Analysis of nanoscale mechanical grasping under ambient conditions

IOP PUBLISHINGJOURNAL OF MICROMECHANICS AND MICROENGINEERINGJ. Micromech. Microeng. 21 (2011) 045009 (12pp) doi:10.1088/0960-1317/21/4/045009Analysis of nanoscale mechanicalgrasping under ambient conditionsHui Xie 1 , Pierre Lambert 2 and Stéphane Régnier 11 Institut des Systèmes Intelligents et Robotique, Université Pierre et Marie Curie/CNRS UMR7222 BC173, 4 Place Jussieu, 75005 Paris, France2 BEAMS Department, Université libre de Bruxelles, CP 165/56, 50 Avenue FD Roosevelt,B-1050 Bruxelles, BelgiumE-mail: xie@isir.upmc.frReceived 11 October 2010, in final form 16 December 2010Published 4 March 2011Online at stacks.iop.org/JMM/21/045009AbstractIn this paper, in order to understand mechanical grasping at the nanoscale, contact mechanicsbetween nanogrippers and nanoobjects is studied. Contact models are introduced to simulateelastic contacts between various profiles of a flat surface, sphere and cylinder for differenttypes of nanoobjects and nanogrippers. Analyses and evaluation instances indicate thatfriction forces, commonly used in macro-grasping to overcome gravity, at the nanoscale areoften insufficient to overcome the relatively strong adhesion forces when picking up thenanoobject deposited on a substrate due to the tiny contact area. For stable nanoscale grasping,nonparallel two-finger grippers with a ‘V’ configuration are demonstrated to have bettergrasping capabilities than parallel grippers. To achieve mechanical nanoscale grasping, ananogripper constructed from two microcantilevers is presented. Experimental results for thepick-and-place manipulation of silicon nanowires validate the theoretical analyses andcapabilities of the proposed nanogripper.(Some figures in this article are in colour only in the electronic version)1. IntroductionGrasping has been widely used to move an object fromone place to another for macro-mechanical manipulationand assembly. Research efforts have been made toscale mechanical grasping down to the microscale withmicrogrippers to integrate functional micro components intomechatronics systems or to perform scientific explorationsin biology [1–6]. Recent work has reported that pickand-placemanipulation at the scale of several micrometershas been achieved with a dual-probe gripper [7]. Somebasic micromanipulation problems attributed to the scaleeffects [8], such as suitable grasping principles and releasetechniques taking adhesion forces into account, have beenidentified. However, the scale effects become more severe atthe nanoscale. In this case, the very tiny size of a nanoobjectto be grasped generates higher requirements for graspingschemes and nanogripper fabricating techniques.In the past two decades, nanomanipulation has usuallybeen restricted to building 2D nanopatterns or in-planenanomaterial characterization through pushing or pullingmanipulation on a single surface with atomic forcemicroscopes (AFM) [9–12]. Nanostructures have, however,been manipulated, assembled and characterized by nanorobotsequipped with manipulators or grippers in scanning electronmicroscopes (SEM) or transmission electron microscopes(TEM) [13–15], in which nanoscale grasping can be performeddue to the vacuum environment and the visual feedback.A liquid medium enables noncontact nanoscale graspingwhere the adhesion forces are greatly reduced, e.g., opticaltweezer [16]. In order to fabricate smart nanogrippers, carbonnanotube (CNT) materials have been used as self-actuatedgripper fingers [17]. However, the alignment of CNT fingersremains a challenge, and the grasping capabilities of theCNT nanotweezer still need further testing and verification,especially for grasping under ambient conditions with thepresence of strong adhesion forces.To achieve mechanical nanoscale grasping, the maindifficulties are fabricating very sharp end-effectors with a sizecomparable to the nanoobject to be manipulated and with0960-1317/11/045009+12$33.00 1 © 2011 IOP Publishing Ltd Printed in the UK & the USA

J. Micromech. Microeng. 21 (2011) 045009 HXieet alenough grasping force output to overcome strong adhesionforces. Moreover, frictional grasping does not work well atthe nanoscale because contact-induced friction forces are oftenless than the relatively large adhesion forces attributable to thesmall contact areas that are favorable for nanoobject release.In order to understand physical interactions duringmechanical nanoscale grasping, based on the Hertz [18], JKR[19], DMT [20] and some other extended theories, variouscontact profiles among a flat surface, sphere and cylinderfor different types of fingers and nanoobjects are analyzed.Grasping capabilities of nanogrippers with single and twofingerconfigurations are discussed and practical designs ofthe nanogripper are proposed. As an example, we presenta prototype of AFM nanomanipulation system, in whichtwo collaborative cantilevers with protruding tips are usedto form a dual-tip nanogripper. In our approach, interactionsbetween the nanogripper and the nanoobject during graspingare analyzed and simulated theoretically. We have used thedeveloped nanogripper to pick-and-place nanowires.2. Contact mechanics of nanoscale grasping2.1. Problems definitionModeling and simulation of contact mechanics between thenanoobject, the gripper and a substrate is necessary toprovide a theoretical estimation of grasping forces, releaseadhesion forces and maximum contact stress for reliablepickup and smooth release operations, as well as protectingthe gripper and the nanoobject from damage. Research on thefollowing aspects associated with nanoscale grasping shouldbe addressed.Adhesion force. The effects of adhesion during a nanoscalegrasping operation are on friction and interfacial wear aswell as a contribution to pickup and release operations, e.g,adhesion forces between a gripper and a nanoobject can beused to counteract substrate adhesion. On the other hand,control and reduction of the adhesion between the gripper andthe nanoobject is essential for the release operation.Nanoscale friction. The empirical da Vinci–Amontons’laws, common knowledge in macroscopic friction as relatedto the outcome of a collective action of a number of asperities,are no longer suitable for nanoscale contact as it is regarded asa single-asperity contact where the friction is dependent on thecontact area [21] and Young’s modulus [22] of the contactinginterface.Contact stress. To protect the brittle gripper and thenanoobject from damage, it is essential to predicate themaximum stress at the contact area and maintain it belowthe contact yield stress. If the adhesion forces, external load,contact area and the stress distribution are known, the contactstress can be accurately estimated.(a)(c)Figure 1. Contact configurations of nanowire/tube and nanoparticle(sphere) grasping with two-finger grippers. In (a) and(c), thenanowire/tube is grasped by grippers with cylindrical andrectangular fingers, respectively. In (b)and(d), the nanoparticle isgrasped by grippers with cylindrical and rectangular fingers,respectively.Grasping. To pick up the nanoobject, the grasping force mustbe greater than the adhesion forces applied on the nanoobjectby the substrate. The grasping force may be comprised ofadhesion forces, friction forces, or other interactive forces.To obtain an adequate and stable grasping force, some basicproblems need to be identified, such as interaction analysis,proper gripper configuration design, and choice of effectivegrasping strategies and techniques.Release. To release the nanoobject in its target position,interactive forces in the releasing direction applied from thegripper should be less than the adhesion forces from thesubstrate. Solutions are needed to reduce the effects of theadhesion forces from the gripper, such as a method to decreasethe contact area, surface modifications, and external actionswith special manipulation schemes.2.2. Contact configurationsFigure 1 shows four configurations of nanoscale graspingconfigurations using two types of two-finger grippers withcylindrical and rectangular fingers. Each configuration usesa nanowire/tube or nanoparticle (sphere). Hence, possiblecontact states between the gripper and the nanoobject beinggrasped can be classified as follows.(i) Contact between two cylinders (C–C), as seen infigure 1(a), where the contact is between a nanowire/tubeand cylindrical fingers.(ii) Contact between a cylinder and a sphere (C–S), as seen infigure 1(b), where a nanoparticle is grasped by cylindricalfingers.(b)(d)2

J. Micromech. Microeng. 21 (2011) 045009 HXieet al(iii) Contact between a flat surface and a cylinder (FS–C).Figures 1(a) and (c) show contacts between a nanowire/tube and the substrate or rectangular fingers.(iv) Contact between a flat surface and a sphere (FS–S), asseen in figures 1(b) and (d) where the contact is betweena nanoparticle and the substrate or rectangular grippers.The contacting states depend on the surface profile ofthe gripper’s fingers and the nanoobject with which they arein contact. Contact mechanics of these four states will bediscussed in detail in the following parts.2.3. Mechanics analyses of different contact profiles2.3.1. Contact mechanics based on the Hertz theory. In ageneral case, i.e. when two surfaces are brought into contactunder a load P, assuming that the contact area is elliptical inshape with semi-axes a and b, the size of the elliptical contactarea S can be calculated from [23]( ) 23PRe3S = πab = π [F1 (e)] 2 (1)4E ∗and the maximum pressure p 0 on the contact area is given byp 0 =3P ( 6PE∗22πab = [Fπ 3 Re2 1 (e)] −2 , (2)where R e is the equivalent radius of the contact area, E ∗are the elastic constants of the contact interface, and F 1 (e)is deduced from complete elliptic integrals of the argumente = (1 − b2) 1 a 2 2 ,b

J. Micromech. Microeng. 21 (2011) 045009 HXieet al(a)(b)Figure 2. Nanoscale grasping with the single-finger gripper.(a) ‘Tip grasping’ method. (b) ‘Side grasping’ method.2.4. Friction forces at the contact interfaceThe well-known Amontons’ law shows that a friction forceis proportional to a normal force applied on two contactingsurfaces with many asperities. However, as the contact areareduces to the nanoscale, the friction with a single asperitycontact is more suitably described by [29]F f = τS (14)where τ is a effective friction coefficient, and S is the contactarea that is associated with the sum of the external forceand the adhesion force. τ is related to the effective shearstress of the contact interface: τ ≈ 0.0345 G ∗ [30], whereG ∗ = ( 2−ν 1G 1+ 2−ν )2 −1,G 2ν and G are the Poisson ratio value andshear strength of each of the contacting surfaces, respectively.3. Nanoscale grasping with different grippersGrippers with one or two fingers are widely proposed inmicro/nano-applications. However, grippers with more thantwo fingers may meet difficulties in finger mounting withina limit space, and in simultaneously aligning several fingerswith nanoscale accuracy, as well as in releasing nanoobjectfrom excessive adhesive contacts. Thus, single- and two-fingergrippers are discussed.3.1. Grasping with the single-finger gripperFigure 2 shows two grasping schemes with a single-fingergripper, namely ‘tip grasping’ with fingertip contact and ‘sidegrasping’ using the finger side.3.1.1. ‘Tip grasping’. As shown in figure 2(a), in orderto grasp and release a nanoobject with a single finger, thefollowing inequalities should hold:{tFa max > s F a , graspt F a < s (15)F a , releasewhere the adhesion forces t F a and s F a , respectively, comefrom the gripper and the substrate. At present, it is difficultto obtain these inequalities since the contact area of the tip–nanoobject contact is often less than that of the nanoobject–substrate contact.Grasping at the atomic scale has been demonstrated usingthe scanning probe microscope (SPM) tip with external energyinduced by electric field trapping [31], tunneling currentinducedheating or inelastic tunneling vibration [32]. Untilrecently, nanoscale grasping using a single AFM tip hasbeen achieved with electro-enhanced capillary forces [33].However, pick-and-place at the nanoscale is still not wellresolved with the ‘tip grasping’ method. Difficulties in thiscase are weak grasping force outputs while controlling overthe applied forces, as well as release accuracy since thesingle finger has no sufficient geometric limits on the graspingoperation.3.1.2. ‘Side grasping’. Figure 2(b) shows the ‘side grasping’method using the following inequalities:{ tFf max > s F a , graspt F f < s (16)F a , releasewhere t F f is the friction force derived from the adhesion forcet F a from the finger side, and s F a is the adhesion force appliedon the nanoobject from the substrate.It is obvious that this method aims to increase t F a throughcontact with the side rather than contact with the tip.t F ashould generally be much larger than s F a to produce enoughfriction force to satisfy the inequalities. In this case, therelease seems more difficult than with the ‘tip grasping.’ The‘side grasping’ scheme is usually used for nanowire/tubegrasping in the SEM, which allows many grasp or releaseattempts using visual feedback. However, ‘side grasping’ hassimilar difficulties to those for the ‘tip grasping’ method, andnanoobject release is to some extent harder.3.2. Nanoscale grasping with two-finger grippersFigure 3 shows nanoscale grasping with two-finger grippers.Grippers with parallel- and nonparallel-finger configurationsare considered.3.2.1. Parallel fingers. Figure 3(a) shows a gripper that hastwo parallel fingers with a tip radius of r. To vertically pickup the nanoobject with a radius R, the following inequalitiesshould hold:R>r andt Ff max > 1 s F2 a . (17)During the grasping operation, the friction force t F f can beadjusted by increasing or decreasing the repulsive force t F p bydriving single or dual fingers. Similarly, for a stable verticalrelease of the nanoobject, the grasping force can be reducedto generate a certain value of t Ff 0 that makes the nanoobjecteasily stick on the substrate with the following inequalities:t Ff 0 < 1 s F2 a andt Ff 0 t Ff ad (18)where t Ffad is the adhesive friction force derived from t F awithout any external clamping forces.4

J. Micromech. Microeng. 21 (2011) 045009 HXieet al(a)Figure 3. Nanoscale grasping with two-finger grippers. (a)Thegripper with parallel fingers. (b) The nonparallel gripper with a ‘V’configuration.MEMS grippers have been used to demonstrate verticallyaligned carbon nanotube pick-up in the SEM [14]. However,this type of MEMS grippers might meet difficulties whengrasping nanotubes attached horizontally to a substrate due totheir coarse fingertips, as well as the small gripper–nanotubecontact area, which cannot provide a sufficient grasping force(friction forces). This fact is explored in detail in the exampleof a CNT nanotweezer evaluation given in section 4.3.2.2. Nonparallel fingers. Fingers with a nonparallelalignment with ‘V’ configuration shown in figure 3(b) aredesigned to increase the grasping capability. For nonparallelfingers, the following inequalities should hold to successfullypick up the nanoobject:R>r and F g = (t Ff max cos φ + t F p sin φ ) > 1 s F2 a .(19)Obviously, the tilted angle φ makes the grasping stronger as aresult of the combined effect of the clamping forces t F p andthe friction forces t F f to overcome the adhesion forces s F a .3.2.3. Release schemes. The release operation of the twofingergripper is more complicated than the single-fingergripper. Particularly a two-finger gripper with rectangularfingers is used, which produces the comparable adhesionforces t F a and s F a . In contrast, cylindrical fingers mayproduce smaller adhesion forces, resulting in easier nanoobjectrelease.As shown in figure 4(a), in the first step of the release, theright finger moves to the right to separate from the nanoobject.In this case, the two opposing adhesion forces from the fingerscancel each other out and the friction force s F f is used to holdthe nanoobject. When the right finger has been separated fromthe nanoobject, the gripper moves upward in figure 4(b)fortherelease operation with the following conditions: t F f < s F a .Note that in this last step, the nanoobject may roll with a smallangle due to t M r derived from t F f . However, apart from therelease accuracy, this scheme may be effective for the twofingergripper release operation.(b)(a)Figure 4. Release operation of the two-finger gripper.(a) Horizontally open the gripper by driving the right finger.(b) Vertically release the nanoobject.Table 1. Interfacial contact parameters of materials used insimulation and experiments.(b)Silicon SiO 2 GoldSurface energy (Jm −2 ) γ 1.4 0.16 1.5Young’s modulus (GPa) E 160 73 79.5Poisson’s ratio ν 0.17 0.165 0.423.3. Grasping capability comparisons of two-finger grippers3.3.1. Grasping force. To compare grasping forces ofthe parallel- and nonparallel-finger gripper, an example ofnanowire grasping in air with a cylindrical gripper is simulated.The nanowire is horizontally deposited on a substrate andvertical grasping makes an orthogonal contact between thegripper and the nanowire. Suppose that the gripper and thenanowire are made of silicon with a very thin SiO 2 coatingand with the same radius of 50 nm. t F f can be estimated by(6), (4) and (14) with mechanics parameters of Si and γ SiO2 =0.16 J m −2 described in table 1. For t F a calculation, the workof adhesion γ at the contact interface is calculated by theDupré equation [34]γ = γ 1 + γ 2 − γ 12 (20)where γ 1 and γ 2 are, respectively, the surface energies of thecontact surface, and γ 12 is the interface energy. For two solidcontact surfaces, γ = 2 √ γ 1 γ 2 . The surface energy of thewater layer on the contact interface γ H2 O = 0.073 J m −2 .The simulated grasping force F g is plotted in figure 5 asa function of the angle φ and the clamping force t F p . Theresult shows that the fingers with a smaller tilted angle tend toproduce a greater grasping force. For example, as seen in pointA, the parallel gripper, with a clamping force t F p = 2897 nN,generates p 0 = 12 GPa, a yield stress of Si at the nanoscale(12–13 GPa) [29]. In comparison, the nonparallel gripper witha tilted angle of 70 ◦ produces eight times more grasping forcethan the parallel gripper with the same clamping force, as seenin point B. In addition, the grasping force from the parallelgripper is still below the pickup critical force that is normallyabout several hundreds of nanonewton.5

tJ. Micromech. Microeng. 21 (2011) 045009 HXieet al6050FS-S contactC-S contactF f[nN]403020100r = 160 nmr = 120 nmr = 80 nmr = 40 nmr = 20 nmFigure 5. Simulated grasping forces F g with different clampingangles.0 40 80 120 160 200R [nm]Figure 6. Simulated adhesive friction forces of the rectangular andthe cylindrical grippers.3.3.2. Releasing capabilities. An example of grasping agold nanoparticle R = 50 nm deposited on a Si substrate witha silicon rectangular gripper is presented. We can calculates F a = t F a = 243 nN using (13) (λ = 3.38) and adhesivefriction forces s F f = t F f = 84 nN using (14) with interfacialcontact parameters of Si and gold described in table 1. Thecalculated results show that the gold nanoparticle can besuccessfully released by the rectangular gripper in air. It can beinferred that nanowire release using this scheme is easier thannanoparticle release because of the stronger adhesion force ofthe nanowire–substrate contact, as calculated from (8).Theoretical calculation verifies that the rectangulargripper can release the nanoparticle. However, the comparablemagnitudes of s F a and t F f introduce uncertainties into therelease process, e.g., the nanoparticle skips from the substrateand sticks to the side of the gripper finger even with weakinterferences. From (14), values simulated for the adhesivefriction forces t F f with the rectangular and cylindrical silicongrippers are plotted in figure 6 as a function of the goldnanoparticle with different radii R. It shows that t F f will begreatly reduced with a cylindrical finger (r in radius) underthe same conditions. Moreover, t F f can be further reduced byusing thinner cylindrical grippers.3.4. In summary: fabricating a practical nanogripperAs discussed above, the following items are recommended tofabricate a practical two-finger nanogripper.(i) Cylindrical fingers (with a circular or elliptical section)with nanoscale radii are proposed for generating lowadhesion forces between the grippers and the nanoobjectfor reliable release.(ii) Enough clamping stiffness of the gripper is required toproduce sufficient clamping forces.(iii) Nonparallel configuration of the finger alignment isrecommended for producing more grasping forces thanthe parallel alignment.Figure 7. Nanoscale grasping with the CNT nanotweezer.4. Evaluations of parallel and nonparallel grippers4.1. Parallel CNT nanotweezerFor further understanding the grasping capability of theparallel-finger gripper, the well-known CNT nanotweezer istaken as an example for grasping a gold nanoparticle depositedon a substrate under ambient environment conditions. For theconfiguration shown in figure 7, the dimensions of the CNTnanotweezer are r = 22.5 nm and L CNT = 5 μm, and theparameters of the contact interface are shown in table 1. Anatural distance between two CNT fingers is d 0 = 100 nm, soa gold nanoparticle radius R = d 0 /2 = 50 nm is adopted.To process the hyperstatic electrode–finger–nanoparticlesystem, as shown in the top inset of figure 7, a force ofconstraint o Fp 1 is applied on the upper finger that makes thedisplacement at the end of the CNT finger equal to zero.Assuming that the voltage V applied between the two CNTfingers produces a clamping force q per unit length, o Fp 1 iscalculated aso F 1 p = 3 8 qL CNT. (21)6

J. Micromech. Microeng. 21 (2011) 045009 HXieet al6030050 d max=d 0250Clamping force [nN]403020100p 0=8.6 GpStiff caseSoft case2001501005006320 30 40 50 60 70Friction force [nN]Figure 8. Frictional grasping force of the CNT nanotweezer.Due to a complicated deflection expression, d max isassumed to be obtained at 5L CNT /8, which is given byd max =615qL4 CNT(22)32768E CNT Iwhere E CNT is Young’s modulus of the CNT, I is the momentof inertia of the finger given by[ (I = πr4 rin) ] 41 − = πr 4 /4 (r ≫ r in ) (23)4 rwhere r in and r are, respectively, the internal and outer radiiof the CNT finger.In ambient conditions, s F a = 58.4 nN and the adhesivefriction t F f = t Ff 1 + t Ff 2 = 20.7 nN are computed,respectively, by (1) and (14) with γ CNT = 0.24 J m −2 , E CNT =1 TPa and ν CNT = 0.165. The result shows that the graspinginequality is unsatisfied with only the adhesive friction forces.However, the clamping stiffness of the CNT nanotweezerk CNT < 0.01 N m −1 , which is too soft to produce sufficientgrasping forces to pick up the gold nanoparticle, as seen fromthe blue line in figure 8.Even assuming k CNT →∞to produce enough clampingforces, e.g., t Fp 1 = t Fp 2 = 234 nN for producing a frictionalgrasping force t F f = t Ff 1 + t Ff 2 = 58.4nNtosuits F a , as seenfrom the red line in figure 8, the maximum contact stress p 0 =8.6 GPa exceeds the yield stress of gold at the nanoscale(around 5–6 GPa at 50 nm due to scale effects [35]) and maydamage the gold nanoparticle. Again, this example proves thatthe main difficulty for nanoscale grasping is the fabrication ofsharp end-effectors with enough grasping force output.4.2. Nonparallel dual-tip nanogripper4.2.1. Nanogripper configuration. Figure 9 shows thesetup of a dual-tip nanogripper that is comprised of twoindividually actuated AFM cantilevers with a protruding tip(see the inset, which are tilted at an angle of 63 ◦ betweenthe front side of tip and the cantilever beam). Nanoscalegrasping with the proposed nanogripper benefits from itsFigure 9. Nonparallel configuration of the dual-tip nanogripper.very tiny tip and a nonparallel ‘V’ configuration, as wellas functions of image scanning and force sensing. Detaileddescriptions of the system setup and manipulation protocolscan be seen in our previous research [36]. In this work,as supplementary contents, mechanics analyses, quantitativecalculation of grasping limit improvement and interactiveforce estimation are detailed based on the contact mechanicsresearch addressed above.4.2.2. Grasping mechanics analysis. Figure 10 shows aschematic diagram of the analysis of the mechanics of acantilever used as a gripper finger. In experiments, thecantilever’s beam length L, beam width w and tip length l aremeasured under an optical microscope. The beam thickness t isdetermined using the forced oscillation method [37]. Thus, thenormal stiffness kb n and the lateral stiffness kl bof the cantilever’sbeam are calculated bykb n = Ewt34L , 3 kl b = Gwt 3(24)3L(l sin φ ′ ) 2where E and G are, respectively, the Young and shear modulusof the cantilever, and φ ′ = 60 ◦ is the tilted angle through therotation axis of the tip relative to the substrate.When a force F is applied at the end of the cantilever’stip, it moves with the displacements δ x , δ y and δ z that dependon the stiffness of the cantilever on each axis in the definedframe. As seen in insets (I) and (II) of figure 10, the decoupleddisplacement on each axis is calculated by[ ] [ ] ⎡ d F xδx sin α sin φ′l sin φ ′ b+ d F ⎤zb⎢=δ ⎣z cos α cos φ ′ l cos φ ′ d F xt + d F z⎥t ⎦ , (25)θ F xb+ θ F zb( 1δ y = F ykbl + 1 )(26)k twhere α is the mounting angle of the cantilever, θ is the angulardeflection of the beam d is the deflection of the cantilever’sbeam and tip (respectively labeled by subscripts b and t, andwith superscripts of forces F x and F z ), and k t is the stiffnessof the cantilever’s tip. The calculation of the deflections isdetailed in appendix B. The tip close to its very end is7

J. Micromech. Microeng. 21 (2011) 045009 HXieet alFigure 11. Force diagram using the nonparallel dual-tipnanogripper.Figure 10. Analysis of the mechanics of the gripper finger (AFMcantilever) during a grasping operation.assumed to be symmetrical in shape; thus, it has the samestiffness on each axis: k t ≈ 20 N m −1 simulated by the finiteelement method. From (24), the overall lateral stiffness of thecantilever is about 19 N m −1 , which makes the tip alignmentmore stable during the grasping operation.4.2.3. Grasping capabilities. Figure 11 shows a forcesimulation for a nanoobject (with a circular section,e.g., nanowires) grasping operation using the proposednanogripper. Equations can be obtained for a staticequilibrium:{Fx = ( t F p sin φ − t F f cos φ)F z = ( t F p cos φ + t F f sin φ) = 1 s (27)F2 awhere φ = 68 ◦ with a mounting angle 5 ◦ of the cantilever.t F f can be calculated by (4) and (14):[ 3( t t F p + t ] 2F a )R3 eF f = τπ [F1 (e)] 2 (28)4E ∗where t F a = o F a , the adhesion force calculated by (3); t F p ,the repulsive force from tips, is significant for estimating themaximum stress on the contacting area using (5) to avoiddamage to both the nanogripper and the nanoobject.The normal force F z and the lateral force F l applied onthe end of the tip can be calculated from the voltage outputsV n and V l of optical levers byk n b δ n = k n b(dF xb+ d F zb)= Vn S n , (29)F y = V l S l (30)where δ n is the normal displacement on the end of the cantileverbeam, S n and S l are, respectively, the normal and lateralsensitivities of the optical lever. When V n is detected andt F a is determined, F x and F z can then be calculated by (27).Figure 12. Simulation of the grasping limit on the size of thenanoobject.As shown in figure 12, for a successful grasping, the angleη as well as the dig-in distance ξ should be positive. Theirrelation is given by⎡ √⎤(R − r)2ξ = R(1 − cos η) = R ⎣1 − 1 − ⎦(R + r) 2 . (31)During the pickup, the beam and tip deflections cause asmaller ξ ′ , and the grasp is probably lost as ξ is reduced tozero. Thus, the minimum radius of the nanoobject R min canbe estimated by assuming that δx max ξ with known s F a ,parameters of the nanogripper and the contacting interface,e.g., Young’s modulus, shear modulus and surface energy.R min can be estimated from the following procedure.(i) Calculate the adhesion forces s F a and t F a withapreestimatedsize limit R 0 of the nanoobject; then calculatethe corresponding t F a and t F f using (27) and (28).(ii) When F x and F z are calculated by (27), R min can thenbe estimated using (25) and (31). Then calculate a newestimation R 1 = (R 0 + R min )/2 and repeat the steps withR 1 until the difference between the estimation R 1 and thecalculated R min in successive iterations is less than thepredefined R = 0.5 nm.8

J. Micromech. Microeng. 21 (2011) 045009 HXieet alR min[nm]36343230282624222034.3 nm28.9 nm21.5 nmincrease k n bincrease k t0 50 100 150 200 250 300Stiffness [N/m](a)k n (k = 20 N/m)b tk tk n b = 4 N/m)R min[nm]36322824201612834.3 nm17.6 nm8 nmincrease t F p0 20 40 60 80 100 120Preload t F p[nN](b)Side contactTip contactFigure 13. Improving grasping limit by (a) increasing stiffness of the cantilever beam k n b or tip k t and (b) t F p preloading.Figure 14. Pre-scanned image of the SiNW. The insets show a 3D topographic image of the tip II and the nanowire height at thelocation A–A.Table 2. Dual-tip nanogripper parameters.k n b k t L l r α s F a4Nm −1 20 N m −1 250 μm 10 μm 8nm 5 ◦ 100 nNGrasping limit can be improved by increasing kb n and k t.As seen in figure 13(a), with parameters described in table 2( s F a is given), the simulated results indicate that a stiffertip is more effective than a stiffer beam for reducing R min .The former provides a lower limit of 21.5 nm and the later28.9 nm.In addition, if a preload of t F p is applied before grasping,the nanogripper will hold the SiNW more tightly with astronger t F f , thereby significantly reducing R min . As shownin figure 13(b), the grasping limit can theoretically equal theradius of the tip apex with a proper preload. However, whenthe radius of the nanoobject decreases to 17.6 nm, it becomesdifficult to grasp because of the contact with the tip apex(sphere–sphere contact). Thus, with this means, R min reachesthe limit of 17.6 nm, while t F p = 83 nN. However, thepreloading involves great risks in damaging the tips as wellas the nanoobjects. In this case, the maximum stress p 0 on thecontact area should be guaranteed less than the yield stress ofthe contact.4.2.4. Silicon nanowires grasping and assembly. Inexperiments, silicon nanowires (SiNWs) were deposited ona freshly cleaned silicon wafer. A pre-scanned image is shownin figure 14, which includes the topographic image of SiNWs,and the local image of tip II (see the zoomed inset). A graspinglocation on the left SiNW is marked A–A, where the SiNWhas a height of 153 nm. The left SiNW will be transportedto the target position where it will be released onto the rightSiNW to build a nanocrossbar.Figure 15 shows an example of contact detection with tipII: seen as icons in the graph, the cantilever is bent upwardresulting in positive forces about 5 nN when it snaps in thenanowire. As the tip makes contact with the SiNW, it starts todig into the root of the SiNW and further movement does notlead to obvious change. During the retraction, the adhesionforces between the tip and the substrate induce a suddendecrease in the bending force to about 48 nN. After the contactbreaks with the substrate, the bending force reaches a positivepeak before the tip pulls off the nanowire. When the tip digsinto the SiNW, the cantilever produces a pre-grasping forceF 1 = 27 nN calculated from a voltage difference of about20 mV and the normal force conversion factor of 1.36 on tipII. The corresponding pre-load on the tip II is estimated ast F p = 26 nN using (27).9

J. Micromech. Microeng. 21 (2011) 045009 HXieet alFigure 15. Contact detection by normal force sensing on tip II.Figure 16 shows the curve of the peeling forcespectroscopy on one of the tips during the pick-and-placemanipulation of the same SiNW. The curve starts fromthe contact state between the nanogripper, the SiNW andthe substrate. As the nanogripper is moved to pick up theSiNW, the cantilever is bent downward creating negativeforces until the cantilever pulls off the substrate with a voltagedifference of 75 mV indicating a pull-off force F 2 = 103 nN.As the nanogripper is moved up further, the force magnitudegradually keeps increasing with the SiNW peeling forceresponses. Retraction leads to a continuous decrease exceptfor a weak fluctuation at 178 nm. Snap-in occurs at 25 nmafter a mild force decrease. With even further retraction, themagnitude of the normal force approaches the prior state beforegrasping.The maximum peeling force occurs at retraction start,where the voltage is about −105 mV indicating a graspingforce of F 3 = 144 nN. At this point, from calculation,t F f = 31 nN that is much smaller than the SiNW–substrateadhesion force, and t F p = 218 nN that generates a maximumcontact stress p 0 = 7.1 GPa, with R ≈ 19.5 nm at the contactFigure 16. Force detection on one of the tips during the graspingand release operation.location of 55 nm from the tip end. Fortunately, this contactstress is still below the yield stress of silicon [29].The post-manipulation image in figure 17 verifies that theSiNW has been successfully transported and piled on anotherSiNW, building a nanocrossbar with a maximum height ofabout 500 nm. During the pick-and-place manipulation, oncethe SiNW was reliably grasped, the nanogripper moved up800 nm at a velocity of 80 nm s −1 ; then the SiNW wastransported a distance of 4.05 μm ontheX-axis at a velocityof 150 nm s −1 and 1.95 μm ontheY-axis at a velocity of72 nm s −1 .The SiNWs can be successfully grasped by the proposednanogripper. However, failure sometimes occurred whengrasping positions were located at the middle part of theSiNWs, which jumped from the substrate and then adhered tothe nanogripper during the pickup or transportation process.To avoid such failures, grasping locations are stronglyrecommended at the end of the nanowires/tubes. In addition,the nanowires/tubes should keep in contact with the substratethroughout the pick-and-place process.(a)(b)Figure 17. Pick-and-place manipulation results for the SiNWs. (a) A post-manipulation image verifies that the manipulated SiNW is piledon another SiNW. (b) 3D topographic image of the manipulation result.10

J. Micromech. Microeng. 21 (2011) 045009 HXieet al5. ConclusionTo understand the interactive phenomena between ananogripper and a nanoobject, contact mechanics weremodeled for different contact profiles. Contact modeling madeit easy to estimate the interfacial adhesion forces, deducecontact friction forces and contact stress, thereby providinga theoretical analysis for the gripper design. To furtherimprove our understanding, grasping strategies with twofingergrippers were discussed. The analysis shows that thegripper with a nonparallel configuration has better graspingcapabilities than the parallel configuration. A homemadenanogripper constructed from two AFM cantilevers with a ‘V’configuration was introduced. The grasping capabilities of theproposed nanogripper were analyzed in detail and ways forimproving the grasping limitation were presented. Contactmechanics between the tip and the silicon nanowire (thecylinder–cylinder contact configuration) has been analyzedwith the modified Hertz model, with which forces appliedon the contact area have been estimated, and results showthat silicon nanowires can be nondestructively graspedby the proposed dual-tip nanogripper. Subsequently, thenanogripper’s capabilities were validated by a successfulpick-and-place manipulation of silicon nanowires to build ananocrossbar. As a result, the theoretical analyses and theexperimental results validate the nanoscale grasping schemesand methods with two-finger grippers.AcknowledgmentsThis work was supported in part by the ANR (French ResearchAgency) through the NANOROL Project under ANR grantno PSIROB07-184846. The authors would like to thankProfessor Sinan Haliyo and Sébastien Alvo for their valuablediscussions.Appendix A. List of selected symbolst F fs F ft F as F ao F aαφφ ′ψηξkbnk tS nS lfriction force on a nanoobject from a tipfriction force on a nanoobject from a substrateadhesion force on a nanoobject from a tipadhesion force on a nanoobject from a substrateadhesion force on a gripper from a nanoobjectcantilever mounting angletilted angle of a tip through its front edgetilted angle of a tip through its rotation axisinclined angle between axes of contact surfaceseffective grasping angle relative to the substrateeffective grasping distancenormal stiffness of a cantilever’s beamstiffness of the cantilever’s tipnormal sensitivity of a optical leverlateral sensitivity of a optical leverAppendix B. Deflections on the cantileverDeflections on the cantilever’s beam and tip can be calculatedbyd F xb= F )x 3l sin φ′(sinkbn α + (B.1)2Ld F zbθ F xbθ F zb= F z(coskbn α +d F xt = F x sin φ ′k td F zt = F z cos φ ′k t= 3F (x sin αkb nL +2= 3F (z cos αkb nL +2)3l cos φ′2L)l sin φ′L(B.2)(B.3)(B.4)(B.5))l cos φ′. (B.6)LSymbolrRSp 0LwtlEGE ∗τPF sγt F po F pDescriptionradius of a tip apexradius of a nanoobject being manipulatedcontact areamaximum pressure on a contact areacantilever beam lengthcantilever beam widthcantilever beam thicknesscantilever tip lengthYoung’s modulusshear moduluscombined elastic moduluseffective friction coefficientexternal loadadhesion forcework of adhesionclamping force of a tiprepulsive force on a gripper from a nanoobject11References[1] Driesen W, Varidel T, Régnier S and Breguet J M 2005Micromanipulation by adhesion with two collaboratingmobile micro robots J. Micromech. Microeng. 15 S259–67[2] Xie H, Rong W B and Sun L N 2007 A flexible experimentalsystem for complex microassembly under microscale forceand vision-based control Int. J. Optomechatronics 1 80–102[3] Walle B L, Gauthier M and Chaillet N 2008 Principle of asubmerged freeze gripper for microassembly IEEE Trans.Robot. 24 897–902[4] Kim K Y, Liu X Y, Zhang Y and Sun Y 2008 Nanonewtonforce-controlled manipulation of biological cells using amonolithic MEMS microgripper with two-axis forcefeedback J. Micromech. Microeng. 18 055013[5] Neild A P, Oberti S, Beyeler F, Dual J and Nelson B J 2006 Amicro-particle positioning technique combining anultrasonic manipulator and microgripper J. Micromech.Microeng. 16 1562–70[6] Zhang Y, Chen B K, Liu X Y and Sun Y 2010 Autonomousrobotic pick-and-place of micro objects IEEE Trans. Robot.26 200–7

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