Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory

NanoSystems MMTAEditorial NanoMMTA

Modeling the tip-sample interaction in atomic forcemicroscopy with Timoshenko beam theory

AbstractA matrix framework is developed for single and mul-tispan micro-cantilevers Timoshenko beam models ofuse in atomic force microscopy (AFM). They are con-sidered subject to general forcing loads and boundaryconditions for modeling tip-sample interaction. Sur-face eects are considered in the frequency analy-sis of supported and cantilever microbeams. Exten-sive use is made of a distributed matrix fundamentalresponse that allows to determine forced responsesthrough convolution and to absorb non-homogeneousboundary conditions. Transients are identied from in-tial values of permanent responses. Eigenanalysis fordetermining frequencies and matrix mode shapes isdone with the use of a fundamental matrix responsethat characterizes solutions of a damped second-ordermatrix dierential equation. It is observed that sur-face eects are inuential for the natural frequency atthe nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantileverbeam actuated by a piezoelectric layer laminated inone side.

KeywordsAtomic force microscopy nanoscale materials andstructures chemical/biological sensors nanomachin-ing microscaled Timoshenko beamsPACS: 07.79.Lh, 62.25.-g, 81.07.-b, 87.85.fk, 46.70.DeMSC: 74H55, 74H10, 35E05, 35L30, 35L35 Versita sp. z o.o.

Julio R. Claeyssen1,2, Teresa Tsukazan1 , LeticiaTonetto1 , Daniela Tolfo1,1 Institute of Mathematics, Universidade Federal do RioGrande do Sul,

Av. Bento Gonalves, 9500, 91509-900, Porto Alegre,RS, Brazil

2 Mechanical Engineering Graduate Program, UniversidadeFederal do Rio Grande do Sul,

Rua Sarmento Leite, 425, 90050-170, Porto Alegre, RS,Brazil

E-mail: [email protected] E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

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Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory

1. IntroductionIn this work, we determine dynamic responses of Timoshenko micro-cantilever beam models of use in the scanningprobe technique of AFM. This AFM technique allows to obtain images of surface topography at the atomic scale, in anoninvasive manner, from a wide variety of samples on a scale from angstroms to 100 microns. Conductive and insulatingsamples of surface structures can be considered in both air and liquid environments. Its predecessor was the stylusproler that magnied, greater than 1000 , the vertical surface of a sample and recorded the motion of the stylus onphotographic paper. The AFM is a mechanical system for sensing force at the nanoNewton level between a sampleand a very tiny tip (< 10nm radius) that is mounted on a microfabricated cantilever ( 100 m). This allows AFM tobe capable of imaging features with a magnication of greater than 106 . Since its invention by Binning et al [5], ithas undergone several developments.A typical AFM consists of a sensitive micro-cantilever with a mounted sharp tip acting as force sensor, a system thatmoves the sample or the sensor in order to probe the sample surface, a detection sensor system of the cantileverdeection, a feed-back system which regulates the force interaction and a controller electronic system which recordsmovements, control the feedback loop and sends the measured data to a computer processing unit (Fig. 1). Interms of the cantilever state of motion during measurement, the two basic types of AFM modes are: static mode(contact, friction or lateral force) and dynamic mode (non-contact, tapping or semi-contact, acoustic, piezoelectric,electrostatic,etc) [33]. They are typically of length 125-450 m, width 28-45 m, thickness 1-8 m, resonantfrequency 12-300 KHz, spring constant 0.1-48 N/m and tip probe height 17m and tip radius less than 10nm. The asso-ciated length scales associated are suciently small to call the applicability of classical continuum models into question.

Laser

Cantilever/ Tip

Scanningsystem

Imaging

FeedbackElectronics

and

Deflectiondetectionsystem

Sample

Fig 1. Schematic of an Atomic Force Microscope operation

The geometry and the material of the cantilever both contribute to the properties that make a cantilever suitablefor any particular imaging modes. Both silicon and silicon nitride micro-cantilevers are commercially availablebut reective back surface coating is used for a better feedback. New generations of nanobeams have includedpiezoelectric materials locally attached at the micro-beam with the role of sensors and/or actuators [21]., The inclusionof smart materials layers will modify material properties between neighboring layers. Active beams for AFM havebeen subject to a variety of tip-sample interaction types models and they can be formulated as a second-ordermatrix dierential equation subject to boundary conditions and compatibility conditions for transversal vibrationsin neighboring segments whenever having a multi-span micro-beam [11], [2], [17], [38]. The use of the AFM, asnanomachining or as a platform for chemical and biological sensors in connection with and surface and thermaleects, make that the eects of transverse shear deformation and rotary inertia on the frequency be signicant.With smaller values of the ratio of the probe length to its thickness, the Timoshenko beam theory is able topredict the frequencies of exural vibrations of the higher modes with higher stiness for the AFM cantilevers [19].As the structural size decreases toward the nanoscale regime, surface/tension eects must be taken into account [20], [15].

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Julio R. Claeyssen, Teresa Tsukazan, Leticia Tonetto, Daniela Tolfo,

In this work, we shall discuss AFM Timoshenko micro-cantilevers models that can involve smart materials or generaldevices for describing tip-sample interaction forces. These models are formulated as second-order evolution systemssubject to initial, forcing and boundary data. Its mathematical study will make an extensive use of distributed matriximpulse response or initial-value Green matrix response. This allows to characterize transients and forced responsesof a variety of AFM models. The vibration modes for general tip-sample interaction will be explicitly formulated interms of a fundamental matrix response of a second-order ordinary matrix dierential equation where the correspondingstiness matrix coecient depends upon the frequency. This matrix response can be determined in closed form in termsof a scalar solution that has a completely oscillatory behavior beyond a critical frequency value [9], [10].This work is organized as follows. In section 2, we formulate the Timoshenko beam model in a matrix form. In section 3,the dynamic response of the matrix model subject to tip-sample interactions and external forcing is given in terms of thedistributed matrix impulse response. Eigenanalysis is discussed in section 4 by solving a non-conservative second-ordermatrix dierential system that depends nonlinearly upon the eigenvalue. The cases of micro-cantilevers involving anelastic append at the free or linearized boundary conditions are discussed in terms of a fundamental response that givenin closed form. In section 5, the Galerkin method is used to obtain reduced-order forced models. Forced responses areapproximated by concentrated responses involving convolution with a concentrated impulse response and a localizedresponse at the end of the beam due to boundary conditions. Finally, in section 6 we discuss the matrix methodologywith modal analysis for the case of harmonic inputs as well as for modulated linear piecewise inputs with compositemicro-cantilever beams.2. Transversal vibrations of AFM using the Timoshenko modelAFM was developed for producing high-resolution images of surface structures. The AFM tip has a vertical resolutionon the order of 1nm or below, and it can detect low-amplitude vibrations corresponding to high frequencies. Nowdaysit is also used to probe properties through interactions between the tip and the sample and to modify surfaces. Thisinteraction process has lead AFM to be used in smart material technology, chemical/biological sensors, tribology andnanomachining, among others elds [23], [38], [12]. Modeling and simulate an AFM microstructural system is a complextask [33]. The sharpness of the tip is often a fundamental resolution-limit parameter [8], [7]. The miniaturized cantileverdevice depends on the accurate extract of static bending and resonant frequency. Test measurements and theoreticalstudies have shown that the vibration behaviour of microstructures at the nanoscale is signicantly size and parametricdependent. As the structural size decreases toward the nanoscale regime, surface eects must be taken into account[20]. This dependence has motivated the use of size-dependent continuum theories in modeling microstructures: couplestress theory [3], surface energy theory [16], [27], nonlocal formulation [26],[36], strain gradient [22], functionally graded[25], among others.In this work, we use the Timoshenko beam-type structure for incorporating shear formation and rotary inertia eects.This allows to consider microbeams with small length-to-thickness aspect ratio. Thick beams have relatively hightransverse shear modulus and the eects of rotary inertia and transverse shear deformation must be used in the dynamicanalysis of such beams. The Timoshenko model corrects the classical beam theory with rst-order shear deformationeects. Also, piezoelectric shearing coecients can be considered in the constitutive relations once shear deformationalso induce an electric displacement [30], [39]. This model rests on the assumptions of small deformations and linearelastic isotropic material behavior. Continuum-based formalisms for nanoscale have been proposed that include theeect of surface properties on the mechanical behavior. Here we shall consider Laplace-Young surface elasticity andresidual surface tension adapted to solid materials [40].We consider the micro-cantilever having length L, width b, thickness 2h and mass density area of the beam . We letI = 2bh33 be the moment of inertia of the cross section area A = 2bh, w(t, x) the exural deection of the beam, (t, x) therotation angle of cross section of the beam, f(t, x) a transverse dynamic load and q(t, x) a moment load. The governingequations are given by [1], [13] Awtt GAwxx + GAx = f(t, x),

Itt EIxx GA (wx ) = q(t, x), (1)

3


whereGA = GA (u + b)b, (2)EI = (EI + 2bh2Es), (3)

are the eective curvature eect and exural rigidity, respectively. Here u and b denote the upper and lower surfacesresidual tensions and Es being a surface elastic modulus. The boundary conditions are those of a cantilever beam orsubject to balance of the moment and shear at the free end x = L. In this work, we shall assume that for a uniform beamthe involved coecients are constant.2.1. Matrix formulationThe coupled Timoshenko model (1) can be written as a second-order dierential equation with matrix coecients

Mv + Kv = F, (4)where

v = w(t, x)(t, x)

, F = ( f(t, x)q(t, x)) , (5)

M = A 00 I

, K = E 2x2 +N x + R, (6)with

E = GA 00 EI , N = 0 GAGA 0

, R = 0 00 GA . (7)

It is frequently found in the literature, that the unforced Timoshenko model is decoupled into the same fourth-order timedierential equation for the deection and rotating angle [35]. However, the boundary conditions actually couple thesystem. Only for very few cases the unforced linear case is completely decoupled. For the case of a forced Timoshenkomodel, the transverse forcing and moment load has to be regular in order to admit dierentiation. These argumentssuggest the convenience of keeping the original second-order physical formulation (4).2.2. Multispan modesMultispan active micro-beams for AFM are used for improving the detection and sensing imaging performance. In thesesituations, the Timoshenko model can be considered in each segment [xi, xi+1], i = 0, N 1 where 0 = x0 x1 x2 , xN = L and to require the fullllment of compatibility conditions of continuity and equilibrium for displacement,rotation and for moment, shear, respectively, in neighbouring segments. In a general setting, we have

E (i)11wi(t, xi) + F (i)11i(t, xi) + G(i)11wi(t, xi) +H(i)11i(t, xi), = E (i)12wi+1(t, xi) + F (i)12i+1(t, xi) + G(i)12w i+1(t, xi) +H(i)12i+1(t, xi),E (i)21wi(t, xi) + F (i)21i(t, xi) + G(i)21wi(t, xi) +H(i)21i(t, xi) = E (i)22wi+1(t, xi) + F (i)22i+1(t, xi) + G(i)22w i+1(t, xi) +H(i)22i+1(t, xi),E (i)31wi(t, xi) + F (i)31i(t, xi) + G(i)31wi(t, xi) +H(i)31i(t, xi) = E (i)32wi+1(t, xi) + F (i)32i+1(t, xi) + G(i)32w i+1(t, xi) +H(i)32i+1(t, xi),E (i)41wi(t, xi) + F (i)41i(t, xi) + G(i)41wi(t, xi) +H(i)41i(t, xi) = E (i)42wi+1(t, xi) + F (i)42i+1(t, xi) + G(i)42w i+1(t, xi) +H(i)42i+1(t, xi).In matrix terms

C1,iwi(t, xi) = C2,iwi+1(t, xi), i = 1, ..., N 1,4


0 x1 x2 xN-1

...1 2 NN-1

L x

z

Fig 2. Multispan micro-beam

wherewi(t, x) =

wi(t, x)i(t, x)w i(t, x)i(t, x)

, C1,i =

E (i)11 F (i)11 G(i)11 H(i)11E (i)21 F (i)21 G(i)21 H(i)21E (i)31 F (i)31 G(i)31 H(i)31E (i)41 F (i)41 G(i)41 H(i)41

, C2,i =

E (i)12 F (i)12 G(i)12 H(i)12E (i)22 F (i)22 G(i)22 H(i)22E (i)32 F (i)32 G(i)32 H(i)32E (i)42 F (i)42 G(i)42 H(i)42

, i = 1, ..., N 1.

Thus for a multispan Timoshenko micro-beam, including proportional damping C = aM+ bK, we have the second-orderblock matrix dierential equationsMv(t, x) + Cv(t, x) + Kv(t, x) = F, 0 < x < L, t > 0, (8)

wherev =

v1v2...vN

, M =

M1 0. . .0 MN , K =

K1 0. . .0 KN

, F =

F1F2...FN

, (9)with

Mj = jAj 00 j Ij

, Kj = jGjAj

2x2 jGjAj xjGjAj x Ej Ij 2x2 + jGjAj

, vj = wjj

,subject to given initial conditions v(0, x) = ro(x), vt(0, x) = r1(x), boundary and compatibility conditions

B1w1 = n0, BNwN = nL, (10)C1,iwi(t, xi) = C2,iwi+1(t, xi), i = 1, ..., N 1. (11)2.3. AFM-tip-cantilever interactionsThe tip interaction with the sample has been usually modeled as being subject to springs or dash-springs or attachedmass for normal and lateral interaction and to an external excitation of the base [19, 28].For instance, when the tip of length h and mass m is subject to normal kN and lateral springs kL and viscous damperscN , cL, the moment and shear conditions at the free end are given by

GA(wx ) = kN cN wt m2wt2 x=L,

EI x = kLh2 cLh2 t mc2 2t2 x=L, (12)5


cNkN

cL

h

x

z

c

kL

Fig 3. AFM tip-sample interaction

where c denotes the distance between the lower edge of the cantilever and the centroid of the cross section.In a general setting, separated boundary conditions at the ends x = 0, L of the micro-beam can be given asa11w1(t, 0) + a121(t, 0) + b11w 1(t, 0) + b121(t, 0) = n11,a21w1(t, 0) + a221(t, 0) + b21w 1(t, 0) + b221(t, 0) = n12; (13)p11wN (t, L) + p12N (t, L) + q11w N (t, L) + q12N (t, L) = n21,p21wN (t, L) + p22N (t, L) + q21w N (t, 0) + q22N (t, L) = n22. (14)Or in matrix form

B1v(t, 0) = Av(t, 0) + Bvx (t, 0) = n1,B2v(t, L) = Pv(t, L) +Qvx (t, L) = n2. (15)More complex descriptions of the tip-sample force include non-linear surface and contact forces at the boundary x = Ldue to Derjaguin-Muller-Toporev (DMT), Johnson-Kendall-Roberts (JKT) [7].2.4. The Timoshenko AFM modelThe Timoshenko microbeam model for AFM operation modes, can be encompassed as the second-order matrix evolutionmodel Mv(t, x) + Kv(t, x) = F, 0 < x < L, t > 0,v(0, x) = vo(x), vt(0, x) = v1(x),B1v(t, 0) = n1,B2v(t, L) = n2,

(16)where F can include driven excitations or hydrodynamic damping and n1, n2 interactions terms with the free end. Forinstance,

n1 = Cv(t, 0)Dvt(t, 0) Evtt(t, 0), (17)n2 = Rv(t, L)Ovt(t, L) Svtt(t, L). (18)

In (12), the given conditions at x = L will haveR = kN GA0 kLh2

, O = cN 00 cLh2 , S = m 00 mc2

. (19)6


Although the unforced governing equation Mv(t, x) + Kv(t, x) = 0 might look to be written in conservative form, theboundary conditions could change such character introducing extra energy terms into AFM system. When using modalanalysis from a micro-cantilever beam with boundary conditions( 1 00 1

) v(t, 0) +( 0 00 0) vx (t, 0) = 0,

(20)( 0 00 1) v(t, 0) +( 0 11 0

) vx (t, L) = 0,the AFM tip-sample interaction can be considered as localized forces at the free end.3. The AFM dynamic responseThe dynamic response of the Timoshenko model (1) or equation (4) can be described in terms of the matrix impulseresponse or matrix Green function h(t, x, ) of the associated homogeneous initial-boundary value problem

Mh(t, x, ) + Kh(t, x, ) = 0, 0 < x, < L, t > 0, (21)h(0, x, ) = 0, Mht(0, x, ) = (x )I,Ah(t, 0, ) + Bhx (t, 0, ) = 0,Ph(t, L, ) +Qhx (t, L, ) = 0,where 0 denotes the 2 2 null matrix and I the 2 2 identity matrix. The Laplace transform of h(t, x, ) with respect totime will be denoted by H(s, x, ) and referred to as the matrix transfer response. Thus

(s2M+ K)H(s, x, ) = (x )I, 0 < x, < L, (22)AH(s, 0, ) + BHx (s, 0, ) = 0,PH(s, L, ) +QHx (s, L, ) = 0.It turns out that h(t , x, ) acts a integrating factor in Lagranges adjoint method for the nonhomogeneous equation

Mv(t, x) + Kv(t, x) = F(t, x), 0 < x, < L, t > 0, (23)v(0, x) = vo(x), vt(0, x) = v1(x),Av(t, 0) + Bvx (t, 0) = n1(t),Pv(t, L) +Qvx (t, L) = n2(t).Multiplying (23) by h(t , x, ) and integrating by parts, it turns out the dynamic response

v(t, x) = t0 L0 (ht(, x, )Mvo() + h(, x, )Mv1())dd+ t0 L0 h(t , x, )F(, )dd + J(v, h)L0, (24)

where J is a term containing eects of the initial-value Green function with values of v at the boundary.The procedure mentioned above is related to the Riemann function method for integrating hyperbolic equations [14], theformula appears in the eld of control of distributed systems [6] and in elastodynamics in connection with vibrations andcracking problems is referred to as the dual integral representation [18]. For homogeneous boundary conditions, theterm J vanishes and (24) becomes a variations of constants formula for a second-order linear matrix dierential equation.

7


If we consider a micro-cantilever beam with a time dependent boundary condition s2(t) at the free end, the term J canbe identied as J = t0 h(t , x, L)ETQn2()d, (25)whereE = ( GA 00 EI

) , Q = ( 0 11 0) . (26)

We can observe that the forced response given by (24) will involve the convolution of the impulse response and distributedor concentrated forcing eects as in (25) and initial-value Green function with values of v at the boundary.3.1. Frequency responseIn practice, when computing the convolution integral for the forced response, we actually have

v(t, x) = vh(t, x) + vp(t, x), (27)where vh(t, x) is a free vibration introduced by the system and whose initial values are a priori unknown. It turns outthat these initial values are supplied by the permanent response vp(t, x) that can be determined by other means.Since the impulse response and its time derivative constitute a basis for the free responses and the forced response in(24) has null initial values at t = 0, the induced system free response due to a permanent response vp(t, x) can be easilydetermined. It turns out

vh(t, x) = L0 h1(t, x, )vp(0, )d L0 ho(t, x, )vp(0, )d, (28)

whereh1(t, x, )() = h(t, x, )M(), ho(t, x, )() = ht(t, x, )t M(). (29)

Harmonic and piecewise linear forcing are of interest in frequency analysis. When seeking a response of the same typethe transfer function is introduced. Given the harmonic inputf(t, x) = eitv(x), (30)

we have the harmonic output response vp(t, x) = eitH(i)v(x), (31)where H(i)v(x) = L0 H(i, x, )v()d. (32)The kernel H(s, x, ) of the transfer operator H is the Laplace transform of the impulse response h(t, x, ). In particular,for a concentrated force [29] at a point x = a of spatial amplitude v(x) = v(x)(x a) we have the permanent responsevp(t, x) = eitH(i, x, a)v(a). (33)

With the initial values vp(0, ) = H(i, , a)v(a), vp(0, ) = ivp(0, ), the induced free response is given byvh(t, x) = L0 r(t, x, , )MH(i, a, )v(a)d, (34)

8


wherer = ht(t, x, ) + ih(t, x, ). (35)

For a pulse amplitudev(x) = vo (Heavisde(x L+ b)Heaviside(x L)) , (36)

the permanent response turns outvp(x) = eit LLbH(i, x, )vod. (37)As before, by substituting the initial values in (28), the induced free response will now be

vh(t, x) = L0 r(t, x, , )MH(i, 0, )v()d, (38)with r given as in (35). In the case of a time linear exponential forcingf(t, x) = exp(t)(ct + d), (39)

we have the particular solution wp(t, x) = exp(t) (tC +D) , (40)whereC = (2M + K )1c, (41)D = (2M + K )1d 2(2M + K )2Mc,

whenever is not an eigenvalue or natural frequency.4. Free transverse vibrationsThe search of exponential solutions

v(t, x) = etv(x), v(x) = ( w(x)(x)) , (42)

of the unforced Timoshenko modelM2vt2 + Kv = 0, (43)subject to general separated homogeneous boundary conditions (15), amounts to determine nontrivial solutions of thesecond-order dierential equation Mv(x) + Cv(x) + K ()v(x) = 0, (44)with matrix coecients

M = GA 00 EI

, C = 0 GAGA 0

, K () = A2 00 2I + GA

, (45)that satisfy the boundary conditions Av(0) + Bv(0) = n1,Pv(L) +Qv(L) = n2. (46)We should observe that if viscous damping forces are considered, then the matrix M has to be modied to include aneigenvalue term. Also, when considering localized linearized tip-sample interactions and viscous damping force actingon a microcantilever beam (clamped-free), the above eigenvalue problem will modify the coecient matrices in (43) butthe boundary conditions (15) will be those of clamped-free beam.

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4.1. The eigenvalue problemIn terms of initial values, the general solution of the second-order matrix dierential equations (44) is given by [9]v(x) = ho(x)v(0) + h1(x)v(0), (47)

where ho(x) = h(x)M+ h(x)C, h1(x) = h(x)M, (48)or, in the more practical form v(x) = h(x)c1 + h(x)c2, (49)for constant 2 1 vectors c1 and c2. Here h(x) is the 2 2 matrix solution of the initial value problem

Mh(x) + Ch(x) + K ()h(x) = 0, (50)h(0) = 0, Ah(0) = I,where 0 denotes the 2 2 null matrix and I the 2 2 identity matrix. The matrix coecients being given as in (45).4.2. Shape modes in closed formFor a micro-cantilever beam of length L, we have the clamped boundary condition v(0) = 0, that is B1 = I. By using theinitial values of h(x) in (49), it turns out that c2 = 0. Thus we have to determine so that

v(x) = h(x, )c (51)satises the boundary condition at the free end x = L. By assuming homogeneous boundary conditions, we have thenonlinear eigenmatrix problem U()c = (Ph(L, ) +Qh(L, )) c = 0. (52)From this, it turns out the characteristic equation

() = det(U) = 0. (53)We should observe that the modes have the same shape, regardless of the conditions at the free end, but the eigenvalue diers according to the boundary coecient matrices P and Q. For a micro-cantilever beam, these matrices are givenin (20). The matrix U() or the characteristic equation (53) can be determined by computing the fundamental matrixsolution h(x).4.2.1. Computing h(x)The fundamental response h(x) can be determined in closed form as follows. Exponential type vector solutions v(x) = ekxuof (44) exist ( u = 0), whenever k is a root of the characteristic polynomial

P(k, ) = det (k2M + kC + K) = 4j=0 jk4j , (54)where 0 = abm, 1 = 0, 2 = (ae2 c2bm a2 + ama), 3 = 0 4 = c2a+ c4e, (55)with

a = GA, c = A, e = I, (56)am = GA = GA (u + b)b,bm = EI = (EI + 2bh2Es).10


The roots of (54) can be easily obtained after writing it asP(k, ) = abm(k4 + g2k2 r4), (57)

whereg2 = g2m + s2, g2m = ( ebm + ca

) 2 s2 = 1bm (am a) , (58)r4 = c2 (a+ e2abm

) .(59)

It turns out that the roots of (57) arek1 = , k2 = , k3 = i, k4 = i,

with = 12

2g2 + 2g4 + 4r4, (60) = 12

2g2 + 2g4 + 4r4. (61)The response h(x) is then given by the formula obtained in [9]

h(x) = 4j=1j1i=0 id(j1i)(x)h4j , (62)

where d(x) = senh(x) sen(x)abm(2 + 2) , (63)is the solution of the initial value problem0d(iv)(x) + 2d(x) + 4d(x) = 0,d(0) = d(0) = d(0) = 0, abmd(0) = 1, (64)

and the matrices hj = h(j)(0) satisfy the matrix dierence equationMhj+2 + Chj+1 + Khj = 0,h0 = 0,Mh1 = I. (65)

By substituting values, we arrive to the closed formulah(x) =

(a+ e2)d(x) bmd(x) amd(x),ad(x) ad(x) + c2d(x), , (66)

where a = GA, am = GA, bm = EI, c = A, e = I.11


4.3. Frequency euqtaion for a supported micro-beamFor a supported Timoshenko model, we have the boundary conditionsu(t, 0) = 0, x (t, 0) = 0,u(t, L) = 0, x (t, L) = 0, (67)

whose coecients written in matrix form (15) areA = F = ( 1 00 0

) , J = Q = ( 0 00 1) . (68)

The matriz U can be reduced to a half size one due to the boundary conditions, that is, c12 = 0, c21 = 0. We thus havethe reduced systemUDc = 0, c = ( c11c22

)T , (69)where

UD = (a e2)d(L) bmd(L) d(L)amad(L) adiv (L) d(L)c2

. (70)The natural frequencies = i can be obtained from (60) or (61) by substituting the roots of the characteristic equation

() = det(UD) = A sin(L) sinh(L) = 0, (71)where

A = ((2bma+ a2 ae2)4 + (4abm + (2c2bm + 2ama)2 c2a+ c4e)2 + (a2 ae2)4 + (c2a c4e)2)a2b2m(2 + 2)2 .We observe that for = nL we can obtain from (61), while for = imL we can use (60). This later kind of frequenciesare associated with the so-called second spectrum [4], [34], [24]. By considering the surface parameter values given in[1], this second spectrum will appear for frequencies above the classical critical frequency 2c = ae .4.4. Frequency equation for a micro-cantileverFor the cantilever Timoshenko model, we have the boundary conditions

w(t, 0) = 0, (t, 0) = 0,x (t, L) = 0, wx (t, L) (t, L) = 0, (72)whose coecients written in matrix form (15) are

A = ( 1 00 1) , J = ( 0 00 0

) ,F = ( 0 00 1

) , Q = ( 0 11 0) .

(73)

12


By following the same reasoning as before, we obtain that due to the boundary conditions, the matrix U can be reducedto a half size one, that is, c21 = 0, c22 = 0. We thus have the reduced systemUDc = 0, c = ( c11c12

)T , (74)where

UD = ad(L) ad(L) + 2cd(L)bmd(L) + 2ed(L) ad(L) 2cd(L) amd(L)

. (75)The natural frequencies = i can be obtained from (60) or (61) by substituting the roots of the characteristic equation

() = 4ce (d(x))2 + 2 [ac d(x) d(x) (ae+ cbm) d(x) d(x)] + (a2 aam) (d(x))2 abm (d(x))2. (76)For a for micro-cantilever beam described by the Timoshenko model with surface eects, Figure 4 below illustrates thesize dependence in the natural frequency of Timoshenko classical model and Timoshenko model including surface eects,The solutions based on classical Timoshenko beam theory and Timoshenko beam theory including surface eects aredenoted by TB and TMB,respectively. The natural frequencies are normalized to fundamental frequency of cantileverEuler-Bernoulli beam. In this gure are considered the parameters utilized in [1] for the same purposes. The material andtypes of surface crystal orientation determine the surface elastic constants. For an anodic alumina Al (Youngs modulusE = 70GPa, Poissons ratio = 0.3 and = 2700kg/m3) are considered two types of crystallographic direction

Al[100] : Es = 7.9253N/m and = 0.5689N/m,Al[111]: Es = 5.1882N/m and = 0.9108N/m.

Fig 4. Inuence of surface eects and size dependence on the normalized fundamental natural frequency of the micro-cantilever for 2h=0.2L,b=0.4L and = 5/6

In the Figure 4 we can observe that for beam length on the order of nanometer to microns, the dierence betweennatural frequencies is apparent and by increasing the length of the microbeam, the results tend to Timoshenko classicaltheory. This same behavior was observed in [1] for a microbeam simply supported. Other observation is that the naturalfrequency of vibration of TB beams is independent of the beam length while for TB this is not occur, that is, the surfaceeects are signicant only in nanoscale.13


5. Modal approximation of dynamic responsesThe Galerkin method [13] can be used for determining approximate dynamic responses of the AFM micro-cantilever beamdescribed by the Timoshenko model. From (24), we actually need to nd an approximation of the fundamental matrixresponse h(t, x, ). For this, we rst introduce the block matrixV(x) = ( v1(x) v2(x) . . . vn(x) ) , (77)

whose columns are the rst n cantilever eigenfunctions (51) corresponding to the micro-cantilever eigenvalues, that is,vj (x) = ( wj (x)j (x)

) = h(x, j )cj , (78)where cj is obtained by nding a nonzero solution of (52) with j = ij . Since the AFM micro-cantilever modes sharethe normal mode property, we can assume that they have been normalized with respect to the mass matrix M. Then weconsider the obtention of an approximate response

v(t, x) .= nj=1 pj (t)vj (x) = V(x)P(t), (79)of the AFM micro-cantilever Timoshenko model (23).For determining the time amplitudes PT (t) = ( p1(t) p2(t) . . . pn(t) ), we substitute (77) into (4), pre-multiply theresulting matrix dierential equation by V()T and integrate in order to apply the normal mode property. It turns outthe n dimensional system P(t) + 2P(t) = f, (80)where

2 =

21 0 00 22 0 0 0 2N

, f = L0 V()TFd. (81)

The above system is subject to the initial conditionsP(0) = L0 V()T vo()d, P(0) =

L0 V()T v1()d. (82)

Thus the solution of (80) with the initial conditions (82) can be written asP(t) = h(t)P(0) + h(t)P(0) + t0 h(t ) (f)d, (83)

where, due to the decoupled character of (80), we have thath(t) = sin(t) =

sin(1t)1 0 00 sin(2t)2 0 0 0 sin(N t)N

. (84)

14


By substituting (83) in the approximated dynamic response v(t, x) = V(x)P(t) of (24), we havev(t, x) = L0 V(x)h(t)VT ()vo()d +

L0 V(x)h(t)VT ()v1()d +

t0 V(x)h(t )VT ()fd. (85)

Consequently, we obtain the spectral approximation for the initial value Green matrix responseh(t, , ) .= V() sin(t) VT () = Nj=1 sin(jt)j vj ()vTj (), (86)

and for the transfer matrix functionH(s, , ) .= V()(s2I+2)1VT () = Nj=1 vj ()v

Tj ()s2 + 2j . (87)We observe that when the probe deection is considered due only to the interaction tip-sample force n2 at the end x = Lof the micro-cantilever, we can use (25) to obtain the approximated response

v(t, ) = t0 h(t , x, L)ETQn(, L)d. (88)6. Numerical simulationsIn this section, we shall consider the eigenvalue problem for a free-free bi-segmented Timoshenko beam and the obtentionof forced responses for a Timoshenko micro-cantilever beam with to a piezoelectric layer above it. The computationswere performed in exact rational arithmetic using the symbolic computation language Maple. Expansions were truncatedwith a small number N of terms, usually between 5 and 10.6.1. Bi-segmented free-free Timoshenko beamIn [31], it was considered the eigenanalysis for a free-free Euler-Bernoulli bi-segmented beam. By using the same data,as given in Table 1.

Properties of beam elementsParameter(unit) Symbol Numerical valuesLength of rst segment (mm) l1 0, 254Length of second segment (mm) l2 0, 140Thickness of rst segment (mm) t1 0, 01905Thickness of second segment (mm) t2 0, 00549Width (mm) w 0, 02545Young Modulus (GPa) E 71, 7Density (Kg/m3) 2830Table 1. Geometrical dimensions and material properties of beam [31]

We have simulated the eigenanalysis of a free-free Timoshenko bi-segmented beam by using the matrix basis generatedby a fundamental matrix response in the study of the eigenvalue problem (52).The corresponding boundary conditionsE1I11(t, 0) = 0, 1G1A1[w 1(t, 0) 1(t, 0)] = 0, (89)E2I22(t, L) = 0, 2G2A2[w 2(t, L) 2(t, L)] = 0,

15


can be written in matrix form as( 0 00 1G1A1)( w1(t, 0)1(t, 0)

)+( 0 E1I11G1A1 0)( w 1(t, 0)1(t, 0)

) = ( 00) , (90)

( 0 00 2G2A2)( w2(t, L)2(t, L)

)+( 0 E2I22G2A2 0)( w 2(t, L)2(t, L)

) = ( 00) . (91)

The compatibility conditions at x = l1 arew1(t, l1) = w2(t, l1),1(t, l1) = 2(t, l1),1(t, l1) = 12(t, l1),w1(t, l1) 1(t, l1) = 1(w2(t, l1) 2(t, l1)),

where 1 = E2I2/E1I1 and 1 = 2G2A2/1G1A1. In matrix form, we have

1 0 0 00 1 0 00 0 0 10 1 1 0

w1(t, l1)1(t, xi)w1(t, l1)1(t, l1)

=

1 0 0 00 1 0 00 0 0 10 1 1 0

w2(t, l1)2(t, l1)w2(t, l1)2(t, l1)

, (92)

or C1w1(t, l1) = C2w2(t, l1) where

C1,i =

1 0 0 00 1 0 00 0 0 10 1 1 0

, C2,i =

1 0 0 00 1 0 00 0 0 10 1 1 0

, Wj (t, x) =

wj (t, x)j (t, x)w j (t, x)j (t, x)

. (93)

In Table 2, we have theoretical and experimental values obtained in [31] for an Euler-Bernoulli beam model (EBT), thoseobtained in this work with a Timoshenko model (TBT) and by applying similar methodology for multispan Euler-Bernoullibeams [37]. We observe that the frequencies obtained for the Timoskenho model are closer to experimental ones. Thecorresponding multispan shape modes for transversal displacement and rotation are illustrated in Figure 5.Freq. Theoretical Experimental This work This work(Hz) (1) (2) (3)1st 292 286 291 292.42 291.772nd 1181 1159 1165 1181.28 1167.893rd 1804 1759 1771 1804.01 1775.94(1): Reference [31] (Euler-Bernoulli beam theory)(2): Reference [31](3): Reference [37] (Euler-Bernoulli beam theory)

Table 2. Natural frequencies of a bi-segmented free-free beam

16


Fig 5. First three matrix shape modes of a bi-segmented free-free Timoshenko beam. Left: transversal deection component w(x). Right: rotationcomponent (x).

6.2. Forced AFM micro-cantilever beam with piezoelectric layerA Timoshenko micro-cantilever beam actuated by a piezoelectric layer laminated on one side of the beam was studied in[32]. The governing equations included viscous damping and the moment at the free end is subject to an applied voltageto piezoelectric layer. The equations and boundary conditions were established for a Timoshenko micro-cantilever witha laminated piezoelectric layer having length L, thickness hp and width b as in Figure (6).z

y

x

z

L

hp

hb

b

Fig 6. Schematic of beam with piezoelectric actuator

By incorporating the boundary condition due to piezoelectricity at the free end as a concentrated forcing into the model,we can describe this later as a forced damped Timoshenko micro-cantilever model. In matrix formulation, we haveMv + Cv + Kv = F, (94)

wherev =

w(t, x)(t, x) , F =

0k11(x)V (t) ,

M = M11 00 M22

, K = K11 K12K21 K22

, C = c1 00 c2

.(95)

Here17


M11 = (phpb+ bhbb), M22 = (pIp + bIb),K11 = 4(pcp55hpb+ bcb55hbb) 2x2 , K12 = 4(pcp55hpb+ bcb55hbb) x ,K21 = 4(pcp55hpb+ bcb55hbb) x , K22 = (cp11Ip + cb11Ib) 2x2 + 4(pcp55hpb+ bcb55hbb),k1 = e13zpmb, u(t) = V (t), b = 10(1+b)12+11b , p = 10(1+p)12+11p .(96)

c1 and c2 are viscous damping constants, zm is the distance between the middle line of the piezoelectric layer and theneutral axis of beam and V (t) is the applied voltage to piezoelectric layer.The beam geometrical and material properties are described in the Tables 3 and 4.

Properties of beam elementsParameter(unit) Symbol Numerical valuesLength (m) L 150Width (m) b 30Thickness (m) hb 10Young Modulus (GPa) cb11 73Density (Kg/m3) b 2200Poisson coecient b 0.17Table 3. Geometrical dimensions and material properties of beam elements

Properties of piezoelectric elementParameter(unit) Symbol Numerical valuesLength (m) lp 150Width (m) b 30Thickness (m) hp 10Young Modulus (GPa) cp11 71Density (Kg/m3) p 7700Poisson coecient p 0.31Table 4. Geometrical dimensions and material properties of piezoelectric element

The rst four obtained natural frequencies are shown in Table 5 for comparison with those of the formulated model in[32]. The micro-cantilever shape matrix modes in Figure 7 are mass normalized. It is observed that the inclusion ofrotatory inertia and shear in beam modeling inuences the rotation component of the forced responses due to a spatialconcentrated and spatial pulse moment excitations that are modulated with a harmonic input.Freq. Reference Present work(KHz) (1)1st 547 5442nd 3314 32983rd 8833 87974th 15951 16210(1): Reference [32]

Table 5. Comparative natural frequencies

18


Fig 7. Mass normalized matrix shape modes of a micro-cantilever beam with a piezoelectric layer. Left: transversal deection componentw(x).Right: rotation component (x). First mode: solid blue line, Second mode: dash-dotted red line, Third mode: dotted black line,Fourth mode: dashed gray line.

19


F (t, x) = col[0 q(t, x)]

Fig 8. Above: transversal deection component w(t, x) due to a concentrated moment q(t, x) = k11(x)V (t) at the free end of the micro-cantileverand proles for several times. Below: rotation component (t, x) and proles for several times.

7. ConclusionsThis paper addresses a matrix formulation for micro-cantilever models in AFM that are subject to quite general tip-sampleinteractions, surface eects and external excitations. Although we have considered a nite length uniform Timoshenkobeam model, the matrix formulation can be used with other beam models. The use of piezoelectric materials as both anactuator and a sensor has motivated to incorporate the matrix treatment of multi-span beams. In this work, it is proposedthe extensive use of fundamental matrix responses such as the distributed matrix impulse response of the micro-cantileverfor predicting forced responses and concentrated matrix responses for determining modes and frequencies of the micro-cantilevers. The eigenalysis involved the solution in closed form of a second-order damped dierential equation withmatrix coecients. The case of a supported micro-beam with surface eects can lead to a second spectrum above a criticalfrequency. For the micro-cantilever case, it was observed the size dependence in the natural frequency of Timoshenkoclassical model and Timoshenko model and that surface eects are signicant only in nanoscale. Simulations wereperformed by using the Galerkin method with micro-cantilever eigenfunctions. The shape matrix modes and frequenciesfor a bi-segmented free-free Timoshenko beam were determined. Forced responses of a piezoelectric micro-cantileverbeam where computed when subject to concentrated and pulse harmonic excitations at the free end.8. AcknowledgmentsWe thanks the reviewers for their important comments and suggestions.

20


F (t, x) = col[0 q(t, x)]

Fig 9. Above: transversal deection component w(t, x) due to a concentrated pulse moment q(t, x) = 0.01V (t)(H(x 6L/8) H(x 7L/8)) at thefree end of the micro-cantilever and proles for several times. Below: rotation component (t, x) and proles for several times.

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Modeling the tip-sample interaction in atomic force microscopy with Timoshenko beam theory