METHODS AND SYSTEMS FOR ANALYZING DETECTION ENHANCEMENT OF MICROCANTILEVERS WITH LONG-SLIT BASED SENSORS

Methods and systems for analyzing the detection enhancement of rectangular microcantilevers with long-slit microsensors. The deflection profile of the microcantilevers can he compared with that of typical rectangular microcantilevers under presence 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 can be obtained based on wave propagation and beam vibration theories.

DETAILED DESCRIPTION

The following Table 1 provides the various symbols and meanings used in this section:

TABLE 1cclearance length for rectangular microcantilever with long-slit (m)dmicrocantilever thickness (μm)EElastic modulus (N μm−2)Fconcentrated force (N)lArea moment of inertia (μm4)kstiffness (N/μm)Ltypical rectangular microcantilever or slit length (μm)Lolength of rectangular microcatilever with long-slit (m)Mmoment (N μm)mmass (kg)nsurface stress model indexPodynamic disturbance force per frequency of disturbance (N.s)tTime variable (s)Wtotal microcantilever width (μm)xaxis of the extension dimension (μm)Yeffective elastic modulus (N μm−2)Zfirst deflection indicatorzdeflection (μm)zdAmplitude of disturbance in deflection (μm)Greek Symbolsχdetection clearness indicatorδslit thickness (μm)γ1the first detection enhancement indicatorγ2the second detection enhancement indicatorλwave length of the dynamic disturbance (μm)vPoisson's ratioσsurface stressωdynamic disturbance frequency (s−1)ωofirst natural frequency of typical rectangular microcantilever (s−1)ωsfirst natural frequency of rectangular microcantilever with long-slit(s−1)SubscriptsddisturbanceFconcentrated force conditionΔσprescribed surface stress conditioneffeffective valueAbbreviationsLBLeft beam of the rectangular microcantilever with long-slitRBLeft beam of the rectangular microcantilever with long-slit1. Theoretical Analysis1.1. The Typical Rectangular Microcantilever

FIGS. 1A and 1Billustrate top and side views100and150of a typical rectangular microcantilever in a corresponding coordinate system, respectively. The properties of this microcantilever are given by the extension length L, width W, thickness d, Young's modulus E, and Poisson's ratio v.

1.1.1. Deflections of the Typical Rectangular Microcantilever

When the length of the microcantilever is much larger than its width, Hooks law for small deflections can be used to relate the microcantilever deflection at a given cross-section to the effective elastic modulus Y of the microcantilever and the bending moment M acting on that section, see Khaled, A.-R. A., and Vafai, K., Analysis of Deflection Enhancement Using Epsilon Assembly Microcantilevers Based Sensors, Sensors, 2011, 11, 9260-9274.

It is given by:

where I is the area moment of inertia of the microcantilever cross-section about its neutral axis. It is given by:

The boundary conditions for Equation (1) are given by:

The magnitude of microcantilever stress at bottom surface (z=d/2) or upper surface (z=−d/2), associated with the bending moment M can be calculated from the following equation:

Concentrated force loading:

If a concentrated force in the direction of the z-axis is exerted on the microcantilever tip located at x=L, then the internal bending moment M at any cross-section is linearly increasing from the tip to the base x=0. The internal bending moment distribution is equal to:

For this case, the effective elastic modulus is the same as the elastic modulus (Y=E). The magnitude of maximum stress occurs at (x,z)=(0,±d/2). It is denoted by σoF. Using Equation (4), σoFcan be shown to be equal to:

The solution of Equation (1), denoted by zF(x), can be expressed as:

The maximum deflection (zF)maxwhich occurs at x=L can be expressed as:

Define the concentrated force deflection indicator ZFas the ratio of the maximum microsensor deflection per maximum stress under constant concentrated force applied at the microcantilever tip. Using Equations (6) and (8), ZFcan be shown to be equal to:

Prescribed differential surface stress:

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 (Δσ). This results in an internal bending moment M at each cross-section. M is related to Δσ through the following equation:

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

Δσ can be considered to vary along the microcantilever length according to the following relationship:

where n is the model index. This variation is expected as the analyte concentration in the surrounding environment increases as the distance from the microcantilever base increases. As such, the solution of Equation (1) denoted by zΔσ(x) subject to boundary conditions given by Equations (3(a,b)) can then be expressed as:

The maximum deflection due to analyte adhesion is obtained from Equation (13) by substituting x=L. It is equal to:

Equation (14) is reducible to Stoney's equation when n is set to be equal to n=0.

Define the prescribed surface stress deflection indicator ZΔσas the ratio of the maximum microsensor deflection per maximum differential surface stress under the given prescribed surface stresses. Using Equations (12) and (14), ZΔσcan be shown to be equal to:

1.1.2. The Disturbance in the Deflections of the Typical Rectangular Microcantilever

The one degree of freedom model that can best describe the disturbance in the tip deflection of the typical rectangular microcantilever, zd, is shown in the following differential equation:

See Khaled, A.-R. A., Vafai, K., Yang, M., Zhang, X., and Ozkan, C. S., Analysis, control and augmentation of microcantilever deflections in bio-sensing systems, Sens. Actuat. B, 2003, 94, 103-115; Rao S. S., Mechanical Vibrations (5th Edition), Prentice Hall, 2010. USA; and Sader, J., Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope, Journal of Applied Physics, 1998, 84, 64-76; where meff,1is the effective mass of the microcantilever at its tip and keff,1is the effective stiffness of the microcantilever at its tip. ω is the frequency of the dynamic disturbance force and t is time variable. Pois the effective amplitude of the dynamic disturbance force at the tip per square of the frequency of dynamic disturbance. Equation (16) is based on the assumption that the microcantilever is excited in the first mode of vibration and that excitations occur without total energy dissipation. meff1and keff,1are given by the following:

See Rao S. S., Mechanical Vibrations (5th Edition), Prentice Hall, 2010, USA; where p is the density of the microcantilever. The particular solution of the differential equation given by Equation (16) is the following:

where ωois the first mode natural frequency which is equal to:

The total maximum deflection of the typical rectangular microcantilever, zt, which is the sum of the deflection due to loading of the microcantilever plus the disturbance in the deflection, can be mathematically expressed as follows:

Define the clearness indicator of the microsensor deflection signal (XD) as the ratio of maximum deflection due to loading type D, where D can be F or Δσ loading types to the sum of that deflection plus the amplitude of the maximum disturbance in the deflection. As such, XFand XΔσcan be shown to be equal to:

1.2. The Rectangular Microcatilever with Long-Slit

FIGS. 2A and 2Billustrate top and side views200and250of a typical rectangular microcantilever with long-slit in a corresponding coordinate system respectively, in accordance with the disclosed embodiments. The thickness of the microcantilever and the slit is d. The slit width is while its length is L where δ<<L. The microcantilever length is Lo. The side beams on left and right sides of the slit have the same width which is equal to W/2. Each side beam has an area moment of inertia I given by:

1.2.1. Deflections of the Rectangular Microcailever with Long-Slit

FIG. 2Cillustrates a deflection profile260of rectangular microcantilever with long-slit with major deflection quantities. The length of the microcantilever with long-slit is considered to be much larger than its width. As such, the Hooks law for small deflections can be used to relate the microcantilever deflection at a given cross-section to the effective elastic modulus Y of the microcantilever and the internal bending moment M acting on that section. See Khaled, A.-R. A., and Vafai, K., Analysis of Deflection Enhancement Using Epsilon Assembly Microcantilevers Based Sensors, Sensors, 2011, 11, 9260-9274. It is given by Equation (1).

Concentrated Force Loadings:

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 F and in the direction of the positive z-axis. On the other hand, the beam on the right side of the long-slit is considered to be loaded by a normal concentrated force of magnitude F, but in the direction of the negative z-axis. When c≦x≦c+L/2, the internal bending moment M distributions on the left side beam (LB) and the right beam (RB) can be shown to be equal to the following:

Accordingly, Equation (1) changes to the following:

The boundary conditions of Equation (27) are given by:

The magnitude of the maximum stress occurs at (x,z)=(c,±d/2). It is denoted by σoF. Using Equation (4), σoFcan be shown to be equal to:

The solution of Equation (1), denoted by zF(x), can be expressed as:

wherex=x/L andc=c/L. The maximum deflections of LB and SB which occurs at x=c+L/2 can then be found. They are equal to the following:

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 (ΔzF)maxas shown inFIG. 2Cwill be:

The deflection of LB and RB in the positive z-direction and negative z-direction, respectively, causes an opening in the long-slit along the x-z plane. This opening has its maximum width along the z-axis equals to (ΔzF)max. The opening maximum length along the x-axis denoted by (ΔxF)maxas shown inFIG. 2Ccan be obtained by the following equation:

By using Equation (30) and the solution of the cubic equation, (ΔxF)max, can be shown to be equal to the following:

As (ΔxF)max>(ΔzF)maxwhen (ΔzF)max|LB>>d/2, the concentrated force deflection indicator ZFof the microcantilever with long-slit can be redefined as the ratio of (ΔxF)maxper maximum stress. It is equal to:

Define the first detection enhancement indicator of the rectangular microcantilever with log-slit due to concentrated force loading γ1,Fas the ratio of ZFindicator of the rectangular microcantilever with log-slit to the ZFindicator of the typical rectangular microcantilever. As such, γ1,Fis equal to:

The following Table 2 provides the maximum value of rectangular microcantilever with long-slit side beams deflection that produces detection enhancement indicator due to concentrated force loading larger than unity.

Prescribed Differential Surface Stress:

When one surface of LB is coated with a thin film of receptor, it will bend if analyte molecules adhere on that layer. This adhesion causes a difference in the surface stresses across the microcantilever section (Δσ). On the other hand, RB will bend in the opposite direction if the receptor coating is placed on the surface opposite to that of the LB coated surface. The relation between the magnitude of the internal bending moment M at each cross-section of LB and RB and Δσ is given by Equation (14). Let Δσ be considered to vary along the microcantilever length according to the following relationship:

The effective elastic modulus for this case is shown in Equation (15). Accordingly, Equation (1) changes to the following:

The boundary conditions of Equation (38) are given by:

The solution of Equation (1) can be expressed as:

wherex=x/L andc=c/L. If the position of the midsection of RB is taken as the datum of the rectangular microcantilever with long-slit, then the maximum deflection in that microcantilever denoted by (ΔzΔσ)maxwill be:

The deflection of LB and RB in the positive z-direction and negative z-direction, respectively, causes an opening in the long-slit along the x-z plane. This opening has its maximum width along the z-axis equals to (ΔzF)max. The opening maximum length along the x-axis denoted by (ΔxF)maxcan be obtained by the following equation:

By using Equation (40) and the solution of the quadratic equation, (ΔxΔσ)maxcan be shown to be equal to the following:

As (ΔxΔσ)max>(ΔzΔσ)maxwhen (ΔxΔσ)max|LB>>d2, the concentrated force deflection indicator ZΔσof the microcantilever with long-slit can be redefined as the ratio of (ΔxΔσ)maxper maximum stress. It is equal to:

Define the first detection enhancement indicator of the rectangular microcantilever with long-slit due to prescribed differential surface stress loading γΔσas the ratio of ZΔσindicator of the rectangular microcantilever with log-slit to the ZΔσindicator of the typical rectangular microcantilever. As such, γΔσis equal to:

The following Table 3 provides the 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.

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, zd, is shown in the following differential equation:

See Khaled, A.-R. A., Vafai, K., Yang, M., Zhang, X., and Ozkan, C. S., Analysis, control and augmentation of microcantilever deflections in bio-sensing systems, Sens. Actuat. B, 2003, 94, 103-115; Rao S. S., Mechanical Vibrations (5th Edition), Prentice Hall, 2010, USA; and Sader, J., Frequency response of cantilever beams immersed in viscous fluids with applications to the atomic force microscope, Journal of Applied Physics, 1998, 84, 64-76: where meff,2is the effective mass of the LB or RB at their midsections, keff,2is the effective stiffness of the LB or RB at their midsections, Pois the effective amplitude of the dynamic disturbance force at LB or RB midsections per square of disturbance frequency, and ω is the frequency of the dynamic disturbance force. The variable t and quantity λ are the time variable and the wave length of the dynamic disturbance, respectively. Equation (46) is based on the assumption that the microcantilever is excited in the first mode of vibration without total energy dissipation. meff,2and keff,2can be shown to be equal to the following:

The particular solution of the differential equation given by Equation (46) is the following:

where ωsis the first mode natural frequency which is equal to:

Thus, the total maximum deflection of LB and RB denoted by ztLand ztR, respectively, is the sum of the deflection due to force loadings of these beams plus the disturbance in the deflection at time equal to t=0. They can be mathematically expressed as follows:

Since the position of RB at its midsection is taken as the maximum total deflection in that microcantilever denoted by (Δzt,F)maxor (ΔztΔσ)maxwill be:

By the inspection of Equations (21) and (53), the maximum ratio of the amplitude of the disturbance in the deflection of the rectangular microcantilever with long-slit to that of the typical microcantilever is lower than 0.01 when the wave length satisfys the following constraints:

The clearness indicator of the deflection signal of the present microsensor (XD) is redefined here as the ratio of maximum deflection due to loading type D, where D can be F or Δσ loading types to the sum of that deflection plus the maximum disturbance in the deflection. As such, XFand XΔσfor this case are equal to:

To compare between the clearness indicator of the rectangular microcantilever with long-slit and that of typical microcantilever, the second detection enhancement indicator (γ2,D) is defined as the ratio of XD- value of the rectangular microcantilever with long-slit to that of the typical rectangular microcantilever where can be either F or Δσ. Mathematically, they are equal to the following:

where C1=P0ω02/F and C2=P0ω02(L/d)/[Δσ0W(1−v)].

2. Results and Discussion

2.1. Validation of the Results

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 end portions of the microcantilever. The deflection profile300for rectangular microcantilever with long-slit with Lo=425 μm, L=415 μm, W=60 μm, d=0.4 μm, c=5 μm, and δ=2 μm under concentrated force loading with F=2×10−9N is shown inFIG. 3. The microcantilever material was taken to be silicon with E=0.1124N·μm−2and a poisons ratio of v=0.28. The maximum deflection of LB is equal to (zF)max=41.4 nm using Equation (31). As can be seen fromFIG. 2C, the average deflection of the mid-section of the LB is about (zF)max=46.4 nm. Notice that the maximum error between the numerical and the derived analytical solutions is less than 11 percent. The previous small percentage difference gives more confidence on the obtained results. The generated results of various defined detection performance indicators are presented graphically inFIGS. 4-11.

2.2. Discussion of the Results

2.2.1. Discussion of the Results of First Detection Enhancement Indicator

FIGS. 4-5illustrate graphs400and500showing the variation of the first detection enhancement indicator of the rectangular microcantilever with long-slit due to concentrated force loading and prescribed surface stress loading (γ1,F, γ1,Δσ), respectively, with maximum side beam relative deflection {(zF)max|LB/d} for different slit profile dimensionless length (L/d). It is noticed that both γ1,Fand γ1,Δσincrease as (zF)max|LB/d decreases and as L/d increases. This indicates the superiority of the rectangular microcantilever with long-slit over typical rectangular microcantilever in detection of low analyte concentration especially when long slits are considered. Both figures show that the detection capability of the rectangular microcantilever with long-slit can be more than 100 times that of the typical rectangular microcantilever. Moreover, it is noticed fromFIG. 4that γ1,Δσis enhanced as n decreases and its maximum value occur when n=0. This means that well-mixed analyte solutions produce better detection capability than weakly mixed ones as n index approaches n=0 when the mixing level increases.

2.2.2. Discussion of the Results of Clearness Indicator of Typical Rectangular Microcantilever

FIGS. 6-7illustrate graphs600and700showing the effect of the dimensionless frequency of dynamic disturbance (ω/ωo) on the dearness indicator of the deflection signal of the typical rectangular microcantilever due to concentrated force loading and prescribed surface stress loading (XF, XΔσ), respectively. It is noticed that both XFand XΔσgoes to zero as the frequency of dynamic disturbance approaches the fundamental natural frequency (ω/ωo=1.0). This indicates that the detection of the typical microcantilever becomes unrecognizable when ω approaches ω=ωo. Moreover, both indicators are expected to decrease as the amplitude of dynamic disturbance excitation force per square of disturbance frequency (Po) increases. This behavior is noticeable inFIGS. 6-7. Furthermore, it is seen inFIG. 6that XΔσis enhanced as n decreases and its maximum values occur when n=0. Again, this confirms that well-mixed analyte solutions produce lower disturbance levels in the detection signal than weakly mixed ones as n approaches n=0 when the mixing level increases.

2.2.3. Discussion of the Results of Second Detection Enhancement Indicator

FIGS. 8-9illustrate graphs800and850showing the effect of the dimensionless dynamic disturbance wave length (λ/W) on the second detection enhancement indicator of the rectangular microcantilever with long-slit under concentrated force loading and prescribed surface stress loading (γ2,F, γ2,Δσ), respectively. The results of this figure are generated with dimensionless frequency of dynamic disturbance equal to (ω/ωo)=0.5. It is noticed that both indicators are always larger than one regardless of the wave length. As such, the detection of the rectangular microcantilever with long-slit is better isolated against dynamic disturbances than the typical rectangular microcantilever. When λ/W≧3.4827, γ2,F, and γ2,Δσare always increasing as the wave length increases. For small wave length dynamic disturbance, there is a chance that the wave disturbances at the center of LB and RB be of the same phase shift, phase shift difference of π or of phase shift difference between 0 and π. In case of the same phase shift, the disturbance in the detection of the rectangular microcantilever with long-slit is eliminated when subtracting the deflection of RB from that of LB. For the case when π is the phase shift difference, the subtraction process leads to agglomeration in the disturbance in the detection signal as dictated from Equations (57) and (58). When the phase shift difference is between 0 and π, the disturbance in the detection signal becomes more significant as the phase shift difference approaches π.FIGS. 8-9demonstrate that larger dynamic disturbance forces obtained by larger Povalues make the rectangular microcantilever with long-slit more superior than the typical rectangular microcantilever since γ2,Fand γ2,Δσincreases as Poincreases. Finally,FIG. 9shows that γ2,Δσincreases as n increases. This is expected because as n increases, the effective force producing the deflection decreases causing a similar effect as that of increasing the C2value.

FIG. 10illustrate graph900showing the effect of the dimensionless frequency of dynamic disturbance (ω/ωo) on the second detection enhancement indicator of the rectangular microcantilever with long-slit due to concentrated force loading (γ2,F) for two different sets of wave lengths. The set shown for solid lines produce same phase shift for the wave disturbances at the center of LB and RB. For this set, the clearance indicator of the rectangular microcantilever with long-slit is equal to one. That is, the detection quantities are unaffected by dynamic disturbances for this set of wave lengths. As such, γ2,Fvalues are maxima for that set of wave lengths. On the other hand, the set of wave lengths shown for dashed lines produce phase shift difference between the wave disturbances at the center of LB and RB equal to π. For this set, the rectangular microcantilever with long-slit will have the minima values of clearance indicators thus, γ2,Fvalues are minima for the dashed lines. According toFIG. 10, γ2,Fis always larger than one regardless of the frequency of dynamic disturbance as long as ω<ωo. When ω>ωo, the one degree of freedom model cannot be used to determine the disturbance in the deflection and more advanced models are required such as the Euler-Bernoulli beam theory. See Khaled, A.-R. A., and Vafai, K., Analysis of Deflection Enhancement Using Epsilon Assembly Microcantilevers Based Sensors, Sensors, 2011, 11, 9260-9274. The applications of these advanced models on the rectangular microcantilever with long-slit have many complications due to complexity of the geometry. Moreover, γ2,Fis always increasing as ω increases. Similar trends are shown inFIG. 11graph950, where the rectangular microcantilever with long-slit is under the prescribed surface stress loading. Furthermore, the superiority of the rectangular microcantilever with long-slit over the typical rectangular microcantilever increases as Poincreases is shown inFIGS. 10-11since both γ2,Fand γ2,Δσincrease as Poincreases.

An investigation on verifying the advantage of using rectangular microcantilevers with long-slit 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 presence 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 maximum deviation less than 10 percent.

It was found that the detection of the rectangular microcantilevers with long-slit based on its 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 wave length, force amplitude, and the frequency of the dynamic disturbance. Moreover, well-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 wave lengths 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.