## Abstract

Design against adhesion in microelectromechanical devices is predicated on the ability to quantify this phenomenon in microsystems. Previous research related the work of adhesion for an adhered microbeam to the beam's unadhered length, and as such, interferometric techniques were developed to measure that length. We propose a new vibration-based technique that can be easily implemented with existing atomic force microscopy tools or similar metrology systems. To make such a technique feasible, we analysed a model of the adhered microbeam using the nonlinear beam theory put forth by Woinowsky–Krieger. We found a new relation between the work of adhesion and the unadhered length; this relation is more accurate than the one by Mastrangelo & Hsu (Mastrangelo & Hsu 1993 *J. Microelectromech. S.*, **2**, 44–55. (doi:10.1109/84.232594)) which is commonly used. Then, we derived a closed-form approximate relationship between the microbeam's natural frequency and its unadhered length. Results obtained from this analytical formulation are in good agreement with numerical results from three-dimensional nonlinear finite-element analysis.

## 1. Introduction

Microelectromechanical systems (MEMS) technology has the capability of connecting digital electronics to the physical world through sensing, actuation or other mechanical means. Current MEMS technologies under research include pressure transducers, accelerometers, microactuators [1,2], biomedical devices [3], optical components [4] and radio frequency (RF) MEMS switches [5]. However, a critical impediment to the full commercialization of MEMS devices is reliability.

Stiction, the unintentional adhesion of compliant microstructure surfaces [6], is notorious for causing serious reliability concerns [7]. Stiction occurs because surface forces (e.g. capillary, electrostatic and van der Waals) dominate at submicrometre scales [8,9]. When elastic restoring forces of structures are unable to overcome these strong adhesive forces, surfaces remain permanently adhered to each other and cause device failure [10] (figure 1).

Stiction may occur at two stages in the life of a MEMS device: fabrication and in-use. Stiction failure during the fabrication stage is usually caused by capillary forces during the release process. Thus, it can be avoided through methods such as dry etching [12], super critical drying [13] and freeze drying [7]. In-use stiction is more difficult to prevent. Straightforward methods for preventing in-use stiction include the stiffening of structures [14] and increasing the gap size between the devices and substrates [15]. However, these two methods may be undesirable for device performance [16]. Other solutions have been reported such as the use of bumps [17], electric force-induced vibration [18–20] and surface texturing [21]. The most promising results have come from research in anti-stiction [22] and self-assembled monolayer coatings [23]. Standardization of these techniques requires the ability to quantify stiction—measuring the work of adhesion *w*^{1} and studying its dependence on parameters such as surface morphology, hydrogen termination [27] and environmental conditions [28].

### (a) Measuring the work of adhesion using microbeam arrays

Previous work [29,30] has related *w* to the unadhered length *a* (figure 2*c*) of the microbeam. This length *a* is a characteristic of the cantilever's geometry, mechanical properties of the cantilever's material and *w* between the surfaces of the cantilever and substrate. Experiments could then be performed to measure *a* and calculate *w*.

Mastrangelo & Hsu [29] developed an interferometric method for finding *a* through an array of cantilevers of increasing length, similar to that shown in figure 1*c*. For this method, they fabricated an array of microbeams on a single chip through sacrificial etching with hydrofluoric acid. Microbeams of length greater than *a* would become adhered due to capillary forces during the drying process, while shorter microbeams would be unadhered. The shortest adhered microbeam with length closest to *a* could then be identified through a change in the interference pattern over the array. This technique has limitations that prevent its widespread application. First, it is required that a whole array of cantilevers be manufactured on a single chip such that the adherence is caused by capillary forces pulling the beams down to the substrate. The combinations of substrate and microbeam materials in such arrays are limited to what can be made on a single chip through such microfabrication techniques. If one desires to characterize adhesion between surface pairs that have been made of different materials, have undergone different surface treatments or have different small-scale geometric features, then multiple arrays must be manufactured. If one wants to calculate statistics on the measurement of *a*, then even more arrays must be made. The resolution of each measurement also depends on the difference in length between each adjacent cantilever, and choice of what range of lengths of cantilevers to manufacture presupposes some knowledge of what the unadhered length *a* could be (knowledge which is not always available since that is the value that the experiment is meant to find).

De Boer & Michalske [30] performed similar experiments but used long microbeams adhered over long attachment lengths. Instead of observing a change in the interference pattern over a whole array of microbeams, a linescan over the top surface of each single beam was performed to acquire the vertical displacements over the whole length of that beam. From plots of the vertical displacements of the beam versus the length of the beam, *a* could then be found. De Boer and Michalske's method provides improvements over that of Mastrangelo and Hsu. For example, issues of the resolution of the measurement depending on the difference in lengths of adjacent cantilevers are avoided. It is also not necessary to make a full array of cantilevers. However, it does require a full linescan of the microbeam such that multiple data points must be processed to find a single experimental value of *a*. This data processing must then be repeated for multiple microbeams if one wanted to calculate statistics on the measured value of *a*.

### (b) Vibration-based technique for measuring work of adhesion

Alternatively, we envision a vibration-based technique which we believe could give a highly accurate estimate of *a* from a single point measurement. The motivation for this idea is that vibration-based techniques of measurement are well established, are known to have high sensitivity and repeatability and are easy to use on a MEMS chip [31,32]. This technique would be implementable on an atomic force microscope (AFM) or related surface metrology tools that involve mechanical contact between a cantilevered structure and a surface. We illustrate how this technique would work in figure 2*a*–*d*. In an AFM system, a commercially available tipless cantilever [33] or one that is specially manufactured [34] can be brought into contact with a surface and then lifted a distance *g*. With the cantilever in its adhered configuration, its natural frequency can then be found by measuring its thermal fluctuations; measurement of the natural frequency through thermal fluctuations is already implemented as part of a well-defined calibration method for AFM cantilevers [35–39]. Besides AFMs, there exist other examples of cantilever-based mechanical measurement systems being used in research [40,41]. Thus, measuring the vibration of a microbeam is a feasible method to find *a*. However, as we detail below, no satisfactory formula connecting the fundamental natural frequency of the adhered microbeam *a* is currently available. Therefore, we derive such a formula in §3b.

### (c) Justification for deriving a new relation between the unadhered length and natural frequency of the adhered microbeam

Study of the vibrations of structures is a well-established subject. Closed-form expressions for the natural frequencies of a number of structural mechanics models can be found in standard textbooks on the subject. These structures include strings, bars, shafts and beams in one dimension and membranes, plates and shells in two dimensions [42]. Owing to the microbeam's high aspect ratio, several researchers have studied both the free and forced vibrations of microbeams using beam theories. For example, Tilmans *et al.* [43–45] studied the natural vibration of a free-standing MEMS microbeam using a modified Euler–Bernoulli beam theory. In the original version of the Euler–Bernoulli theory, the structure only transmits bending moments and shear forces along its length, whereas in the modified theory used by Tilmans *et al.*, the structure additionally transmits a constant tensile force. Ghayesha *et al.* [46–51] and Farokhi *et al.* [52,53] studied the nonlinear dynamics of microbeams by considering the size effect. They obtained size-dependent frequency–response curves of both Euler–Bernoulli beams and Timoshenko beams through Galerkin and pseudo-arclength continuation techniques. Zhang & Zhao [54] studied the forced vibration of an adhered MEMS microbeam. The forcing was applied through a time varying voltage between the microbeam and substrate, and the adhered beam was modelled using a nonlinear beam theory. In addition to the bending energy, the model included two additional terms in the elastic potential energy of the beam that the authors refer to as the ‘stretching energy’ terms. They used Galerkin and Newton–Raphson numerical methods to solve the governing equations of their model. However, they provided neither a closed-form expression for the adhered microbeam's fundamental natural frequency nor any theoretical analysis on the frequency's dependence on the problem parameters. Such a closed-form expression is critical for determining *a* from

Therefore, we derive a closed-form expression relating *a* to

*Outline of paper*. In §2, we review previous theory by Mastrangelo & Hsu [29,26] relating *a* to *w* of the microbeam. The full derivation of our formula connecting *a* to

## 2. Previous work connecting the unadhered length to the work of adhesion

Mastrangelo & Hsu [26,29,56] previously studied the adhered shapes of microbeams whose geometries are shown in figure 3.

### (a) Cantilevered microbeams

The microbeam shown in figure 3*a* is a cantilevered beam, as in, one of its ends is fixed while the other is free. This is the geometry of micromachined AFM probes (figure 2) and those of other cantilever-based metrology systems. It is therefore relevant to our proposed vibration-based method for measuring *w*. Figure 3 also shows the vectors *i*=1,2,3, which form an orthonormal set of Cartesian basis vectors. The origin of the coordinate system, marked *a*) as the reference configuration ^{2} and occupies the cuboidal region [0,*L*]×[−*H*/2,*H*/2]×[−*W*/2,*W*/2]. That is, it is straight with length *L* and has a rectangular cross section of width *W* and height *H* that is perpendicular to the *g* above it.

Mastrangelo & Hu [26,29,56] and other researchers [30,58,59] have previously studied the mechanics of adhered microbeams using a configurational force balance approach. This approach was pioneered by Griffith [60]. The techniques of configurational force balance have since been greatly expanded [61] and have been applied to problems such as the adhesion of thin films, the peeling of lap joints and double torsion tests [62]. Per this perspective, a configuration is considered to be locally stable (metastable) if and only if infinitesimal perturbations around that configuration lead to an increase in the system's potential energy *Π*. For the adhered microbeam, this requirement implies that
*A*_{a} is the magnitude of the area over which the microbeam and the substrate are in contact. Generally, it is assumed that the contact region formed between the cantilever and the substrate is simply connected and its delamination front is straight and parallel to the *a* is the unadhered length of the microbeam (figure 3*c*).

For the adhered microbeam, the total potential energy *Π* consists of two terms: the adhesion energy ^{3} is generally taken to be
*E* is the Young's modulus, *I* is the second moment of area of the microbeam's cross section, *X*_{1} is the Cartesian coordinate corresponding to the *c*). By substituting (2.5) into (2.4), we find *Π* by summing the expressions for *w* and *a* to be

### (b) Fixed–fixed microbeams

Mastrangelo & Hsu [26] also studied the adhered microbeam in the fixed–fixed configuration shown in figure 3*b*. These types of microbeams are found in RF MEMS capacitive switches (figure 1*a*). This geometry is not relevant to our proposed experimental method. Nonetheless, we still discuss it because our results apply to it.

In its reference configuration, the cantilever microbeam that we use for our derivations is equivalent to the reference configurations of both the left and right halves of the fixed–fixed microbeam. Following Mastrangelo & Hsu [26], we assume that the fixed–fixed beam is symmetric about its midsection even in its adhered configuration. Owing to this assumpton and the manner in which our cantilever microbeam comes into contact with the substrate (figure 2*a*–*d*), even when adhered, the cantilever microbeam we study is equivalent to both the left and right halves of the fixed–fixed microbeam.

Mastrangelo and Hsu analysed the fixed–fixed microbeam using a nonlinear beam theory. Per that theory, the elastic potential energy of half of the fixed–fixed microbeam is^{2}
*Π* by summing *w* and *a* are related by^{4}
*w* to *a*.

## 3. Nonlinear model for the adhered microbeam

We model the adhered microbeam using Woinowsky–Krieger beam theory [55], which is a geometrically nonlinear beam theory. We derive the equations governing its motion using Lagrangian mechanics. The potential and kinetic energies of the microbeam as per Woinowsky–Krieger theory are
*U*_{1}(*X*_{1},*t*) and *U*_{2}(*X*_{1},*t*) are the displacements of the material point *X*_{1} on the beam's centroidal axis at time *t* in the *A* is the area of the beam's cross section and *ρ* is the density of the material of the beam.

We introduce the following non-dimensional variables: *ξ*:=*X*_{1}/*a*, *ζ*:=*U*_{1}/*H*, *η*:=*U*_{2}/*H*, *τ*:=*tω*_{0}, ^{5}

It is challenging to derive a general, closed-form solution to the nonlinear partial differential equations (PDEs) (3.3a)–(3.3b). However, recall that we do not need to know the general dynamical behaviour of the adhered microbeam. We are only interested in the vibratory motion that the microbeam may execute about a static, adhered configuration (figure 4), which is relevant within the context of our proposed experimental method. Therefore, we attempt to solve (3.3a)–(3.3b) approximately by making the ansatz that the vibratory solution that we seek admits the asymptotic expansion
*ζ*_{0}(*ξ*) and *η*_{0}(*ξ*) describe the static shape of the adhered microbeam assumed in the absence of any dynamical motion, and *ε* is the non-dimensional amplitude of the microbeam's vibratory motion, and *O*(*ε*^{3}) in (3.5b) and (3.5a) denotes all terms in the solution that vanish at a rate that is faster than or equal to *ε*^{3} as *ε*→0. Since vibratory motion is by definition of infinitesimal magnitude, we limit our analysis to the special case of *ε*→0.

Substituting the asymptotic forms (3.5a)–(3.5b) into the nonlinear PDEs (3.3a)–(3.3b) and the boundary conditions (3.4a)–(3.4b) and then integrating the resulting equations with respect to *τ* for 0 to *ζ*_{0} and *η*_{0} satisfy the nonlinear ordinary differential equations
*ϵ*_{0} is related to *ϵ*_{0} as a function of *ϵ*_{0} can be ascertained from figure 5, in which the approximate values of *ϵ*_{0} given by (3.10) are compared with the exact values of *ϵ*_{0} that we obtained by numerically solving (3.9).

### (a) A new relation between the unadhered length and the work of adhesion

Substituting *ζ*_{0} and *η*_{0} given by (3.8a)–(3.8b) into (3.5a)–(3.5b) and then substituting the resulting asymptotic expansions for *ζ* and *η* into (3.2a), we get

The dimensional, elastic potential energy of the adhered microbeam, *H*^{5}*WE*/(24*a*^{3}). Combining the thus obtained *Π*, substituting that result into the configurational force balance equation (2.2), then taking the limit *ε*→0, we get
*a*, an approximate value for *w* can be calculated using (3.13).

### (b) Relating the fundamental, natural frequency of an adhered microbeam to its unadhered length

We solve for

Substituting the functions *ζ*_{0} and *η*_{0} given by (3.8a)–(3.8b) into (3.5a)–(3.5b), substituting the resulting asymptotic expansions into (3.2a)–(3.2b), evaluating the integrals in the resulting equations and simplifying, we get the maximum changes in the non-dimensional potential and kinetic energies to be
*ε*^{2}, and taking the limit *ε*→0, we get that

Equations governing *ζ*_{1} and *η*_{1} can be derived using a procedure similar to that employed for deriving the governing equations (3.6a)–(3.6b) for *ζ*_{0} and *η*_{0}. However, we were unable to solve those equations analytically. Consequently, we derive an approximate expression for *ω*_{B} by making a reasonable choice for *ζ*_{1} and *η*_{1} in (3.15). This step is similar to the process of choosing an approximate mode shape in Rayleigh's energy method.

Considering the boundary conditions (3.4a)–(3.4b), a reasonable choice for *η*_{1} is the fundamental mode shape of a straight fixed–fixed beam, which can be described as
*b*, approximately equal to 4.730, is the first non-trivial root of
*ζ*_{1} and *η*_{1} are only approximate, i.e. they do not exactly satisfy (3.3a)–(3.3b), then the corresponding estimate for *ζ*_{1} to make the numerator of the expression on the right-hand side of (3.15) as small as possible and denominator as large as possible. In the light of this knowledge, a good choice for *ζ*_{1} is
*η*_{0} and *η*_{1} are, respectively, given by (3.8b) and (3.16). This is because, for this choice of *ζ*_{1}, the second term in the numerator of (3.15) vanishes. Also, the expression for *ζ*_{1} given by (3.17) satisfies the essential boundary conditions stipulated by (3.4a)–(3.4b).

Substituting the approximate *ζ*_{1} and *η*_{1} given by (3.16) and (3.17) into (3.15) and simplifying we get that
*c*_{0}≈0.22. Again, the exact relation between *ϵ*_{0} and

## 4. Discussion

### (a) Comparison of the static elastic potential energy given by (eqn3.12) with numerical results

In figure 6, we compare a numerically computed

We numerically computed the static elastic potential energy *a*. We assumed hyperelastic material behaviour. Specifically, we assumed a compressible, neo-Hookean material model in which
** S** is the second Piola–Kirchhoff stress tensor,

**is the right Cauchy–Green deformation tensor,**

*C**J*is the Jacobian determinant,

**is the identity tensor, the parameters**

*I**λ*

_{0}and

*μ*

_{0}are the Lamé constants and (⋅)

^{−1}is the inverse operator.

The right Cauchy–Green deformation tensor and the Jacobian determinant are defined as
^{T} is the transpose operator. The static, adhered configuration of the microbeam was obtained by solving the Cauchy momentum equation
*U*_{1},*U*_{2},*U*_{3})=(0,0,0), while that on the right face were everywhere fixed to be (*U*_{1},*U*_{2},*U*_{3})=(0,−*g*,0).

The governing equations (4.1)–(4.5) were discretized using standard finite-element procedures to obtain a system of nonlinear algebraic equations [65]. We used eight-node linear brick elements in the finite-element mesh. The system of nonlinear algebraic equations were solved using the Newton–Raphson iterative procedure.

From the numerical solution, the static, elastic potential energy, *Ω* is an infinitesimal volume element belonging to

### (b) Comparison of the fundamental natural frequency ω B given by (eqn3.18) with numerical results

We also compared the value of *b* shows that the difference between the analytical and numerical calculation for beams with width *ν*=0.22 is less than 1% when

Let

This vibratory motion leads to oscillations of the form ** F**, such that

**∥ denotes the norm of Δ**

*F***and the symbol**

*F**o*(∥Δ

**∥) denotes all terms which vanish at a rate that is faster than ∥Δ**

*F***∥ as ∥Δ**

*F***∥→0. The amplitude of the second Piola-Kirchhoff stress change during vibratory motion**

*F***as**

*U*For the constitutive law (4.1),
** U** satisfies the equation

*ρ*is the density of the material composing the microbeam. The boundary conditions on

**stipulate that as Δ**

*U***vanish on both ends of the microbeam.**

*U*Equation (4.9) is a linear partial differential equation in Δ** U**. We discretized (4.9) using standard, finite-element procedures to get a linear, matrix–vector equation. However, in that matrix–vector equation,

^{4}(for

^{5}(for

### (c) Asymptotic behaviour of the *w*–*a* equation (eqn3.13)

As *w*–*a* relation (3.13), which we derived using Woinowsky–Krieger theory, to reduce to the *w*–*a* relation (2.7), which was derived by Mastrangelo & Hsu [29] using Euler–Bernoulli theory. We find that this is in fact the case. For example, if we expand *w*–*a* relation (3.13), the relation attains the aymptotic form
*w*–*a* relation (2.7) reads as
*w*–*a* relation (3.13) and the *w*–*a* relation (2.7) are the same up to

Interestingly, the *w*–*a* relation (2.9) derived by Mastrengelo & Hus [26] using a nonlinear beam theory does not match the Euler–Bernoulli *w*–*a* relation (2.7) in the limit *w*–*a* relation (2.9) in terms of non-dimensional variables and expanding the right-hand side in powers of *w*–*a* relation (2.9) were to match the *w*–*a* relation (2.7) exactly in the limit *w*–*a* relation are consistent with Euler–Bernoulli theory in the limit

## 5. Conclusion

We believe that the flexibility of a vibration-based method allows it to be applied to a wider variety of problems. Beyond the reliability of MEMS, the topic of adhesion at submicrometre scales is important in its own right. For example, some of the unique capabilities of biological materials, such as insect wings [66] and the adhesive toe pads of geckos [67], are thought to arise through adhesion at small scales. In addition, the adhesion between solids is generally measured using axisymmetric, contact mechanics-based methods [68]. However, surface roughness is known to cause considerable difficulties in unambiguously measuring *w* using such methods [69–71]. Therefore, it would be interesting to see how competitive the proposed vibration-based method for measuring *w* would be in comparison to the contact mechanics-based methods.

Implicit in our and previous models of the adhered microbeam is the assumption that the interbody adhesion forces are infinitesimally short ranged. This is similar to what is assumed in, for example, the Johnson–Kendall–Roberts (JKR) adhesive contact model [72]. Equation (2.3), which states that *Π*_{s}=−*w*(*L*−*a*)*W*, is a consequence of this assumption. However, studies have shown that adhesive forces (which, at submicrometre scales, are primarily due to van der Waals interactions [73]) can act over long distances [74] and have been measured to act over distances as large as a micrometre [75]. Therefore, the forces on the adhered microbeam can act over its full length and are spatially non-uniform. It remains to be seen how important an effect such non-uniformity creates and if the assumption of the interbody adhesion forces being infinitesimally short ranged is an acceptable approximation. We plan to explore this effect in future experiments.

## Data accessibility

This work does not have any experimental data.

## Author contributions

W.F., J.M. and H.K. carried out the theory portion of the research. W.F. and H.K. carried out the computational part of the research. H.K. designed the research. W.F., J.M. and H.K. wrote the paper. All authors discussed the results and gave final approval of the manuscript.

## Competing interests

We declare that we have no competing interests.

## Funding statement

This work was supported by the National Science Foundation through the Mechanics of Materials and Structures Program award number 1562656. J.M. was supported by a Presidential Fellowship from Brown University.

## Acknowledgements

We acknowledge Prof. Rudra Pratap of the Indian Institute of Science, Bangalore for introducing us to the fascinating problem of adhesion in MEMS devices. We also thank Prof. Suhas Mohite, Government College of Engineering, Karad, India, for some initial discussions on this problem.

## Appendix A. Derivation of (3.10)

We arrive at (3.10) through approximating the adhered beam's static displacements using the solution given by Mastrangelo & Hsu [29], which is based on Euler–Bernoulli beam theory. Specifically, using (2.5) and the fact that longitudinal displacements are assumed to be of negligible magnitude in the Euler–Bernoulli theory, we approximate *ζ*_{0} and *η*_{0} as
*ζ*_{0} and *η*_{0} to approximate *ϵ*_{0}. To obtain a better approximation for the dependence of *ϵ*_{0} on *ζ*_{0} and *η*_{0} by solving equations (3.6a)–(3.7b) under the assumption that *ζ*_{0} and *η*_{0} into (A 3) to finally arrive at (3.10).

## Footnotes

↵1 The work of adhesion

*w*is defined as*w*=*γ*_{1}+*γ*_{2}−*γ*_{12}, where*γ*_{1}and*γ*_{2}are the energies needed to create new surfaces of each respective material and*γ*_{12}is the energy required to create a unit area of interface between the two surfaces [24]. In the case where the two materials are the same,*γ*_{12}is zero and the*w*reduces to*w*=2*γ*_{1}=2*γ*_{2}. Other notations may simply refer to this case as*w*=2*γ*with*w*also called the interfacial surface energy [25]. In our formulation, we refer to*w*in the general sense, whereas in the work of Mastrangelo & Hsu [26], the materials of the beam and substrate are assumed to be the same so that*w*=2*γ*.↵2 In [56], Mastrangelo and Hsu give the elastic potential energy of the fixed–fixed microbeam to be (512

*g*^{2}*EI*/5(2*a*)^{3})[1+*T*(2*a*)^{2}/42*EI*+(256/735)(*g*/*H*)^{2}(*a*/*L*)], where*T*=*WHσ*_{R}is the axial residual tensile force and*σ*_{R}is the internal residual tensile stress. Such residual stresses generally arise as a consequence of the microfabrication processes used for manufacturing the microbeams [57]. However, such stresses are likely to be absent in the AFM microcantilevers that will be employed in our proposed, new experimental method (figure 2*a*–*d*). Thus, we ignore residual stresses in our current work. The expression for the elastic potential energy given in (2.8) was obtained by putting*σ*_{R}=0 in the expression given by Mastrangelo & Hsu in [56,26]. Also, there is a difference of a factor of 1/2 between the two expressions. This is because the expression in (2.8) corresponds to only one half of the symmetric, fixed–fixed microbeam, whereas that given by Mastrangelo & Hsu [56,26] corresponds to the full beam.↵3 In this case, Mastrangelo and Hsu [29] followed the assumption that the two surfaces, substrate and beam, were of the same material; then,

*w*is equivalent to 2*γ*.↵4 Mastrangelo & Hsu [26] give the relation between

*w*and*a*as*w*=(8*Eg*^{2}*H*^{3}/5*a*^{4})[1+(256/2205)(*g*/*H*)^{2}]. We believe that the numerical factor 256/2205 in this equation is an error. Based on the expressions for the adhesion and elastic potential energies given in (2.3) and (2.8), respectively, the numerical factor should instead be 512/2205 (cf. (2.9)).↵5 Owing to the manner in which the microbeam is brought into contact with the substrate (figure 2

*c*–*d*) there are no displacements in the microbeam until it comes into contact with the substrate. After the microbeam and the substrate make contact, we assume that there is no slippage, i.e.*ζ*=0, in the adhered portion of the microbeam as its base moves to a height*g*above the substrate (figure 2*b*–*d*). Recall that once the microbeam's base reaches a height*g*, it is held fixed at that height. We assume that after the microbeam's base reaches the height*g*, the unadhered length*a*also remains fixed irrespective of any dynamical behaviour that the microbeam may display. Finally, we assume that there is no slippage in the adhered portion of the microbeam after the microbeam's base reaches the height*g*. The boundary condition*ζ*=0 at*ξ*=1 is a consequence of these assumptions.

- Received September 5, 2017.
- Accepted February 5, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.