observe the deflection. In practice, the accuracy in the deflection measurements not only depends on the actual deflection occurred but also on the signal-to-noise ratio. Most of the noise in the signal can be attributed to the thermal drift. To improve the signal-to-noise ratio, the resonant frequency of the cantilever should be made as large as possible. Thus, to increase the overall cantilever sensitivity, we should select a design that shows both higher deflection and higher resonant frequency. The sensitivity of a cantilever can be changed by changing the cantilever material, shape, size, or profile. Polymeric materials such as polyethylene terephthalate (PET) [

To increase simultaneously the deflection and resonant frequency of a microcantilever, this paper investigates the deflection and vibration characteristics of rectangular and trapezoidal profile microcantilevers having three different shapes. These cantilevers can be used as the sensing element in biosensors. First, we separately analyze the effect of cantilever profile change and the effect of cantilever shape change, and then combine the profile change with the shape change to investigate the deflection and resonant frequency of the microcantilevers. All the cantilevers were investigated for maximum deflection occurred, fundamental resonant frequency, and maximum induced stresses. The surface-stress induced deflection in the microcantilever is modelled by an equivalent in-plane tensile force acting on the top edge of the cantilever, in the length direction. A commercial finite element method (FEM) software ANSYS is used in this analysis.

Microcantilever biosensors exploit surface-stress induced deflections to assay the target molecules. When the target molecules attach onto the functionalized top surface of the cantilever, the surface stress distribution on this surface is changed, resulting in a differential stress across the top and bottom surfaces of the cantilever. The differential stress ultimately generates deflections in the cantilever. For a rectangular profile microcantilever, the differential surface stress (Δ

The fundamental resonant frequency (_{0}) for a rectangular profile cantilever of mass density (

As can be seen from _{0}) as:

Thus, instead of increasing deflection or resonant frequency individually, it is more practical to increase the overall sensitivity predicted by _{0} value, with more inclined towards the deflection.

For a microcantilever of trapezoidal profile, _{l} + (_{0} – _{l}) _{0} and _{l} are the thicknesses of the cantilever at the fixed and free ends. This study used _{l} = _{0}/2. Hoffman and Wertheimer [

In this equation, _{1}, _{2} and _{3} are tapering-ratio dependent mass distribution parameters.

The surface-stress induced deflection in a microcantilever can be modelled by applying a lengthwise in-plane tensile force at the free edge of the top surface of the cantilever (^{−1}(∼2.5 μM) myoglobin protein onto the functionalized surface of the microcantilever. The surface stress resulted in a maximum deflection of 0.89 μm at the cantilever free end. The cantilever size was 500×100×0.5 μm, and the elastic modulus and Poisson ratio was 130 GPA and 0.28, respectively. This cantilever is used as a reference in this analysis.

_{l} = _{0}/2. For simulations, a FEM software ANSYS Multiphysics was used to calculate the deflection, fundamental resonant frequency and maximum stress induced. The simulations were performed on three-dimensional FE models of the cantilevers, under linear, static conditions. The FE models were meshed by SOLSH190 elements. As shown in ^{−6} m = 5×10^{−6} N/m was applied to the top free edge of all the six models.

To ascertain the validity of modelling surface-stress induced deflection by in-plane tensile force,

In

^{6} MPa). The maximum induced stresses range from a minimum of 0.41 MPa for Model #1 to a maximum of about 2 MPa for Models #5 and #6. A stress comparison between Models #1 and #4 shows that profile change alone increased the stresses from 0.41 MPa to 0.79 MPa. The deflection and stress values for Models #2 and #3 are almost equal. Similar observation is true for #5 and #6. This behaviour is expected because from mechanics of material point of view, Models #2 and #3 are identical because they have same flexural stiffness, i.e. their resistance to bending is equal. Same observation holds for #5 and #6. However, it should be noted that Models #3 and #6 have better torsion resistance than #2 and #5, and therefore should be preferred.

The changes in the cantilever profile or the cantilever shape will lead to a change in the area which will introduce sharp corners in the cantilever. The sharp corners in the microcantilever models can raise the stress concentration factors by many folds. As we can see in the

_{0}), sensitivity (Δ_{0}), and maximum stress induced (_{max}) for all the six models shown in

From

Another approach to simultaneously increase the deflection and frequency is to change the cantilever profile. Comparing the deflection shown by Model #1 to Model #4, we observe about 57% increase in deflection and about 9% increase in the resonant frequency. Furthermore, it can be easily observed in _{0} value, making it most suitable cantilever. However, since this cantilever also has the least value of the resonant frequencies, it will have the smallest signal-to-noise ratio. Therefore, the reference cantilever is unsuitable under dynamic conditions. Except for Model #4, all the models suggest any increase in deflection is accompanied with a decrease in frequency. Thus, Model #4 seems most suitable to be used as sensing element in microcantilever biosensor. However, if dynamic properties of the cantilever are not a major concern, Models #5 and #6 show highest deflection and sensitivity values, and are more appropriate to increase the overall sensitivity of the biosensor.

As mentioned earlier,

In the experimental case, the cantilever thickness is 0.5 μm and the deflections predicted by analytical and simulation is about 1.1 μm (

The dynamic properties of microcantilevers used in biosensors are critical in accurate measurement of deflections. In practical applications, there can by thermal, structural, or flow induced excitations that can interfere with and hence produce noise in the signals. Therefore, it is vital to eliminate or isolate the noise in the signal, and to insure that the deflections induced are solely due the surface stress change. To prevent noise, a cantilever should have high signal-to-noise ratio, which can be achieved by making the resonant frequency of the cantilever as large as possible. The fundamental resonant frequency of a rectangular cantilever is given as:
_{0} values (

From structural dynamics point of view, reduction in spring constant is undesirable, because it will decrease the resonant frequency of the cantilever. Therefore, another way to reduce the mass, while keeping the spring constant unchanged, is to change the cantilever profile. As mentioned above, geometric properties at the fixed end of the cantilever define its behaviour. Therefore, we may change the cantilever profile in a manner keeping the fixed-end thickness same and changing the thickness far from it. This scheme can be easily realized by trapezoidal profile cantilever (Model #4). It can be readily calculated that Model #4 has 25% less mass than #1 (

Thus far we discussed the deflection and vibration characteristics pertaining to the static mode of the microcantilevers. Static mode is used for determining the surface stress, diffusion or biomolecular recognition, whereas the dynamic mode is used as microbalance, thermogravimetry, or determining the amount of biomolecules adsorbed onto the cantilever. Dynamic mode uses the mass change induced resonant frequency change to calculate the amount of molecules adsorbed onto the functionalized surface of the microcantilever. Using dynamic mode, mass changes in the picogram range can be observed [_{0} and _{1} are the fundamental resonant frequencies before and after the mass addition. It can be readily observed from

To improve the sensitivity of microcantilevers used in sensors, this study investigated rectangular and trapezoidal profile microcantilevers. For each profile, three cantilever designs were analyzed. The surface stress was successfully modelled by an in-plane tensile force applied to the top surface of the cantilevers. The finite element analysis investigation indicated that by changing the profile from rectangular to trapezoidal, the cantilever sensitivity is increased by 71%. Further, for each cantilever type, if we change only the shape, the sensitivity is increased by 17% for the rectangular and 7% for the trapezoidal cantilevers. However, if we combine the profile change with the shape change, the overall sensitivity is improved by 77%. Stress analysis showed that compared to the ultimate strength of silicon, the maximum stress induced in the models is negligible. We also showed that the high fundamental resonant frequency is a basic requirement for high sensitive cantilevers used in either static or dynamic mode. Based on the results of this investigation, we can conclude that the trapezoidal profile cantilevers has better deflection and resonant frequency characteristics than the rectangular, and hence can be efficiently used as the sensing element in microcantilever sensors.

This study was supported by Inha University.

_{2}detection using microcantilever based potentiometry

Modelling the surface stress induced deflection in a microcantilever by in-plane tensile force acting on the top surface.

Schematic designs for the rectangular and trapezoidal profile cantilevers. All the models have same length and fixed-end thickness.

Von Mises stress distribution in the microcantilever models. Models #1, #2 and #3 have rectangular profiles, and Models #4, #5 and #6 have trapezoidal profiles.

Verification results comparing the experimental, analytical and simulation results.

0.05 | 0.89 | 1.11 | 1.14 |

Comparison between analytical and simulation results for uniform width rectangular and trapezoidal cantilevers.

#1 | 0.28 | 0.28 | 4.79 | 4.91 |

#4 | 0.43 | 0.44 | 5.59 | 5.33 |

Comparison between simulation values for maximum deflection, fundamental resonant frequency, sensitivity and maximum induced stress.

Δ |
_{0} |
Δ_{0} |
_{max} | |
---|---|---|---|---|

#1 | 0.28 | 4.91 | 1.37 | 0.41 |

#2 | 0.53 | 3.04 | 1.61 | 1.13 |

#3 | 0.53 | 3.04 | 1.61 | 1.21 |

#4 | 0.44 | 5.35 | 2.35 | 0.79 |

#5 | 0.66 | 3.68 | 2.43 | 2.02 |

#6 | 0.67 | 3.68 | 2.46 | 1.99 |

Ref. | 1.14 | 2.45 | 2.79 | 0.86 |