78 Chapter 2 where h is the distance measured about the z-direction between the microcantilever and the xy detection plane. Figure 2.9 indicates the in-plane position of the attached mass. The torsion moment produced by the added mass is: such that the tip torsion angle is: Figure 2.8 Experimental system for in-plane mass detection Figure 2.9 In-plane position of added mass 2. Microcantilevers, microhinges, microbridges 79 By coupling Eqs. (2.36) through (2.40) it is possible to determine the amount of deposited mass as well as its in-plane position defined by and Example 2.4 Determine the added mass as well as its position and on a variable cross-section microcantilever when the tip slopes and and the deflection are known. Solution: When the cross-section of the microcantilever is variable, the tip slope (which is given in Eq. (2.35) for a constant cross-section) can be found by means of Castigliano’s displacement theorem as: where: is the compliance of the microcantilever portion which is comprised between the point of application of the deposited mass and the fixed root. In a similar manner, the tip deflection can be expressed as: where the new partial compliance is: The torsion-related tip slope is expressed as: with the partial torsional compliance being: 80 Chapter 2 Equations (2.41), (2.43) and (2.45) can be solved for the unknown amounts and by also employing Eqs. (2.42), (2.44) and (2.46). 2.2.2.2 In atomic force microscopy (AFM) reading applications, a microcantilever such as the one sketched in Fig. 2.10 is designed to determine the amounts of force that are applied on its tip about the x-, y- and z-directions, and thus describe (read) a three-dimensional variable geometry. The experimental setup pictured in Fig. 2.8 can again be utilized to evaluate the tip rotations and Figure 2.10 Tip forces detected by a microcantilever in atomic force microscopy Example 2.5 Evaluate the forces and that act on the tip of the constant rectangular cross-section microcantilever of Fig. 2.10 by using the available experimental data. Solution: It can simply be shown that the tip slope which is produced through bending by the combined action of and as well as the slope which is generated by torsion due to can be calculated as: If the only experimental information consisted of the tip slopes and the bending-produced deformation can globally be interpreted as generated by an apparent tip force, such that: Three Dimensional Force Detection in Atomic Force Microscopy Applications 2. Microcantilevers, microhinges, microbridges 81 By equating the first Eq. (2.47) to Eq. (2.48) results in: which indicates that by approximating the force according to this equation will always result in either an overestimation, when acts in unison with or in an underestimation, when is directed in the opposite direction. Assuming that the force can be expressed as a fraction of the real namely: the apparent force of Eq. (2.49) can be written as: The relative error between the force calculated by Eq. (2.51) and the real force is: and Fig. 2.11 is a plot giving the relative errors for the case where c = 0.5. Figure 2.11 Errors between the real force and the apparent one in AFM measurements 82 Chapter 2 In order to render this problem determinate, another set of experimental measurements, for instance the deflection at a point placed at a distance from the free end, is needed. In this case, the deflection at the experimental point of detection is: Equations (2.47) and (2.53), together with Eqs. (2.38), enable solving for and from experimental measurements. All these calculations are valid for and have been applied thus far to a constant cross-section microcantilever. The following example will solve this problem for the cases where the cross-section of the microcantilever is variable. Example 2.6 Determine the force components and that are applied at a microcantilever’s tip when contacting a three-dimensional surface. The tip slopes and tip deflection are experimentally available. Assume that the microcantilever’s cross-section is variable and neglect the axial deformation. Solution: The Castigliano’s displacement theorem is utilized again to express the tip displacements. The torsion, for instance, is produced by the force which is offset by the quantity h and the corresponding tip slope is given by the equation: where the torsional compliance has been defined in Eq. (2.12). The tip slope in bending is produced by the combined action of the forces and and is: whereas the tip deflection is: Equations (2.54), (2.55) and (2.56) enable finding the tip force components when and are available experimentally. 2. Microcantilevers, microhinges, microbridges 83 2.2.3 Solid Trapezoid Design The geometry of a trapezoid microcantilever is sketched in Fig. 2.12. Figure 2.12 Geometry of solid trapezoid microcantilever The variable width depends linearly on the abscissa x, namely: By utilizing this equation in the generic compliance formulation, the stiffnesses of a solid trapezoidal microcantilever can be computed by inverting the compliance submatrices or terms. The bending-related stiffnesses corresponding to the sensitive axis are: The stiffnesses that describe bending about the other bending axis (the z- axis) are: 84 Chapter 2 The axial stiffness and torsional stiffness are: For the particular condition where the trapezoid becomes a rectangle, and consequently by taking the limit in Eqs. (2.58) through (2.65), the stiffness equations for a constant, rectangular cross- section cantilever – Eqs. (2.27), (2.28), (2.29) and (2.30) – should be retrieved, which indeed occurs, as it can easily e checked. Example 2.7 Compare a constant cross-section, rectangular microcantilever to one of trapezoid configuration, which has the same length and by analyzing the and stiffnesses. Consider that and c spans the [1, 5] range. Also consider that the microcantilever’s cross-section is very thin (t << w). Solution: The following stiffness ratios can be formulated: where the * superscript indicates the constant rectangular cross-section microcantilever. By utilizing Eqs. (2.27), (2.30), (2.58) and (2.65), together with the given width relationship, Eqs. (2.66) transform into: 2. Microcantilevers, microhinges, microbridges 85 which are functions of only one variable, the parameter c. Figure 2.13 shows the variation of these stiffness ratios, and it can be seen that, as expected, the trapezoid design becomes stiffer in both bending about the sensitive axis and torsion as the ratio of the maximum width to the minimum width increases. Figure 2.13 Stiffness comparison between constant rectangular and trapezoid microcantilevers: (a) bending about the sensitive axis; (b) torsion Example 2.8 Determine the deformations at the tip of a trapezoid microcantilever that is acted upon by the forces and as indicated in Fig. 2.10. Consider that the member is constructed of a material with E = 160 GPa and and that its geometry is defined by: Solution: The displacements that are related to y-axis bending can be expressed in terms of compliances as: It has been shown that the compliance matrix corresponding to bending about the y-axis is the inverse of the related stiffness matrix, which is given in Eq. (2.16). The terms entering the stiffness matrix of Eq. (2.16) are calculated by means of Eqs. (2.58), (2.59) and (2.60). They have the following values: It follows that the stiffness matrix of Eq.(2.16) is: 86 Chapter 2 By inverting Eq. (2.69) it is found that the related compliance matrix is: and therefore and such that, according to Eqs. (2.68), and By applying a similar procedure for the bending about the z-axis, which is produced by the force the corresponding tip displacements are: and It can be seen that although the force is 10 times larger than the force the displacements produced by are approximately one order of magnitude smaller than those generated by 2.2.4 Filleted Microcantilevers 2.2.4.1 Circularly-Filleted Design Microcantilevers that are filleted at their root by means of two circular portions are customary designs, particularly in mass detection applications. The circularly-filleted area is a way of reducing the stress concentrations, but sometimes is a technological by-product, as sharp corners at a microcantilever’s root are difficult to obtain through certain microfabrication procedures. However, when the fillet radius is small compared to the length and width, the fillet area is usually neglected in analytical calculations. On occasion, the fillet radius can be relatively large, as a means of increasing the root area, and therefore increasing the torsional stiffness for instance. In such situations, neglecting the contribution of the fillet zone to the various stiffnesses defining the microcantilever would amount to unacceptable error levels. Closed form compliance equations will be provided here (as also given in Lobontiu and Garcia [8], where a more generic model has been proposed) for two filleted designs, namely: one with circular areas, and the other with elliptical areas. A circularly-filleted microcantilever is shown in Fig. 2.14, together with the defining geometry. The filleted area extends over the entire length of the microcantilever and the length is equal to the radius of the circular fillet. The circular fillet is tangent to both the horizontal and vertical lines that meet at the root, and therefore this particular configuration is called right circularly- filleted microcantilever. The variable width of this configuration is: 2. Microcantilevers, microhinges, microbridges 87 The eight compliances that characterize the elastic behavior of this design can be calculated according to their definitions, as defined in Eqs. (2.8) through (2.12), by using the variable width of Eq. (2.71). The bending about the sensitive axis (the y-axis, which is contained in the plane of the figure) is defined by the following compliances, which are calculated based on their definition of Eqs. (2.8) through (2.10): Figure 2.14 Geometry of a right circularly-filleted microcantilever The corresponding stiffnesses, and can be determined through inversion of the compliance matrix, which comprises, according to Eq. (2.6), the individual compliances of Eqs. (2.72), (2.73) and (2.74). The compliances that are connected to bending about the z-axis can similarly be calculated and they are: [...]... cantilever of length 1 The two segments, 1-2 and 4- 5 , are actually combined in parallel by means of the rigid coupler 2 -4 , and therefore the compliances of the parallel combination are always half the compliances of one single member The moment produces a torsion angle, according to: where the torsion stiffness is: The in-plane compliances can be determined by applying two forces, and and a moment (none of. .. motion of this device is an out -of- the-plane bending about an axis contained in the plane of the microflexure Figure 2.23 Double-symmetry circular corner-filleted microflexure The microhinges are generally slender portions (notches) that can sustain axial and shearing deformations in addition to bending and torsion A microhinge is modeled as a fixed-free member, exactly as the microcantilever was, and. .. about the z-axis can be determined by inverting the compliance submatrix, Eq (2.6), containing the terms of Eqs (2.75), (2.76) and (2.77) The axial compliance is: and the torsional compliance for a very thin cross-section (t . compliances can be separated into two subcategories, namely out- of- the-plane and in-plane. The force and moments and generate deformations that are out of the xy plane, as sketched in Fig. 2.18. Application. elliptically-filleted one when and is the friction force between the microcantilever tip and the investigated three- dimensional surface. The microcantilevers have the same length thickness width w = 1 /4 and. where c = 0.5. Figure 2.11 Errors between the real force and the apparent one in AFM measurements 82 Chapter 2 In order to render this problem determinate, another set of experimental measurements,