Abstract:
An assembly of microcantilever-based sensors with enhanced deflections. A deflection profile of an ε-assembly can be compared with that of a rectangular microcantilever and a modified triangular microcantilever. Various force-loading conditions can also be considered. A theorem of linear elasticity for thin beams is utilized to obtain the deflections. The obtained defections can be validated against an accurate numerical solution utilizing a finite element method with a maximum deviation of less than 10 percent. The ε-assembly produces larger deflections than the rectangular microcantilever under the same base surface stress and same extension length. Also, the ε-microcantilever assembly produces a larger deflection than a modified triangular microcantilever. The deflection enhancement increases as the ε-assembly&#39;s free length decreases for various types of force loading conditions. The ε-microcantilever can be utilized in microsensing applications to provide a favorable high detection capability with a reduced susceptibility to external noises.

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This patent application is a Continuation-in-Part of U.S. patent application Ser. No. 12/793,826, entitled “Methods and Devices Comprising Flexible Seals, Flexible Microchannels, or Both for Modulating or Controlling Flow and Heat” filed on Jun. 6, 2010, which in turn is a continuation of U.S. patent application Ser. No. 11/184,932, entitled “Methods and Devices Comprising Flexible Seals, Flexible Microchannels, or Both For Modulating or Controlling Flow and Heat,” which was filed on Jul. 20, 2005, the disclosure of which is incorporated herein by reference in its entirety. U.S. patent application Ser. No. 11/184,932 in turn is a continuation-in-part of U.S. patent application Ser. No. 10/840,303, filed 7 May 2004, which claims the benefit of U.S. Provisional Patent Application No. 60/470,850 filed 16 May 2003, which are herein incorporated by reference in their entirety. 
     
    
     TECHNICAL FIELD 
       [0002]    Embodiments generally relate to sensors with enhanced deflections comprising microcantilever-based sensors. 
       BACKGROUND OF THE INVENTION 
       [0003]    The rapid growth of nanotechnology has led to developments 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. This is in order to screen a patient for the presence of a disease at an early stage. Microcantilevers based microsensors have been proven to be very sensitive and accurate. See Wu G, Ji H, Hansen K, Thundat T, Datar R, Cote R, Hagan M F, Chakraborty A K, and Majumdar A (2001) Origin of Nanomechanical cantilever motion generated from biomolecular interactions, PNAS, 98:1560-1564. 
         [0004]    The changes in the physical properties of the microcantilever are considered to indicate or detect changes in the environment surrounding it. The most often considered physical property is the deflection. The deflection of the microcantilever was first used for atomic force microscopy. However, it is mainly used to indicate the presence or absence of a certain analyte. See Alkamine S, Barrett R. C., and Quate C. F. (1990) Improved atomic force microscope images using microcantilevers with sharp tips, Appl. Phys. Lett., 57:316 and Raiteri R, Nelles G, Butt H-J, Knoll W, and Skladal P (1999) Sensing of biological substances based on the bending of the microfabricated cantilevers, Sens. Actuators B, 61:213-217. 
         [0005]    The magnitude of microcantilever deflection is of the order of nanometers and it is usually measured using an optical method. The noise level in the surrounding environment affects the performance of the microcantilever as a sensing device. For example, Raiteri R, Nelles G, Butt H-J, Knoll W, and Skladal P (1999), Sensing of biological substances based on the bending of the microfabricated cantilevers, Sens. Actuators B, 61:213-217 reported that the microcantilever deflection due to flow disturbances and due to thermal effects could reach 5-10 times that due to analyte sensing. Accordingly, further developments in microcantilever technology is necessary in order to magnify the deflection signal due to the sensing effect so that its signal can be easily distinguished from the noise signal. See Fritz J, Bailer M K, Lang H P, Rothuizen H, Vettiger P, Meyer E, Guntherodt H-J, Gerber Ch, and Gimzewski J K (2000) Translating biomolecular recognition into nanomechanics, Science, 288:316-8, Yang M, Zhang X, Vafai K, and Ozkan C S (2003) High sensitivity piezoresistive cantilever design and optimization for an analyte-receptor binding, J. Micromech. Microeng., 13:864-72, and Khaled A.-R. A. and Vafai, K. (2004) Optimization modelling of analyte adhesion over an inclined microcantilever-based biosensor, J. Micromech. Microeng., 14:1220-29. As such, Khaled A-R A, Vafai K, Yang M, Zhang X, and Ozkan C S (2003) Analysis, control and augmentation of microcantilever deflections in bio-sensing systems, Sens. Actuators B, 94:103-115, pointed out the necessity of establishing special microcantilevers assemblies for this purpose. 
       BRIEF SUMMARY 
       [0006]    The following summary is provided to facilitate an understanding of some of the innovative features unique to the disclosed embodiment and is not intended to be a full description. A full appreciation of the various aspects of the embodiments disclosed herein can be gained by taking the entire specification, claims, drawings, and abstract as a whole. 
         [0007]    It is therefore one aspect of the disclosed embodiments to provide for an epsilon-shaped microcantilever and assembly thereof that can be employed in a variety sensing, cooling, and/or microfluidic applications. 
         [0008]    It is another aspect of the disclosed embodiments to provide for an epsilon-shaped microcantilever and assembly thereof having at least a first beam, a second beam, and an intermediate beam. 
         [0009]    It is yet another aspect of the disclosed embodiments to provide for an epsilon-shaped microcantilever and assembly thereof with enhanced deflections for sensing applications. 
         [0010]    The aforementioned aspects and other objectives and advantages can now be achieved as described herein. An epsilon-shaped microcantilever is disclosed, which includes a first side beam, a second side beam, and an intermediate beam, wherein the first side beam comprises an end attached to an end of the second side beam. The intermediate beam comprises an end attached to the ends of the first and second side beams such that the intermediate beam is positioned between the first and second side beams. Additionally, the ends opposite to the attached ends of the first side beam, second side beam, and intermediate beam are left free and are force-loaded. The first side beam, the second side beam, and the intermediate beam each possess a top surface and a bottom surface. The epsilon-shaped microcantilever can also include a receptor coated on the top surfaces of the first and second side beams and on the bottom surface of the intermediate beam. 
         [0011]    Varying force loadings including at least one of a concentrated force, a concentrated moment, and a constant surface stress can be utilized with respect to the epsilon-shaped microcantilever assembly disclosed herein. In general, a deflection theory of thin beams can be utilized to obtain a deflection profile with respect to the epsilon-shaped microcantilever assembly disclosed herein. Obtained deflections of the epsilon-shaped assembly can be validated against an accurate numerical solution utilizing a finite element method with a maximum deviation of less than approximately ten percent. Additionally, the deflection enhancement increases as the free length decreases for various types of force loading conditions with respect to the disclosed epsilon-shaped microcantilevers assembly. 
     
    
     
       DESCRIPTION OF THE DRAWINGS 
         [0012]    This invention is further understood by reference to the drawings wherein: 
           [0013]      FIG. 1  is a schematic diagram and the coordinate system of Rectangular microcantilever assembly. 
           [0014]      FIG. 2  is a schematic diagram and the coordinate system of modified triangular microcantilever assembly. 
           [0015]      FIG. 3  is a schematic diagram and the coordinate system of ε-microcantilever assembly. 
           [0016]      FIG. 4  shows the deflection profile for assembly of  FIG. 1  using numerical solutions with L=385 μm, W=30 μm, t=20 nm, M=10 −12  Nμm, E=0.185 Nμm −2 , and v=0.33. 
           [0017]      FIG. 5  shows the effects of the relative dimensions of the modified triangular and ε-microcantilevers assemblies on the first performance indicators γ bF  and γ cF . 
           [0018]      FIG. 6  shows the effects of the relative dimensions of the modified triangular and ε-microcantilevers assemblies on the first performance indicators γ bM  and γ cM . 
           [0019]      FIG. 7  shows the effects of the relative dimensions of the modified triangular and ε-microcantilevers assemblies on the first performance indicators γ bΔσ  and γ cΔσ . 
           [0020]      FIG. 8  shows the effects of the relative dimensions of the modified triangular and ε-microcantilevers assemblies on the second performance indicators λ cF . 
           [0021]      FIG. 9  shows the effects of the relative dimensions of the modified triangular and ε-microcantilevers assemblies on the second performance indicators λ cM . 
           [0022]      FIG. 10  shows the effects of the relative dimensions of the modified triangular and ε-microcantilevers assemblies on the second performance indicators λ cΔσ . 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0023]    The following Table 1 provides the various symbols and meanings used in this section: 
         [0000]    
       
         
               
             
               
               
               
             
               
             
               
               
               
             
               
             
               
               
               
             
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
             
             
               
                 Nomenclature: 
               
             
          
           
               
                   
                 B 
                 Base length of the microcantilever assembly 
               
               
                   
                 E 
                 Elastic modulus (N μm −2 ) 
               
               
                   
                 F 
                 concentrated force (N) 
               
               
                   
                 I 
                 Area moment of inertia (μm 4 ) 
               
               
                   
                 L 
                 microcantilever or assembly extension length (μm) 
               
               
                   
                 M 
                 moment (N μm) 
               
               
                   
                 n 
                 surface stress model index 
               
               
                   
                 t 
                 microcantilever thickness (μm) 
               
               
                   
                 W 
                 microcantilever width (μm) 
               
               
                   
                 x 
                 axis of the extension dimension (μm) 
               
               
                   
                 Y 
                 effective elastic modulus (N μm −2 ) 
               
               
                   
                 z 
                 deflection (μm) 
               
             
          
           
               
                 Greek symbols: 
               
             
          
           
               
                   
                 γ 
                 first deflection indicator 
               
               
                   
                 λ 
                 second deflection indicator 
               
               
                   
                 ν 
                 Poisson&#39;s ratio 
               
               
                   
                 σ 
                 surface stress 
               
             
          
           
               
                 Subscripts: 
               
             
          
           
               
                   
                 F 
                 concentrated force condition 
               
               
                   
                 M 
                 moment condition 
               
               
                   
                 Δσ 
                 constant differential surface stress condition 
               
             
          
           
               
                 Abbreviations: 
               
             
          
           
               
                   
                 IB 
                 the intermediate beam of ε-assembly 
               
               
                   
                 SB 
                 the side beams of ε-assembly 
               
               
                   
                   
               
             
          
         
       
     
         [0024]    The accompanying figures, in which like reference numerals refer to identical or functionally similar elements throughout the separate views and which are incorporated in and form a part of the specification, further illustrate the present invention and, together with the detailed description of the invention, serve to explain the principles of the disclosed embodiments. 
         [0025]      FIG. 1  illustrates a schematic diagram depicting the geometry of a rectangular microcantilever (MC)  10 . Specifying the extension length L, width W, thickness t, Young&#39;s modulus E, and Poisson&#39;s ratio can summarize the properties of the rectangular microcantilever 10 v. When the length of the microcantilever is much larger than its width, Hooks law for small deflections can be utilized to relate the microcantilever deflections to the effective elastic modulus Y and the bending moment M, and can be represented by the following equation (1): 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         2 
                       
                        
                       z 
                     
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   
                     M 
                     YI 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   1 
                 
               
             
           
         
       
     
         [0000]    In the formulation above, the variable z is the deflection of the microcantilever at any section located at a position x from the base surface. I is the area moment of inertia of the microcantilever cross-section about its neutral axis. For a rectangular cross-section with its neutral axis coinciding with its centroidal axis, I is given by equation (2) below: 
         [0000]    
       
         
           
             
               
                 
                   I 
                   = 
                   
                     
                       1 
                       12 
                     
                      
                     
                       Wt 
                       3 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     2 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The boundary conditions for Equation (1) are given by 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                      
                     
                       ( 
                       
                         x 
                         = 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                            
                           z 
                         
                         
                            
                           x 
                         
                       
                        
                       
                         | 
                         
                           x 
                           = 
                           0 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     Eqs 
                     . 
                     
                         
                     
                      
                     3 
                   
                    
                   
                     ( 
                     
                       a 
                       , 
                       b 
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0026]    For a concentrated force exerted on the rectangular microcantilever tip (x=L), the solution of Equation (1), denoted by z aF (χ), subject to boundary conditions given by Equations 3 (a, b) can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       aF 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           6 
                            
                           
                             FL 
                             3 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               x 
                               L 
                             
                             ) 
                           
                           2 
                         
                         - 
                         
                           
                             1 
                             3 
                           
                            
                           
                             
                               ( 
                               
                                 x 
                                 L 
                               
                               ) 
                             
                             3 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     4 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The above result is based on a realistic linearly increasing bending moment from the base prescribed by: 
         [0000]    
       
         
           
             
               
                 
                   M 
                   = 
                   
                     FL 
                      
                     
                       ( 
                       
                         1 
                         - 
                         
                           x 
                           L 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     5 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    For thin cross-sections, the surface stress, σ, can be calculated from the following equation: 
         [0000]    
       
         
           
             
               
                 
                   σ 
                   = 
                   
                     
                       M 
                       I 
                     
                      
                     
                       ( 
                       
                         t 
                         2 
                       
                       ) 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     6 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The surface stress at x=0 (base surface) denoted by σ aFo  is equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     aFo 
                   
                   = 
                   
                     
                       6 
                        
                       FL 
                     
                     
                       Wt 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     7 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The maximum deflection which occurs at the microcantilever tip (x=L) can be expressed as indicated by equation (8) below: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     
                       aF 
                        
                       
                           
                       
                        
                       max 
                     
                   
                   = 
                   
                     
                       4 
                        
                       
                           
                       
                        
                       
                         FL 
                         3 
                       
                     
                     
                       EWt 
                       3 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
         [0027]    For a bending moment M exerted on the rectangular microcantilever tip (x=L), the solution of Equation (1), denoted by z aM (χ), subject to boundary conditions given by Equations 3 (a, b) can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       aM 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           6 
                            
                           
                               
                           
                            
                           
                             ML 
                             2 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       
                         ( 
                         
                           x 
                           L 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The surface stress at the base section which is denoted by σ aMo  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     aMo 
                   
                   = 
                   
                     
                       6 
                        
                       M 
                     
                     
                       Wt 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The maximum deflection which is the deflection at the microcantilever tip is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     
                       aM 
                        
                       
                           
                       
                        
                       max 
                     
                   
                   = 
                   
                     
                       6 
                        
                       
                           
                       
                        
                       
                         ML 
                         2 
                       
                     
                     
                       EWt 
                       3 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
         [0028]    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 M [1, 8] is given by equation (12) below: 
         [0000]    
       
         
           
             
               
                 
                   M 
                   = 
                   
                     
                       Δ 
                        
                       
                           
                       
                        
                       σ 
                        
                       
                           
                       
                        
                       Wt 
                     
                     2 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    wherein Δσ is the difference between the surface stresses of the top and bottom sides of the microcantilever. The solution of Eq. (1), denoted by z aΔσ (χ), subject to boundary conditions given by Equations 3 (a, b) can then be expressed by equation (13) below: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       
                         a 
                          
                         
                             
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         σ 
                       
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         6 
                          
                         
                           ( 
                           
                             1 
                             - 
                             v 
                           
                           ) 
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         
                           σ 
                           o 
                         
                          
                         
                           L 
                           2 
                         
                       
                       
                         
                           
                             Et 
                             2 
                           
                            
                           
                             ( 
                             
                               n 
                               + 
                               1 
                             
                             ) 
                           
                         
                          
                         
                           ( 
                           
                             n 
                             + 
                             2 
                           
                           ) 
                         
                       
                     
                      
                     
                       
                         ( 
                         
                           x 
                           L 
                         
                         ) 
                       
                       
                         n 
                         + 
                         2 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     13 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    This is because the effective elastic modulus for this case is given by Y=E/(1−v). Also, Δσ is considered to vary along the microcantilever length according to the following relationship of equation (14): 
         [0000]    
       
         
           
             
               
                 
                   Δσ 
                   = 
                   
                     
                       
                         Δσ 
                         o 
                       
                        
                       
                         ( 
                         
                           x 
                           L 
                         
                         ) 
                       
                     
                     n 
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    wherein n is the model index. This variation is expected as analyte concentration in the surrounding environment and is expected to increase as the distance from the microcantilever base increases. The maximum deflection due to analyte adhesion is obtained from Equation 205 by substituting x=L. It is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       
                         a 
                          
                         
                             
                         
                          
                         Δσ 
                          
                         
                             
                         
                          
                         mmax 
                       
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       6 
                        
                       
                         ( 
                         
                           1 
                           - 
                           v 
                         
                         ) 
                       
                        
                       
                         Δσ 
                         o 
                       
                        
                       
                         L 
                         2 
                       
                     
                     
                       
                         
                           Et 
                           2 
                         
                          
                         
                           ( 
                           
                             n 
                             + 
                             1 
                           
                           ) 
                         
                       
                        
                       
                         ( 
                         
                           n 
                           + 
                           2 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    Equation (15) is reducible to the Stoney&#39;s equation when n is set to be equal to zero. 
         [0029]      FIG. 2  illustrates a schematic diagram depicting the geometry of a microcantilever assembly  12 . The microcantilever assembly  12  shown in  FIG. 2  includes an SB (Side Beam)  14  and an SB  16 . The microcantilever assembly  12  depicted in  FIG. 2  constitutes a modified triangular MC assembly, as opposed to the rectangular MC  10  depicted in  FIG. 1 . 
         [0030]    Equation (1) is changeable to the following when the center line of the free end (x=L) is loaded by a normal concentrated force of magnitude F: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         2 
                       
                        
                       
                         z 
                         bF 
                       
                     
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           FL 
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                     × 
                     2 
                      
                     
                       
                         ( 
                         
                           1 
                           - 
                           
                             x 
                             / 
                             L 
                           
                         
                         ) 
                       
                       / 
                       
                         
                           cos 
                           3 
                         
                          
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     16 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    Note that I for each beam is I=Wt 3 /12. Note that θ is half the triangular tip angle. The cosine of the angle θ is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     cos 
                      
                     
                       ( 
                       θ 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         1 
                         + 
                         
                           
                             0.25 
                              
                             
                               [ 
                               
                                 
                                   B 
                                   / 
                                   L 
                                 
                                 
                                   1 
                                   - 
                                   
                                     0.5 
                                      
                                     
                                       ( 
                                       
                                         W 
                                         / 
                                         L 
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           2 
                         
                       
                       ] 
                     
                     
                       
                         - 
                         1 
                       
                       / 
                       2 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     17 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The boundary conditions for Equation 208 are given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       b 
                     
                      
                     
                       ( 
                       
                         x 
                         = 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                            
                           
                             z 
                             b 
                           
                         
                         
                            
                           x 
                         
                       
                        
                       
                          
                         
                           x 
                           = 
                           0 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     
                       Eqs 
                       . 
                       
                           
                       
                        
                       18 
                     
                      
                     
                       ( 
                       
                         a 
                         , 
                         b 
                       
                       ) 
                     
                   
                    
                   
                       
                   
                 
               
             
           
         
       
     
         [0000]    The solution of Equation (16), denoted by z bF (χ), subject to the above boundary conditions is the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       bF 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             3 
                              
                             
                               FL 
                               3 
                             
                           
                           
                             EWt 
                             3 
                           
                         
                         ) 
                       
                        
                       
                         [ 
                         
                           
                             
                               ( 
                               
                                 x 
                                 L 
                               
                               ) 
                             
                             2 
                           
                           - 
                           
                             
                               1 
                               3 
                             
                              
                             
                               
                                 ( 
                                 
                                   x 
                                   L 
                                 
                                 ) 
                               
                               3 
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       ( 
                       
                         1 
                         
                           
                             cos 
                             3 
                           
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     19 
                   
                    
                   
                     ( 
                     a 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    Using Equation (6), the surface stress at x=0, σ bFo , is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     bFo 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           FL 
                         
                         
                           Wt 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       [ 
                       
                         1 
                         
                           cos 
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The maximum deflection occurs at the tip (x=L). It is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     
                       bF 
                        
                       
                           
                       
                        
                       max 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           
                             FL 
                             3 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       { 
                       
                         2 
                         
                           3 
                            
                           
                               
                           
                            
                           
                             
                               cos 
                               3 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     21 
                     ) 
                   
                 
               
             
           
         
       
     
         [0031]    For a bending moment M about x-axis exerted on the centerline of the free end of the assembly (b) (at x=L), Equation (3) is changeable to the following form: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         2 
                       
                        
                       
                         z 
                         bM 
                       
                     
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           M 
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                     × 
                     
                       2 
                       / 
                       
                         cos 
                          
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     22 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    the solution of Equation (22), subject to boundary conditions given by Equation 18 (a, b) is the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       bM 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           2 
                            
                           
                             ML 
                             2 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       
                         
                           ( 
                           
                             x 
                             L 
                           
                           ) 
                         
                         2 
                       
                        
                       
                         [ 
                         
                           1 
                           
                             cos 
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     23 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    As such, the maximum deflection is expected to be equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     
                       bM 
                        
                       
                           
                       
                        
                       max 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           
                             ML 
                             2 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       { 
                       
                         1 
                         
                           cos 
                            
                           
                             ( 
                             θ 
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     24 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    Using Equation (6), the surface stress at x=0, σ cMo , is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     bMo 
                   
                   = 
                   
                     
                       
                         3 
                          
                         M 
                       
                       
                         Wt 
                         2 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     25 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    When a receptor layer is coated on one side of assembly (b)-side beams (SB), Equation (3) changes to the following form after the analyte adhesion on these coatings: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         2 
                       
                        
                       
                         z 
                         
                           b 
                            
                           
                               
                           
                            
                           Δσ 
                         
                       
                     
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           6 
                            
                           
                             ( 
                             
                               1 
                               - 
                               v 
                             
                             ) 
                           
                            
                           
                             Δσ 
                             o 
                           
                         
                         
                           Et 
                           2 
                         
                       
                       } 
                     
                     × 
                     
                       
                         
                           ( 
                           
                             x 
                             / 
                             L 
                           
                           ) 
                         
                         n 
                       
                       / 
                       
                         
                           cos 
                           2 
                         
                          
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     26 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The solution of Equation (26), subject to boundary conditions given by Equation (18) (a, b) is the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       
                         b 
                          
                         
                             
                         
                          
                         Δσ 
                       
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           6 
                            
                           
                             ( 
                             
                               1 
                               - 
                               v 
                             
                             ) 
                           
                            
                           
                             Δσ 
                             o 
                           
                            
                           
                             L 
                             2 
                           
                         
                         
                           
                             
                               Et 
                               2 
                             
                              
                             
                               ( 
                               
                                 n 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                             ( 
                             
                               n 
                               + 
                               2 
                             
                             ) 
                           
                         
                       
                       } 
                     
                      
                     
                       
                         
                           ( 
                           
                             x 
                             L 
                           
                           ) 
                         
                         
                           n 
                           + 
                           2 
                         
                       
                        
                       
                         [ 
                         
                           1 
                           
                             
                               cos 
                               2 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     27 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The maximum deflection due to analyte adhesion is then equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     z 
                     
                       b 
                        
                       
                           
                       
                        
                       Δσ 
                        
                       
                           
                       
                        
                       m 
                        
                       
                           
                       
                        
                       max 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           6 
                            
                           
                             ( 
                             
                               1 
                               - 
                               v 
                             
                             ) 
                           
                            
                           
                             Δσ 
                             o 
                           
                            
                           
                             L 
                             2 
                           
                         
                         
                           Et 
                           2 
                         
                       
                       } 
                     
                      
                     
                       { 
                       
                         
                           1 
                           / 
                           
                             
                               cos 
                               2 
                             
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         
                           
                             ( 
                             
                               n 
                               + 
                               1 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               n 
                               + 
                               2 
                             
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     28 
                     ) 
                   
                 
               
             
           
         
       
     
         [0032]    Define the first deflection indicator γ pU  as the ratio of the microcantilever deflection at the tip (x=L) per surface stress at the base for the microcantilever of type (p) due to force loading of type U to the corresponding value for the rectangular microcantilever. The type (p) can be either the microcantilever type shown in  FIG. 2  or the type depicted in  FIG. 3 . The force loading of type U can be either a concentrated force loading (F), an external bending moment (M) or a constant surface stress (Δσ o ). As such, γ bF , γ bM  and γ bΔσ     o    are equal to 
         [0000]    
       
         
           
             
               
                 
                   
                     γ 
                     bF 
                   
                   = 
                   
                     1 
                     / 
                     
                       
                         cos 
                         3 
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     29 
                   
                    
                   
                     ( 
                     a 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     γ 
                     bM 
                   
                   = 
                   
                     1 
                     
                       
                         cos 
                         2 
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     29 
                   
                    
                   
                     ( 
                     b 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     γ 
                     
                       b 
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       
                         σ 
                         o 
                       
                     
                   
                   = 
                   
                     1 
                     / 
                     
                       
                         cos 
                         2 
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     29 
                   
                    
                   
                     ( 
                     c 
                     ) 
                   
                 
               
             
           
         
       
     
       The Microcantilever ε-Assembly 
       [0033]      FIG. 3  illustrates a schematic diagram of an epsilon (ε) shaped microcantilever or MC assembly  13 , in accordance with the disclosed embodiments. The MC assembly  13  shown in  FIG. 3  includes a SB  15  and a SB  17  along with an IB (Intermediate Beam)  19 , which together form the epsilon (ε) shaped microcantilever or MC assembly  13 . In order to configure the geometry of the microcantilever assembly  13  shown in  FIG. 3 , the centerline of the assembly free end (x=L) can be loaded by a normal concentrated force of magnitude F. Additionally, the free end of the intermediate beam (IB)  19  can be loaded by the negative of the previous load (−F). Accordingly, Equation (3) changes to the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         2 
                       
                        
                       
                         z 
                         cF 
                       
                     
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           FL 
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                     × 
                     
                       [ 
                       
                         
                           
                             
                               
                                 2 
                                 / 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               , 
                             
                           
                           
                             
                               ( 
                               
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 SB 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 
                                   - 
                                   4 
                                 
                                  
                                 
                                   ( 
                                   
                                     x 
                                     / 
                                     L 
                                   
                                   ) 
                                 
                               
                               , 
                             
                           
                           
                             
                               ( 
                               
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 IB 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     30 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    wherein SB stands for the side beams of the assembly. The boundary conditions of Equation (30) can be given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       cSB 
                     
                      
                     
                       ( 
                       
                         x 
                         = 
                         0 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                            
                           
                             z 
                             cSB 
                           
                         
                         
                            
                           x 
                         
                       
                        
                       
                         | 
                         
                           x 
                           = 
                           0 
                         
                       
                     
                     = 
                     0 
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     31 
                   
                    
                   
                     ( 
                     a 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       z 
                       cSB 
                     
                      
                     
                       ( 
                       
                         x 
                         = 
                         L 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       z 
                       cIB 
                     
                      
                     
                       ( 
                       
                         x 
                         = 
                         L 
                       
                       ) 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     31 
                   
                    
                   
                     ( 
                     b 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       
                          
                         
                           z 
                           cSB 
                         
                       
                       
                          
                         x 
                       
                     
                      
                     
                       | 
                       
                         x 
                         = 
                         L 
                       
                     
                   
                   = 
                   
                     
                       
                          
                         
                           z 
                           cIB 
                         
                       
                       
                          
                         x 
                       
                     
                      
                     
                       | 
                       
                         x 
                         = 
                         L 
                       
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     31 
                   
                    
                   
                     ( 
                     c 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The solution of Equation 30, denoted by z cF (χ), is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         z 
                         cSBF 
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         ( 
                         
                           
                             3 
                              
                             
                               FL 
                               3 
                             
                           
                           
                             EWt 
                             3 
                           
                         
                         ) 
                       
                        
                       
                         
                           
                             ( 
                             
                               x 
                               L 
                             
                             ) 
                           
                           2 
                         
                          
                         
                           [ 
                           
                             1 
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     32 
                   
                    
                   
                     ( 
                     a 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       z 
                       cIBF 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           
                             FL 
                             3 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       { 
                       
                         
                           
                             - 
                             
                               ( 
                               
                                 2 
                                 3 
                               
                               ) 
                             
                           
                            
                           
                             
                               ( 
                               
                                 x 
                                 L 
                               
                               ) 
                             
                             3 
                           
                         
                         + 
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 1 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               x 
                               L 
                             
                             ) 
                           
                         
                         + 
                         
                           D 
                           1 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     32 
                   
                    
                   
                     ( 
                     b 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    wherein D 1  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     1 
                   
                   = 
                   
                     - 
                     
                       { 
                       
                         
                           1 
                           
                             cos 
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         + 
                         
                           4 
                           3 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     32 
                   
                    
                   
                     ( 
                     c 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The surface stress at the base section σ cFo  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     cFo 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           3 
                            
                           FL 
                         
                         
                           Wt 
                           2 
                         
                       
                       ) 
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   33 
                 
               
             
           
         
       
     
         [0034]    The second deflection indicator λ dU  can be defined as the ratio of the IB-free end deflection z cIBU (χ=0) to that at the assembly free end z cU (χ=L) due to force loading of type U. The force loading of type U can be either the current described force loading (F), external bending moment loading (M) or the constant surface stress (Δσ o ) loading. The last two types of force loadings will be described later on. As such, λ cF  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     cF 
                   
                   = 
                   
                     
                       
                         
                           zc 
                           dIBF 
                         
                          
                         
                           ( 
                           
                             x 
                             = 
                             0 
                           
                           ) 
                         
                       
                       
                         
                           z 
                           cF 
                         
                          
                         
                           ( 
                           
                             x 
                             = 
                             L 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       - 
                       
                         { 
                         
                           1 
                           + 
                           
                             
                               4 
                               3 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   34 
                 
               
             
           
         
       
     
         [0035]    Now, let a bending moment M be exerted on the free end centerline of the MC assembly  13  and let another bending moment of same magnitude be exerted on the IB-free end at x=0. The deflection equations for this assembly under the current moments loading is given by the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                          
                         2 
                       
                        
                       
                         z 
                         cM 
                       
                     
                     
                        
                       
                         x 
                         2 
                       
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           6 
                            
                           M 
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                     × 
                     
                       [ 
                       
                         
                           
                             
                               
                                 2 
                                 / 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               , 
                             
                           
                           
                             
                               ( 
                               
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 SB 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 - 
                                 2 
                               
                               , 
                             
                           
                           
                             
                               ( 
                               
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 IB 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   35 
                 
               
             
           
         
       
     
         [0000]    The boundary conditions are given by Equations 31 (a-c). The solution of Equation 35 is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       z 
                       cSBM 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         cos 
                          
                         
                           ( 
                           θ 
                           ) 
                         
                       
                     
                      
                     
                       ( 
                       
                         
                           6 
                            
                           
                             ML 
                             2 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       
                         ( 
                         
                           x 
                           L 
                         
                         ) 
                       
                       2 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     36 
                   
                    
                   
                     ( 
                     a 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       z 
                       cIBM 
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       ( 
                       
                         
                           6 
                            
                           
                             ML 
                             2 
                           
                         
                         
                           EWt 
                           3 
                         
                       
                       ) 
                     
                      
                     
                       { 
                       
                         
                           - 
                           
                             
                               ( 
                               
                                 x 
                                 L 
                               
                               ) 
                             
                             2 
                           
                         
                         + 
                         
                           2 
                            
                           
                             ( 
                             
                               
                                 1 
                                 
                                   cos 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                               + 
                               1 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               x 
                               L 
                             
                             ) 
                           
                         
                         + 
                         
                           D 
                           2 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     36 
                   
                    
                   
                     ( 
                     b 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    wherein D 2  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     D 
                     2 
                   
                   = 
                   
                     - 
                     
                       [ 
                       
                         
                           1 
                           
                             cos 
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                         
                         + 
                         1 
                       
                       ] 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     36 
                   
                    
                   
                     ( 
                     c 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    the surface stress at x=0, σ cMo , is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     cMo 
                   
                   = 
                   
                     
                       
                         6 
                          
                         M 
                       
                       
                         Wt 
                         2 
                       
                     
                      
                     
                       cos 
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   39 
                 
               
             
           
         
       
     
         [0000]    The second deflection indicator for assembly (c) (i.e.,  FIG. 3 ) for the current moments loading λ cM  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     cM 
                   
                   = 
                   
                     
                       
                         
                           z 
                           cIBM 
                         
                          
                         
                           ( 
                           
                             x 
                             = 
                             0 
                           
                           ) 
                         
                       
                       
                         
                           z 
                           cM 
                         
                          
                         
                           ( 
                           
                             x 
                             = 
                             L 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       - 
                       
                         [ 
                         
                           
                             cos 
                              
                             
                               ( 
                               θ 
                               ) 
                             
                           
                           + 
                           1 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   40 
                 
               
             
           
         
       
     
         [0036]    If the top surfaces of the side beams of MC assembly  13  can be coated with a receptor while the receptor coating on the intermediate beam is on its bottom surface, then the deflection equations of the MC assembly  13  changes to: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         cos 
                         2 
                       
                        
                       
                         ( 
                         θ 
                         ) 
                       
                     
                     × 
                     
                       
                         
                            
                           2 
                         
                          
                         
                           z 
                           
                             cSB 
                              
                             
                                 
                             
                              
                             Δ 
                              
                             
                                 
                             
                              
                             σ 
                           
                         
                       
                       
                          
                         
                           x 
                           2 
                         
                       
                     
                   
                   = 
                   
                     
                       - 
                       
                         
                           
                              
                             2 
                           
                            
                           
                             z 
                             
                               cIB 
                                
                               
                                   
                               
                                
                               Δ 
                                
                               
                                   
                               
                                
                               σ 
                             
                           
                         
                         
                            
                           
                             x 
                             2 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           6 
                            
                           
                             ( 
                             
                               1 
                               - 
                               v 
                             
                             ) 
                           
                            
                           
                             Δσ 
                             o 
                           
                         
                         
                           Et 
                           2 
                         
                       
                        
                       
                         
                           ( 
                           
                             x 
                             L 
                           
                           ) 
                         
                         n 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   41 
                 
               
             
           
         
       
     
         [0037]    The solution for Equation 41 subject to boundary conditions given by Equations 223(a-c) is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         z 
                         
                           cSB 
                            
                           
                               
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           σ 
                         
                       
                        
                       
                         ( 
                         x 
                         ) 
                       
                     
                     = 
                     
                       
                         { 
                         
                           
                             6 
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 v 
                               
                               ) 
                             
                              
                             Δ 
                              
                             
                                 
                             
                              
                             
                               σ 
                               o 
                             
                              
                             
                               L 
                               2 
                             
                           
                           
                             
                               E 
                                
                               
                                 ( 
                                 
                                   n 
                                   + 
                                   1 
                                 
                                 ) 
                               
                             
                              
                             
                               t 
                               2 
                             
                           
                         
                         } 
                       
                        
                       
                         
                           ( 
                           
                             x 
                             L 
                           
                           ) 
                         
                         
                           n 
                           + 
                           2 
                         
                       
                        
                       
                         { 
                         
                           
                             1 
                             / 
                             
                               
                                 cos 
                                 2 
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             n 
                             + 
                             2 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     42 
                   
                    
                   
                     ( 
                     a 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     
                       z 
                       
                         cIB 
                          
                         
                             
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         σ 
                       
                     
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       { 
                       
                         
                           6 
                            
                           
                             ( 
                             
                               1 
                               - 
                               v 
                             
                             ) 
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           
                             σ 
                             o 
                           
                            
                           
                             L 
                             2 
                           
                         
                         
                           
                             E 
                              
                             
                               ( 
                               
                                 n 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                            
                           
                             t 
                             2 
                           
                         
                       
                       } 
                     
                      
                     
                       { 
                       
                         
                           
                             
                               - 
                               1 
                             
                             
                               ( 
                               
                                 n 
                                 + 
                                 2 
                               
                               ) 
                             
                           
                            
                           
                             
                               ( 
                               
                                 x 
                                 L 
                               
                               ) 
                             
                             
                               n 
                               + 
                               2 
                             
                           
                         
                         + 
                         
                           
                             [ 
                             
                               1 
                               + 
                               
                                 1 
                                 
                                   
                                     cos 
                                     2 
                                   
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                             [ 
                             
                               
                                 x 
                                 L 
                               
                               - 
                               
                                 
                                   n 
                                   + 
                                   1 
                                 
                                 
                                   n 
                                   + 
                                   2 
                                 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   
                     Eq 
                     . 
                     
                         
                     
                      
                     42 
                   
                    
                   
                     ( 
                     b 
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    The deflection indicator for MC assembly  13  due to the alternating analyte adhesion on the surfaces λ cΔσ  is equal to: 
         [0000]    
       
         
           
             
               
                 
                   
                     λ 
                     
                       c 
                        
                       
                           
                       
                        
                       Δ 
                        
                       
                           
                       
                        
                       σ 
                     
                   
                   = 
                   
                     
                       
                         
                           z 
                           
                             cIB 
                              
                             
                                 
                             
                              
                             Δ 
                              
                             
                                 
                             
                              
                             σ 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             = 
                             0 
                           
                           ) 
                         
                       
                       
                         
                           z 
                           
                             cSB 
                              
                             
                                 
                             
                              
                             Δ 
                              
                             
                                 
                             
                              
                             σ 
                           
                         
                          
                         
                           ( 
                           
                             x 
                             = 
                             L 
                           
                           ) 
                         
                       
                     
                     = 
                     
                       - 
                       
                         
                           ( 
                           
                             n 
                             + 
                             1 
                           
                           ) 
                         
                          
                         
                           [ 
                           
                             
                               
                                 cos 
                                 2 
                               
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                             + 
                             1 
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                    
                   43 
                 
               
             
           
         
       
     
         [0000]    The deflection indicators γ cF , γ cM  and γ cΔσ     o    can be shown to be equal to the following: 
         [0000]      γ cF =1.5/cos 2 (θ)  Eq. 44(a)
 
         [0000]      γ cM =1/cos 2 (θ)  Eq. 44(b)
 
         [0000]      γ cΔσ     o   =1/cos 2 (θ)  Eq. 44(c)
 
       Validation 
       [0038]    Note that 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 x=L. The deflection contours for assembly (c) with L=385 μm, W=30 μm and t=20 nm under concentrated moment condition described in section 2.2.II with M=10 −12  Nμm is shown in  FIG. 4 . The microcantilever material was taken to be silicon with E=0.185 Nμm −2  and a poisons ratio of v=0.33. The assembly deflection at x=L is equal to z cM (χ=L)=0.028 μm using Equation (36)(b). Also, the deflection at the intermediate beam&#39;s free end can be shown to be equal to z cIBM (χ=0)=0.048 μm. 
         [0039]    As can be seen from  FIG. 4 , the corresponding numerical values of those deflections are equal to 0.026 μm and 0.045 μm, respectively. Notice that the maximum error between the numerical and the derived analytical solutions is less than 10 percent. Also, notice that the numerical values of deflections are smaller than those predicated by the analytical methods. This is because the geometrical constraints imposed on the assemblies impede the deflections. 
         [0040]      FIG. 5  illustrates a graph  50  depicting plotted data indicative of the variation of the performance indicators γ bF  and γ cF  with the relative dimensions of assemblies  12  and  13 . It is noticed that all the values of γ bF  and γ cF  are larger than one which indicates that assemblies  12  and  13  produce larger deflections than rectangular microcantilevers under same surface stress at the base and same length L. Moreover, both indicators increase as both the microcantilever width W and the assembly width B increase. Similar findings are noticed for the performance indicators γ bM , γ cM , γ bΔσ , and γ cΔσ  as can be seen from graphs  60  and  70  shown respectively  FIGS. 6 and 7 . On the other hand, an increase in B causes the effective free length of the assembly to increase, which makes the assembly more pronounced to external noises. 
         [0041]      FIG. 8  illustrates a graph  80  depicting data indicative of the variation of the second performance indicator λ cF  with the relative dimensions of the epsilon shaped microcantilever assembly  13 . It is noticed that all values of λ cF  are smaller than minus one. This indicates that IB-free end deflection is always larger than that of the assembly tip deflection. Moreover, the absolute value of λ cF  is noticed to increases as both W and B decreases. Similar findings are noticed for the performance indicators λ cM  and λ cΔσ  as can be seen from graphs  90  and  100  depicted respectively in  FIGS. 9 and 10 . As a result, the epsilon shaped microcantilever assembly  13  can provide larger deflections than the modified triangular microcantilever assembly  12 , while it is less affected by external noise. This is because its deflection increases as B decreases, which results in a reduction of the assembly&#39;s free length. Moreover, the absolute values of λ cΔσ  increases as n increases as can be shown using Equation 234. This indicates the advantage of assembly  13  in microsensing applications as compared to rectangular cantilevers or triangular cantilevers. 
         [0042]    Advantages of utilizing microcantilever assemblies including the ε-assembly in microsensing applications have been explored, as discussed herein. Various force loadings conditions that can produce noticeable deflections such as the concentrated force, moment and constant surface stress, which can be due to analyte adhesion, are considered. The linear elasticity theory for thin beams is used to obtain the deflections. Different deflection indicators are defined and various controlling variables are identified. The performance of different microcantilever assemblies is compared with the performance of rectangular microcantilevers in order to map out conditions that produce magnification of the sensing deflection relative to the noise deflection. 
         [0043]    To the extent necessary to understand or complete the disclosure herein, all publications, patents, and patent applications mentioned herein are expressly incorporated by reference therein to the same extent as though each were individually so incorporated. 
         [0044]    Having thus described exemplary embodiments of the present invention, it should be noted by those skilled in the art that the within disclosures are exemplary only and that various other alternatives, adaptations, and modifications may be made within the scope of the present invention. Accordingly, the present invention is not limited to the specific embodiments as illustrated herein, but is only limited by the following claims.