A Comprehensive Model for Stiffness Coefficients in V-Shaped Cantilevers

Transcription

1 Int. J. Nanosci. Nanotechnol., Vol., No., March. 06, pp A Comprehensive Moel for Stiffness Coefficients in V-Shape Cantilevers A. H. Korayem *, A. K. Hoshiar, S. Barlou, an M. H. Korayem. Rootic Research Laoratory, Center of Excellence in Experimental Soli Mechanics an Dynamics, School of Mechanical Engineering, Iran University of Science an Technology, Narmak, Tehran, I. R. Iran. Faculty of Inustrial an Mechanical Engineering, Islamic Aza University, Qazvin Branch, Qazvin, I. R. Iran (* Corresponing author: hkorayem@iust.ac.ir (Receive: 0 Nov 04 an Accepte: 0 Jan. 06 Astract During past ecae the AFM ase nanomanipulation has een focus of attention as the promising nano farication approach. The main challenge in this process is the real-time monitoring. Consequently, the ynamic moels have een propose as a solution to the existing challenge. In the moeling approach the magnitues of the forces are proportional to the stiffness coefficients of cantilevers. The precise calculation of these coefficients has een introuce in numerous works. The propose stiffness coefficients for the V-shape cantilevers fail to present in all commercial cantilever geometry. The geometrical eviation has a consierale impact on the magnitue of stiffness coefficients. Therefore, in this paper the existing moel has een moifie to inclue the commercial cantilever an take into account the effect of geometry variation. FEM simulation has een use to investigate the effect of geometry change an the results of these simulations have een exerte to the moel which resulte in propose comprehensive moel. The propose new stiffness moel covers a wie range of commercial V- shape cantilevers an makes the process more practical. Keywors: Atomic force microscope; V-shape cantilever; stiffness coefficients.. INRODUCTION Atomic Force Microscope (AFM is one of the primary instruments for investigation of surface topographies; however, it can also act as a nanomanipulator to push nano particles an manufacture nano structure; thus, it is very important to know the ynamics of the AFM in orer to achieve a reliale an efficient manipulation process []. In the controlle manipulation of nanoparticles, tip of cantilever pushes the particle till it reaches the esire location. Throughout the process of manipulation y the AFM, the manipulation forces are measure ase on the eformation of the cantilever. Therefore, to have a successful process, it is vital to measure spring constants precisely. One of the parameters involve in the accuracy an precision of nanoparticles isplacement operations is the stiffness coefficient of cantilevers. The moeling of stiffness coefficient for rectangular, V- shape an agger-shape cantilevers, has een stuie y innumerous researches. Experimental, numerical an theoretical techniques have een applie to formulate the stiffness moels. In this paper, a comprehensive stiffness moel for practical V-shape cantilevers has een introuce. The computation of static eformations of cantilever plates is a funamental principle in the use of atomic force microscope (AFM. Static test methos have alreay een consiere for otaining a cantilever s stiffness coefficient. In one metho, a mass is hung from the free en of the cantilever an the static isplacement is measure. Then y using the force-isplacement relationship, the normal stiffness coefficient of every cantilever with aritrary shape is 7

2 etermine []. Also, the isplacement of the cantilever tip is etermine comparing with a reference cantilever, an the normal stiffness coefficient of the consiere cantilever is otaine [3-6]. By using the plate eformation theory, Saer an White presente approximate formulas for the static isplacement of cantilevers an they evaluate the otaine results y the theoretical solution through the finite element metho [7]. Using the ynamics of ae mass to cantilever an the resonant frequency an analysis of thermal noise in a oy are the ynamic an nonestructive testing methos that are use for fining the normal stiffness of cantilevers with aritrary geometries [8- ]. In another technique, the theory of parallel eam approximation (PBA has een use. Despite its limitations, ecause of its simplicity of analytical computations, this metho is commonly use in fining the spring constant of the V-shape cantilever in the atomic force microscope []. Different methos of caliration of AFM cantilevers with respect to accuracy an precision have een compare [3, 4]. Neumeister offere an analytical moel for etermining the longituinal, lateral an normal stiffness coefficients of a V-shape cantilever an compare the results with the values otaine from the finite element metho an showe that the presente equations are very accurate for etermining the stiffness coefficients for the cantilever of specific geometry [5]. Green an Saer have attempte to improve the accuracy of the V-shape cantilever s stiffness coefficients [6]. Regaring the stiffness coefficients of the V-shape cantilever, Cliffor an Seah have concentrate more on the cantilever s triangular section [7]. Saer et al. employe the hyroynamic functions in experiments to calculate the stiffness values of nonrectangular an irregular cantilevers which are actually use in practice [8]. In recent research works, Korayem an Daeinai investigate the V-shape, rectangular an aggershape cantilevers an explore the effects of all the influential factors such as thickness, with, length an tip-to-free en istance of the ieal cantilevers on the stiffness coefficients in the main irections [9, 0]. The stiffness of V-shape cantilever has een stuie using an experimental setup y Rui et.al []. The moeling of V-shape cantilever an its effect on critical forces on manipulation of iological particles has een stuie y Daeinai et.al []. The effect of piezo layers on stiffness an ynamic has een investigate y Koreyem an Ghaeri [3]. The stiffness coefficients moel of V- shape cantilevers is limite to the ieal form so in this paper the exciting moel has een improve to present the moel for ifferent from of V-shape cantilevers. Due to the analytical moel restrictions, FEM simulations have een use to evelop the new moel. In this paper, a general comprehensive moel for stiffness coefficients has een evelope. The stiffness coefficients of V- shape cantilevers in ifferent geometries are otaine using the FEM metho. These coefficients are then compare with the values otaine in analytical reports an finally y using the correction coefficients, the moeling of stiffness coefficients is improve. Figure..The V-shape cantilever use in the AFM consists of the triangular section (I an two olique rectangular eams (II [5]. 8 Korayem et al

3 . MODELLING THE FORCES APPLIED ON THE V-SHAPED CANTILEVER In this section, y using the moel of a V-shape cantilever, the eformations of the cantilever uner the effect of force is investigate an a 3D moel is presente. In this analysis, three forces are applie to the proe tip along the three X, Y an Z irections (Figure. [5, 9]... Moeling of the normal stiffness coefficient ( Kz Consiering the metho conucte y Neumeister, the normal stiffness coefficient of the cantilever (in Z irection will e as follows [5,9]: eformation an the ening angle in the junction of the two parts (II&I, respectively an =[ L tan( - w ], w L L(, w LI an tan L L L. I II.. Moeling of the cantilever s lateral stiffness coefficient ( Kx The spring constant of the V-shape cantilever in the X irection is calculate from the ening an twisting cause y force Fx an the ening stiffness resulting from this force is otaine as follows [5,9]: K Z F sin [ ( ] z Z Z ( Kx en Et L ( sin cos(ln( (6 6FZ L ( L Z ( [ Ewt 3 L ( L (ln( L ln( ] L FZ L L Z ( [ Et 3 cos cos 3( cot cos r sin ] 3F ( ( z L ( r cot Et 3 cos sin ( (3 (4 The rotation of the triangular plate aroun the X axis which is prouce y moment T T Fx L is as follows: ( tip 3T ( ( log( Et3 tan sin 3TL( cos ( [ (6L sin 3 Et 3 ( cos 3 ( cos ] 8L (7 (8 ( L r tan ( sin ( cos (5 ( ( cos Where, an respectively are the half of the angle of tip, the with of the two ases an istance from en of the cantilever, L I is the length of the cantilever s triangular plate, is Poisson's ratio, Z I is the eformation of the triangular plate (I, Z, are the K Et3 Kx tors ( L tip 3( L tip log( sin cos 3sin ( L tan 8 Kx Kx Kx en en Kx Kx tors tors (9 (0 International Journal of Nanoscience an Nanotechnology 9

4 I an II inicates the rotation of the triangular plate (I an olique eams (II. The twisting stiffness ecomes: T K ; an total lateral stiffness ( will e as follows:.3. Moeling of the cantilever s longituinal stiffness coefficient ( Ky Finally, the effect of the force in the Y irection ( Fy is presente. The longituinal eformation of the cantilever has een isregare, ecause it is very small isplacement; so only the coefficient of stiffness resulting from net moment M Fy L tip ue to force Fy along the Y axis is consiere [5, 9]: 6M ( log( Et3 tan sin 6ML( Et3( ( cos ( ( In the aove relation I an II are the rotations of the triangular plate an the two olique eams, respectively. 6 KY {[ log( Et3 tan sin 6L( ] L tip} Et 3( ( cos (3 3. INVESTIGATING THE NEW GEOMETRICAL MODELS In this paper, the finite element simulation has een employe to otain the static isplacements. As it is emonstrate in Figure the X parameter varies etween 0 to 50 m. Then, the force applie on the tip of the cantilever an the constraints applie to the other en. Using the eformation results an applie forces, the stiffness coefficients ( Kx, Ky an Kz have een otaine (3D, 0 noe structural element has een use for FEM simulation. To calculate the cantilever stiffness coefficients in the three irections of X, Y an Z, the forces of Fx, Fy an Fz are respectively applie on the tip of the V-shape cantilever an the static isplacements in these three irections are otaine using FEM metho. Now, since the isplacement is equal to ( K F, y having the values of the force an the isplacement, the stiffness coefficients of Kx, Ky an Kz are etermine. Consiering the geometrical variations in the cantilever accoring to Fig., the changes of stiffness coefficients Kx, Ky an Kz have een presente (Fig. relative to the other stiffness coefficients, the lateral stiffness coefficient Kx is highly sensitive to the changes of the geometrical variations. Also, as it has een pointe out, stiffness coefficient Kz has a lower sensitivity to the geometrical changes of the V-shape cantilever [7]. The effects of variation in thickness, length an with have also een investigate using FEM metho. For the ieal cantilever (X=0 the result of FEM simulation in X, Y an Z irections are in harmony with Neumeister moel. However, the moel faile to estimate the accurate stiffness coefficient with the change in cantilever geometry (parameter X ut as it is emonstrate in Figs. 3 an 4 the rate of change in stiffness coefficient for oth FEM metho an the moeling, are similar [5] 4- VALIDATING THE MODELING OF FORCES APPLIED ON THE CANTILEVER In orer to valiate the presente simulation, a finite element moel of the V-shape cantilever with the imensions shown in Tale has een use. For valiation purposes, the results otaine from this simulation are compare with those of the Neumeister an Ducker s moel. 30 Korayem et al

5 Figure. The presente moel, Geometrical changes of the finite element moeling an changes of Kx, Ky an Kz versus X Figure3. Stiffness coefficients of variation for the cantilever thickness (X is in terms of micrometer International Journal of Nanoscience an Nanotechnology 3

6 X Displacement (m Y Displacement (m Z Displacement (m Figure4. Stiffness coefficients of variation for the cantilever with (X is in terms of micrometer Tale. Geometrical values an mechanical properties of the V-shape cantilever Geometric features Physical features L W t α E Ν 4.(µm 33(µm 0(µm 0.6(µm (µm 8(Deg. 43(G.Pa x 0-0 Analytical Dis FEM Dis 5 x Analytical Dis FEM Dis.4 x 0-6. Analytical Dis FEM Dis F x (N x F Y (N x F z (N x 0-8 Figure5. Comparison etween force-isplacement iagrams along the X, Y an Z irection otaine y the analytical an finite element methos Consiering the force-isplacement iagrams for the ieal V-shape cantilever in Fig. 5, it is oserve that the X-irection isplacements otaine y the analytical an finite element methos are ientical. Regaring the force-isplacement iagram along the Y irection, the two iagrams have a slight ifference, which coul e ue to not consiering the cantilever s longituinal eformation in the Y irection. It is also oserve that the two iagrams in the Z irection are in harmony. 3 Korayem et al

7 5. CORRECTION FACTORS DETERMINATION Diviing the stiffness coefficient resulte from FEM simulation an ieal stiffness coefficient, the correction factors for the cantilever in three irections are epicte in Figure. 6. These correction factors are imensionless (X is the geometry parameter presente in Figure.. x Ky NEW [ ( ] Ky x 6 W [ ( ]{[ * log( Et3 tan sin 6L( ]* L tip} EW t 3( ( * cos (5 x x Kx NEW [.5( ] Kx [.5( ] [[ ] [ ] ] x Kx en Kx tors [.5( L [[ Et( sin cos(ln( ] 3 log( Et [( ( sin 3( L tan L tip cos 3sin 8 ] ] (4 To make the propose imensionless correction factor, (X/ has een use as the variale, an a first, secon an thir orer polynomials have een propose to approximate the result in Figure. 6. These correction factors are use as multiplier terms to the stiffness values otaine y Neumeister an Ducker for ieal cantilevers an yiel a new comprehensive moel, which is feasile for practical V- shape cantilevers. The new comprehensive moel for practical V- shape cantilevers, with respect to geometrical changes (equivalent to the changes in the value of X, are as follows ( is cantilever with an X is the changing geometrical parameters (shown in Figures an, respectively. Similarly, the correcte factors are multiplie y the longituinal stiffness coefficient ( Ky an normal stiffness coefficient ( Kz, respectively, an yiel geometry-epenent stiffness coefficients moel. x 3 x Kz 3 NEW [ ( ] Kz [ ( ] F [ ( ] z Z Z sin x 3 6L ( L [ ( ] {( [ Ewt 3 L ( L (ln( L ln( ] L L L ( [ 3( cot Et 3 cos cos 3L( cos r sin ] ( Et 3 cos ( r cot( } sin sin (6 Finally, accoring to Figure. 7, the new stiffness coefficients in three main coorinate irections have een presente with respect to geometrical changes an compare with the results otaine from the FEM simulation.for stiffness coefficient Kx, the value otaine in the ieal case ( X 0, the ata otaine y the finite element metho an y equation (4 are almost ientical. As is oserve in Figure. 7, the values of Kx are higher relative to the other two stiffness values. For the new Ky, at X 5m, the changes occur with a mil slope, an for a larger geometrical change, we will witness a high sensitivity to this geometrical parameter. In the propose Kz a minor change in stiffness has een oserve. International Journal of Nanoscience an Nanotechnology 33

8 Figure6. Correction factor for stiffness coefficient Kx, Ky an Kz Figure7.Comparison etween the moifie stiffness coefficient of Kx, Ky, Kz an the result otaine from the finite element analysis As is oserve, the stiffness values otaine from the new theory correlate very closely with the ata otaine from the finite element approach (there is a minor eviation which is in acceptale limits; consequently, oth results are in harmony. The gaps forme in the iagrams are ue to FEM simulation errors an also the assumptions in the theoretical formulas which can e neglecte. 6. CONCLUSION The real-time monitoring limitation is the most important arrier in AFM ase nano manipulation. Consequently, to evelop a practical manipulation strategy, moeling an simulation of the process have een heavily investigate in numerous researches. For a successful manipulation y the means of AFM nano root, it is of great importance to have an accurate ynamic moel an to have precise 34 Korayem et al

9 ynamic moel, stiffness coefficient shoul e etermine accurately. The cantilever geometry varies as a result of existing manufacturing errors in commercial cantilevers which have a huge impact on cantilever stiffness. Base on the simulations 44%, 3% an 3% change of Kx, Ky an Kz is otaine, respectively. Therefore, in this paper the exciting stiffness moel with the use of FEM simulation has een further improve to inclue effect of cantilever geometry. With the use of presente moel, it is possile to preict the stiffness coefficients ( Kx, Ky an Kz for a eviation of X etween 0 to 50 m. The evelope moel enales us to preict stiffness coefficients for a wie range of commercial cantilevers. The evelope moel shows acceptale consistency (less than.5 % eviation an enales us to moel stiffness coefficient precisely. REFERENCES. Binnig, G., Quate, C.F., Gerer, C. ( 986. Atomic force microscope Phys. Rev. Lett,. 56: Senen, T., Ducker, W., (994. Experimental Determination of Spring Constants in Atomic Force Microscopy Langmuir,. 0: Tortonese, M., Kirk, M., (997. Characterization of application-specific proes for SPMs. Micromachining an Imaging Micromachining an Imaging, 53: Cumpson, P.J., Zhan, P., Heley, J. (004 Caliration of AFM cantilever stiffness: a microfaricate array of reflective springs Ultramicroscopy, 00: Cumpson, P.J., Heley, J., Zhan, P. (003 Accurate force measurement in the atomic force microscope: a microfaricate array of reference springs for easy cantilever caliration Nanotechnology,. 4: Gison, C.T., Watson, G.S., Myhra, S., (996 Determination of the spring constants of proes for force microscopy /spectroscopy Nanotechnology, 7: Saer, J.E., White, L., (993 Theoretical analysis of the static eflection of plates for atomic force microscope applications Journal of Applie Physics, 74: Clevelan, P., Manne, S., Bocek, D., Hansma, P.K. (993 A nonestructive metho for etermining the spring constant of cantilevers for scanning force microscopy. Review of Scientific Instruments, 64: Saer, J.E., Chon, J.W.M., Mulvaney, P., (999 Caliration of rectangular atomic force microscope cantilevers Review of Scientific Instruments, 70: Hutter, J.L., Bechhoefer, J., (993 Caliration of atomic-force microscope tips Review of Scientific Instruments,. 64: Green, C. P., Lioe, H., Clevelan, J. P., Proksch, R., Mulvaney, P., Saer, J.E., (004 Normal an torsional spring constants of atomic force microscope cantilevers Review of Scientific Instruments, 75: Saer, J. E., (995 Parallel eam approximation for V-shape atomic force microscope cantilevers Review of Scientific Instruments, 66: Burnham, N. A, Chen, X., Hoges, C. S., Matei, G. A., Thoreson, E. J., Roerts, C. J., Davies, M. C., S. J. B. Tenler (003 Comparison of caliration methos for atomic-force microscopy cantilevers Nanotechnology, 4: Gates R. S. (0 Atomic Force Microscope Cantilever Flexural Stiffness Caliration: Towar a Stanar Traceale Metho National Institute of Stanar Technology 6: Neumeister, J. M., Ducker, W. A. (994 Lateral, normal, an longituinal spring constants of atomic force microscopy cantilevers Review of Scientific Instruments,. 65: Saer, J. E., Green, C. P., (004 In-plane eformation of cantilever plates with applications to lateral force microscopy Review of Scientific Instruments, 75: Clifforan, C. A., Seah, M. P. (005 The etermination of atomic force microscope cantilever spring constants via imensional methos for nanomechanical analysis, Nanotechnology, 6: Saer, J. E., Sanelli, J. A., Aamson, B. D., Monty, J. P., Wei, X., Crawfor, S. A., Frien, J. R. Marusic, I., Mulvaney, P., Bieske, E. J. (0 Spring constant caliration of atomic force microscope cantilevers of aritrary shape Review of Scientific Instruments, 0. 83: Daeinai, K., Korayem, M. H., Yarijani, S. A. (0 Spring constant analysis of the AFM rectangular, V- shape an agger cantilever proes Micro an Nano Letters, 6: Daeinai, K., Korayem, M. H., Yarijani, (0 Transucer Moeling of Rectangular, V-shape, an Dagger Cantilever Proes Base on Atomic Force Microscopy Instrumentation Science an Technology, 40: International Journal of Nanoscience an Nanotechnology 35

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