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Micro cantilever pre-stress

Dear all,

Im a PhD student in Cambridge Uni, UK working in the field of MEMS, and as part of my work, of late Ive been looking at deriving materials properties of MEMS thin film materials by means of resonant testing. The basic outline of the experiment is first creating free standing rectangular cantilevers of the material under test, evaporate with gold to increase reflectivity (when needed), then (under reasonable vacuum) applying a base excitation using a chirp signal into a piezo actuator and logging the cantilever tip response using a laser/photodetector setup. The frequency response is then calculated and the modal frequencies noted.

To determine the materials' properties, both an analytical model (with bending/torsion modes) and finite element model (using 2D mindlin) are created with similar geometry as the sample, and by minimising the squared relative error between the measured modes and those from the models, the value of Young's modulus (known density) and poissons ratio may be determined iteratively. These yield fairly consistent results.

To take the work further I now feel I should also include the effects of residual stress in the cantilevers. The method Ive been looking at is by using finite element (via COMSOL) - the beam geometry is created and loaded with the stress model ('surface stress' as a force tangential to the top boundary, and gradient stress as a 'tangential' force that varies from +F to -F from top to bottom boundary of the cantilever). The model is solved statically, and the deformed shape is then saved as the linearisation point for the next model, which then computes the eigenfrequencies. Btw I can only do this in 3D FE, which makes computation times quite long hence using iteration to quantify this stress highly unlikely.

In any case, is there an analytical model I can use to model the effect of this stress on multiple modes of a cantilever. Id like to verify whether the FE is giving me anything close to ball park numbers before I work out a means to compare them with experimental results. I was thinking of using the Rayleigh method by representing the effect of prestress as an additional term in the potential energy. The original mode shapes, with some modification will be used to evaluate the two energy integrals. The potential energy due to stress is worked out by measuring the static deflected shape using a zygo inteferometer - some rough model is used, with the beam curvature and peak deflection as input to work out the amount of this energy. Not having much experience in mechanics (i was an electronics undergrad!), Im not sure how good an estimate this would be, if at all its a useable or even possible one. Will the extra energy factor in to the torsion modes just the same?

What with the recent discussion on stress in microcantilevers for biosensing, I thought I should somewhat have a clear idea as to what is out there in literature before I jump into the problem! Please feel free to suggest any references that may be of help. 

All in all I hope someone can give me pointers as to whether or not Im approaching the problem the right way. Whatever it is, Id appreciate any comments you may have. Many thanks for reading, regards, Iskandar

Wei Hong's picture

You might want to take a look at the discussion and related literature in this blog post:

Microcantilever for biomolecular detections

If both the substrate and film are linear elastic (within the range), a constant residual stress will not change the resonant frequencies of the cantilever.

More importantly, a 'surface stress' is a set of self-balanced internal force, or a residual stress just as you mentioned, rather than "a force tangential to the top boundary".

As far as I know, the residual stress due to mismatch of mechanical properties for each layer of cantilevers (e.g. piezoelectric layer, Si layer, etc) may not affect the resonance behavior of a cantilever.

For biomolecular detection, until recently, several researchers (e.g. Thundat et al., Mohanty, et al.) suggested the role of surface stress on the resonance behavior of a cantilever (Ref: Cherian GY, Thundat T, Wachter EA, Warmack RJ, J. Appl. Phys. 77, 3618, 1995; Dorignac J, Kalinowski A, Erramilli S, Mohanty P, Phy. Rev. Lett. 96, 186105, 2006). However, with use of surface stress, it might be not a good approach to understand the resonance behavior of biomolecular behavior (e.g. protein-protein binding). In recent days, I am more focusing on the molecular model for proteins and/or DNA as well as continuum model for a cantilever, for insight into resonant frequency shift as a function of molecular interactions (without using any surface stress!).

What I have felt these days is that the multiscale modeling on a topic of "microcantilever for biomolecular detection" may be much better approach than a continuum model, by using a potential field for biomolecular interaction. Specifically, when the potential field for a nano-bio-system (e.g. cantilever functionalized by biomolecules) is given, then it is easy to compute the stiffness of a system (that may allow us to calculate the resonant frequency of a system).  When I have some result, I will try to share my further results on this topic as well as some discussions at iMechanica.

Wei, thank you for your comments that helped me to pursue the further refined (multiscale) model on my topic. Thanks.


Zhigang Suo's picture

Dear Wei:  I mentioned the discussions on Kilho Eom's paper to Jim Rice the other day, and he showed me how residual stress can affect vibration frequency and wave speed in the setting of finite deformation. The effect can be pronounced in slender bodies such as beams even when the stress is small, as noted in a previous comment.

I summarized the general formulation and the example of beams in my lecture notes on finite deformation. Once you've downloaded the Word file, look for the section titled "Infinitesimal, inhomogeneous deformation superimposed on a homogeneous field". I have not thought through how these ideas are related to surface stress. Would you be willing to take a look and let us know how you think about the problem? Thank you.

Pradeep Sharma's picture

Certainly, inclusion of nonlinear elasticity should cause residual stresses to play a role but can we realistically expect this effect to be appreciable? As a an interesting side note, Professor ZP Huang recently co-authored a paper where he showed that (provide nonlinear elasticity is used), residual surface stress will impact the overall effective elastic modulus of a composite or porous material (obviously as we discussed in previous post, linearized elasticity cannot show this). Reference to the paper is here.

Hi all, and thanks for the replies so far.

Actually I read the posts regarding the (bio) micro cantilevers prior to posting my entry. One thing that caught my attention the most was how the relative shift for all modes was found to be the same for a given amount of prestress/surface stress. This is counter intuitive when considering the Rayleigh method (well at least my understanding of it), the residual strain in the cantilever as a result of the residual stress manifests itself by producing a curvature/deflection. This strained geometry has a potential energy associated with it and remains constant in the material when in harmonic motion. And thus the original Rayleigh relationship differs from the original E-B frequencies only by adding the extra 'static' potential energy due to the aforementioned residual strain.

From here, its simple to see that the mode frequencies will not vary by the same relative amount, but instead as a function of kinetic energy of that particular mode.

I noticed very few people deal with multi mode vibration of cantilevers, but for my purposes I am considering about 10 each time, 2-3 torsional, the rest bending. Btw my findings with finite element somewhat agree with the Rayleigh method in that the resonant frequency shits are large for lower modes, and vice versa.

 For me such a result would be helpful in estimating the gradient residual stress in films as well as other mechanical properties - where I am most interested in Young's Mod, poisson's ratio measurements.

Thanks again, regards, Iskandar

Here are a few simple examples how these can occur.

Nat. freqs. of a finite structure are a function of elastic properties and geometry. Internal/residual stress change geometry, therefore they can change nat. freqs.

This can easily be demonstrated with simple spring-mass systems. Imagine any symmetric 2D system and an internal force breaking its symmetry.

The speed of propagation of out-of-plane displacement in a taut string is a function of the tension in the string. 

These are all related to Prof. Suo's finite deformation argument.

Wenbin Yu's picture

If you model the microcantilever as a composite beam, the GEBT (namely geometrically exact beam theory) plus VABS should be able to get you all the natural frequencies needed for finite deformation.

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