L12. Mechanics of Nanostructures: Mechanical Resonance. Outline

Transcription

1 L1. Mechaics of Naostructures: Mechaical Resoace Outlie 1. Theory. Mechaical Resoace xperimets 3. Ruoff group work. Summary SiO aowires Quartz Fibers Crystallie Boro Naowires

2 Part Oe: Mechaical Resoace Method Simple Beam Theory The mechaical resoace testig of aostructures has to date bee based o simple beam theory. z L x h b z y Assumptios o geometry: Log ad thi ( L >>b,h) Loadig is i Z directio (o axial load) Assumptio o deformatio: Plae sectios remai plae ad perpedicular to the mid-plae after deformatio Natural Frequecy f = I ml b β : eigevalue b : elastic modulus I : momet of iertia L : beam legth m : mass per uit legth Morse, P. Vibratio ad soud, d editio, New York ad Lodo, McGraw-Hill,(198)

3 Resoace Test Priciple Accordig to simple beam theory, the atural frequecy of a catilevered circular cross-sectio beam is give by: Natural Frequecy β D b f = β: costat π L 16ρ β0 = D: diameter β L: legth 1 =.69 ρ: desity β = π ρ L b : elastic modulus b = f β = β D Bedig Modulus: The bedig modulus ca be calculated through measuremet of dimesio ad resoace frequecy. lectrical xcitatio The mechaical resoace of a catilevered structure ca be excited by applyig a periodic load whose frequecy equals the atural frequecy of the structure. V ac V dc F(t) = α( V + V = α( V + V dc ) dc + V 1 + V ac ac cosωt) + α ( V + V dc αv cosωt + cosωt ) V ac ac

4 Mechaical xcitatio The mechaical resoace of a catilevered structure ca also be mechaically excited through the vibratio of its substrate. Piezoelectric Actuator The workig frequecy rage of the mechaical excitatio method is here limited by the frequecy respose of the piezoelectric actuator. Part Two: Mechaical Resoace xperimets

5 Mechaical Resoace: Carbo Naotube (top) lastic properties of aotubes (left) lectromechaical vibratio of a MWCNT (A) thermal vibratio (B) Fudametal resoace (C) First overtoe resoace Pocharal,P., et al, Sciece, 83, (1999) Mechaical Resoace: DLC Pillar Schematic of mechaical vibratio experimetal setup SM image of the vibratio Fujita,J. et al, J.Vac.Sci.Techol. B 19(6), (001)

6 Mechaical Resoace: ZO Naobelt Bai et al, App. Phys. Lett. 8(6) (003) Mechaical Resoace: ZO aobelt (co t)

7 Mechaical Resoace: Work Fuctio Measuremet X. D. Bai,. G. Wag, P.X. Gao ad Z. L. Wag, Nao Letters, 3 (003) Parametric Resoace For example: boro aowire is drive electromechaically Sigal icludes DC ad AC compoets xperimetal stage is mouted iside SM for visualizatio

8 lectromechaical Drivig: Mathieu quatio The force due to the electrical field is similar to that o a catilever betwee capacitor plates: The voltage-iduced force o the aowire icreases as the tube beds closer to the electrode: quatio for each mode takes the form of the damped Mathieu equatio Stability of the Mathieu quatio d Y dt Stable dy + µ + y dt ( a + ε cost) = 0 a = ε = 0.05 µ = 0.01 Ustable a = ε = 0.05 µ = 0.01 From: Beder ad Orszag, Advaced Mathematical Methods for Scietists ad gieers. McGraw-Hill, p.56

9 d Y dt Parametric Resoace dy + µ + y dt ( a + ε cost) = 0 Mathieu equatio has ustable solutios for: Resoace is observed at drivig frequecies that give the ustable values of a: Mi-Feg Yu, Gregory J. Wager, Rodey S. Ruoff, Mark J. Dyer, Realizatio of parametric resoaces i a aowire mechaical system with aomaipulatio iside scaig electro microscope, Phys. Rev. B 66, (00). d Y dt Istability Regios for Vibratig Naowire dy + µ + y dt ( a + ε cost) = Small shifts i frequecy or drivig voltage ca cause switch from ustable to stable behavior From: Beder ad Orszag, Advaced Mathematical Methods for Scietists ad gieers. McGraw -Hill, p.56 ε V=1.5 V=.0 V=.5 Theory Mi-Feg Yu, Gregory J. Wager, Rodey S. Ruoff, Mark J. Dyer, Realizatio of parametric resoaces i a aowire mechaical system with aomaipulatio iside scaig electro microscope, Phys. Rev. B 66, (00) a This istability ca be used to sese chages i the eviromet of a vibratig aowire

10 Part Three: Ruoff Group Work xperimet Tool: Naomaipulator Four-degree of freedom (x,y,z liear motio ad rotatio) Two separate stages (X-Y stage, Z-θ stage) Sub-aometer motio resolutio

11 (a) lectrical xcitatio xperimetal Setup Z Stage Couter lectrode Coductive AFM Catilever Piezo Bimorph X-Y Stage V ac V dc (b) Mechaical xcitatio AFM Catilever Naostructure Piezo Bimorph X-Y Stage V ac Mechaical Resoace of SiO Naowire D. A. Diki, X. Che, W. Dig, G. Wager, R. S. Ruoff, Resoace vibratio of amorphous SiO aowires drive by mechaical or electrical field excitatio, Joural of Applied Physics 93, 6 (003).

12 SiO Naowire: Source Ultrasoically dispersed SiO aowire TM image (iserts: High resolutio image ad diffractio patter) Sythesized by Z.W. Pa (J.Am.Chem.Soc. 0) SiO Naowire: Mechaical Resoace lectrical xcitatio Mechaical xcitatio W wire (couter electrode) W wire W wire

13 SiO Naowire: Charge Trappig xperimet µm 51 -beam modes, time betwee scalies: Hitachi S500 TV mode 0.06 ms 3-d mode 17 ms -th mode 50 ms D. A. Diki, X. Che, W. Dig, G. Wager, R. S. Ruoff, Resoace vibratio of amorphous SiO aowires drive by mechaical or electrical field excitatio, Joural of Applied Physics 93, 6 (003). SiO Naowire: Charge Trappig xperimet Cotr lectrode 8pox v (x) d ( a x) a = 3π 3 Ampl. of vibratio, µm a Legth of NW uder loadig, µm D. A. Diki, X. Che, W. Dig, G. Wager, R. S. Ruoff, Resoace vibratio of amorphous SiO aowires drive by mechaical or electrical field excitatio, Joural of Applied Physics 93, 6 (003).

14 SiO Naowire: Bedig Modulus µm lectrical drivig Calculated f i i d = β π L 1ρ β 1 = y x y ( x) = A [si β x sih β x B (cosβ x cosh β x)] A sih β x si βx = (cosh β x + cos β x) B cosh βx + cos βx = sih β x si β x Legth, um Diameter, m Natural, GPa (± 0.) (± 5) frequecy, khz ± ± ± 8. The desity of the SiO is 00 kg/m 3. Mechaical Resoace of Quartz Fiber Xiqi Che, Suli Zhag, Gregory J. Wager, Weiqiag Dig, ad Rodey S. Ruoff, Mechaical resoace of quartz microfibers ad boudary coditio effects, Joural of Applied Physics, 95 (9), 83-88, 003

15 Quartz Fibers Quartz fibers were home-made by pullig a fused quartz rod (G Quartz, Ic) o a wide flame. Typical sample geometry: diameter: µm, legth: 5-10 mm Quartz Fiber: Mechaically Iduced Resoace Optical microscope pictures of the first four modes of resoace of a quartz microfiber. The isets are the theoretical displacemet curves.

16 Quartz Fiber: Correctio for No-uiform Diameter Due to the pullig process, the quartz fiber diameter is ot quite uiform. The resoace frequecy chage for a beam of circular cross-sectio, with liearly varyig diameter, was calculated accordig to the followig equatio: α=(d 1 -D 0 )/D 0 α is geerally small. =0: f = f 0 (1-0. α) =1: f = f 0 ( α) =: f = f 0 ( α) f 0 is the correspodig resoace frequecies of a beam with uiform diameter D 0. D 0 fixed ed D 1 free ed Quartz Fiber: Youg s Modulus # L, mm D 0, um D 1, um f 0, Hz f 1, Hz f 1 /f 0 Corrected 0 Corrected Note: L: legth, D 0 : the clamped side diameter, D 1 : the free ed diameter, f 0 : the fudametal resoace frequecy, f 1 : the first overtoe resoace frequecy.

17 Mechaical Resoace of Crystallie Boro Naowires W. Dig, L. Calabri, X. Che, K. Kohlhaas, R.S. Ruoff, Mechaics of Crystallie Boro Naowires, mauscript i preparatio Boro Naowire: Source SM image of boro aowires o alumia substrate TM image of a boro aowire Otte et al, J.Am. Chem. Soc.,1 (17), 00

18 Boro Naowire: Resoace First two modes of resoace of a catilevered BNW Typical frequecy respose of the 1 st mode resoace Boro Naowire: Legth Determiatio SM images oly give a twodimesioal projectio of the catilevered aowire. It is critical to have accurate legth measuremet: A parallax method was used to recostruct the correct three-dimesioal represetatio of the aowire based o two SM images take from differet agles. b b 6 = L L D f Schematic represetatio of a wire beig partitioed ito N segmets, before (a) ad after (b) rotatio. Huag, Diki, Dig, Qiao, Che, Fridma, Ruoff (00), Joural of Microscopy, 16, 06.

19 Boro Naowire: Legth Determiatio (co t) Top view ad 5 o tilted view of a aowire 3-D recostructio result Boro Naowire: Oxide Layer ( ) ) )( ( ) ( D T D D T D L A I I L A I L f o o B o O B o o B B ρ ρ ρ π β ρ π β ρ π β + + = + = = ) 1 (1 1)) 1 ( (1 1 α α ρ ρ α + + == o beam B o B ρ π β 16 Beam L D f = 6 B beam f D L β ρ π = Without cosiderig oxide layers: Cosiderig oxide layers: T D B O Defie: α=(d-t)/d

20 Boro Naowire: Results Boro Naowire: Curved Wire A curved circular cross-sectio catilevered beam ca vibrate i two perpedicular directios: (1) i plae ad () out of plae. I-plae vibratio Out-of-plae vibratio

21 FA Aalysis The simple beam theory is based o the assumptio that the beam deflectio is due to bedig oly ad that trasverse shear, rotatory iertia, ad axial extesio effects are egligible; for curved beams these assumptios are ot correct. Modal aalysis was performed o several curved catilever aowires with ANSYS. The FA model was based o the 3-D recostructio of the aowire cofiguratio. Legth (µm) Diamete r (m) Frequecy (khz) (out) (i) Modulus assume straight (GPa) Modulus FA modelig (GPa) (i) (i) (out) (i) (i) Summary 1. Mechaical Resoace method based o simple beam theory, ad some work has bee doe with FA. There are two commoly used methods to excite the mechaical resoace of catilevered aostructures: electrical excitatio ad mechaical excitatio. 3. It is critical to have accurate geometry measuremet.. Mechaical resoace method is a odestructive ad effective way to determie the elastic modulus of aostructures; oe aspect deservig careful scrutiy i the future is the low values for the modulus ofte obtaied compared to the bulk material.