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ELECTROMAGNETIC-RESONANCE-ULTRASOUND MICROSCOPY (ERUM) FOR QUANTITATIVE EVALUATION OF LOCALIZED ELASTIC CONSTANTS OF SOLIDS J. Tian, H.Ogi, T. Tada and M. Hirao Graduate School of Engineering Science, Osaka University, Osaka, Japan ABSTRACT: This paper presents a new acoustic-resonance microscopy, electromagnetic- resonance- ultrasound microscopy (ERUM), to evaluate the localized elastic constant of a solid material. It is based on the resonance-frequency shift of a vibrating oriented langasite (La3Ga5SiO14) crystal contacting the specimen via a sphere tip. The vibration of langasite is excited and detected by a dynamic electric field from a sorrounding solenoid coil. Thus, the acoustic coupling between the oscillator and specimen is made only through the tip, isolating the oscillator from any other contacts. (They may otherwise obscure the effect of point contact with specimen and annihilate the quantitative evaluation.) Selection of such a vibration mode that is sensitive to Young's modulus of the specimen is achieved by a calculation using the Rayleigh- Ritz method. On the basis of the calibration for the specimen's contact effective stiffness, the local elastic stiffness is determined from resonance-frequency shifts of the oscillator. As an illustrative example, we measured the elastic-stiffness distribution of an Nb-Ti/Cu resin superconductive wire. We compared our measurements with both static-contact and dynamic- contact models. INTRODUCTION: Recently, multiphase composites have been widely produced as functional and intelligent materials, which often contain microscale and nanoscale components. Mechanical properties of individual components are required because they govern the composite's macroscopic mechanical properties. Especially, elastic constants represent indispensable structural design parameters. Many small-scale components show inhomogeneous microstructure and elastic anisotropy, thus a distribution of elastic constants. For example, silicon-carbide fiber shows a multilayer microstructure, and its elastic constants vary strongly along the radial direction.1 Thus, measurement of elastic constants in a local area within a component assumes a central importance. One candidate measurement method is acoustic microscopy using a high-frequency surface wave.2 However, surface roughness and material's anisotropy make it difficult to determine accurately wave velocities. Also, it needs a couplant such as water. Recently, ultrasonic-atomic-force microscopy has been developed to measure the elastic property in microscale and nanoscale regions.3,4 This method uses flexural vibration of a microscale cantilever. The tip attached at the free end taps the specimen surface with an applied force. The cantilever's resonance-frequency shift is related to the specimen's elastic properties. However, the resonance frequency depends strongly on many factors, including the contacting piezoelectric transducer and gripping condition at the fixed end, preventing one from deducing the elastic constant of the specimen quantitatively. Here, we present an alternative acoustic resonance microscopy, electromagnetic-resonant- ultrasound microscopy (ERUM), to evaluate a material's local stiffness. This method uses the resonance-frequency shift of a rectangular-parallelepiped piezoelectric crystal (langasite, La3Ga5SiO14) touching the specimen only at an anti-nodal point. The piezoelectric material's vibrations are excited and detected through dynamic electric fields using a surrounding solenoid coil. Thus, neither electrode nor mechanical contact is required for the acoustic coupling. Such noncontact excitation and detection of ultrasonic vibrations eliminates the measurement errors associated with contact coupling and ambiguous boundary conditions at the supports. Scanning the object surface with this probe and measuring the resonance frequency then provides an image of elastic-stiffness distribution. We consider the elastic-coefficient mapping of an Nb-Ti/Cu- resin-composite superconductive wire. ELECTROMAGNETIC-RESONANCE-ULTRASOUND MICROSCOPY: Figure 1 shows the measurement setup of ERUM. An oriented rectangular-parallelepiped langasite crystal was located within a solenoid coil, whose crystallographic X-axis (a axis) is oriented vertically. The dimensions of the crystal along the X, Y, and Z axes are 4.954, 5.769, and 4.016 mm, respectively. Langasite possessed trigonal symmetry with point group 32, showing six elastic constants Cij, two piezoelectric coefficients eij, and two dielectric coefficients εij as given in Ref.[5]. Langasite's piezoelectric coefficients (two times larger than those of quartz) promise effective generation of vibration5 by applying a contactless dynamic electric filed. Furthermore, its elastic constants show weak temperature dependence (of the order of 10-5 K-1), assuring stable resonance frequencies.6 Received signal Solenoid coil Specimen Superheterodyne spectrometer Weight Tone bursts Langasite Computer X-Y Stage Fig.1 Because langasite's Young's modulus along the X and Y directions is smaller than that in the Z direction, we kept the X faces parallel to the specimen's surface fixed on a stage to obtain high sensitivity. (As shown below, a smaller oscillator's Young's modulus shows higher sensitivity to the specimen's Young's modulus.) The crystal touches the specimen through a tungsten-carbide spherical bearing with radius R bonded at the center of the bottom surface. A biasing force F0 acts on the upper surface through needles to increase the sensitivity. (The sensitivity increases with increasing biasing force.) To eliminate their influence on vibration, the needles are located on the nodal line of vibration. An oscillating electric field is generated by applying tone bursts to the solenoid coil, which excite vibrations of the crystal by the converse piezoelectric effect.5 The vibrations are received by the same solenoid coil with the piezoelectric effect after the excitation. A superheterodyne spectrometer extracts the in-phase and out-of-phase components of the receiving signal to calculate the amplitude from the root of the sum of their squares.7, 8 Thus, a frequency scan provides a resonance spectrum (Fig. 2). Moving the stage by a stepping motor keeping the crystal/specimen contact produces a two-dimensional image of elastic-stiffness distribution. 3.0 Ag-1 2.5 u nits) 2.0 Ag-3 Am plitude (a rb. 1.5 1.0 0.5 0.00.30 0.35 0.40 Frequency (MHz) Fig. 2 Free-vibration resonance spectrum of a langasite probe measured with electric coupling to a solenoid coil. MODAL ANALYSIS: We consider the sensitivity of the resonance frequency to the specimen's elastic constants. We assume Hertzian contact3,4 for the tip-specimen contact and replace it with elastic springs. The previous ultrasonic atomic force microscopy (UAFM) studies3,4 used a simple static Hertz-contact model to calculate the effective contact stiffness as a G k s * 22 8 = = , k s11 (1a) 3 2 k s = 33 6 RE 0 F *. (1b) Here ( ) ( )12 2 1 1 1 1 * 2 2 ν . E , , and G ν represent Young's modulus, shear modulus, and Poisson's ratio, respectively. Subscripts 1 and 2 indicate the bearing and the specimen, respectively. a= 3 denotes the contact radius. R is the bearing's radius. Recently, we showed experimentally and theoretically that this static model failed to explain the contact stiffness because of larger resistance of the interface deformation to the dynamic response associated with mass density and viscosity of the contacting materials. * G ν ν G and ( ) ( )12 2 2 1 1 2 1 1 * 1 - - + - = G - - - E ν - = E E 1 - - + - 30 4E R F 9 Thus, the dynamic spring constants are usually larger than the static spring constants, although their analytical derivation is much complicated. We therefore approximate the spring stiffness as s = , k11 = k 11 22 α k (2a) k33 k33 β = . (2b) s We consider the Lagrangian L in a piezoelectric material to calculate the resonance frequencies and vibration modes: = dS u u K dV u u ∫∫∫( S C S i j ij + 2 e S ρω - φ ε φ - φ , , 1 1 2 , L (3) Here Sij, ρ, ui, and i φ denote the strain tensor, mass density, displacement, and electric potential, respectively. Ω and Γ denote the volume and the surface of the piezoelectric crystal, respectively. Kij denotes the spring constant per unit area, thus, k11=K11δ(x-x0)dS, for example; δ denotes Kronecker's delta and x0 denotes the position of the point contact. Approximating the displacement and electric potential in terms of linear combinations of the basis functions consisting of normalized Legendre polynomials, the stationary point of the Lagrangian provides the resonance modes (Rayleigh-Ritz method).5, 8, 10 Equation (3) shows that increasing the specimen's elastic stiffness increases the resonance-frequency of the piezoelectric crystal. 2 ij i k ijk ij kl ijkl ij 2 , j i i ) + ∫∫Ω Γ 7 6 5 4 3 2 1 0 Ag-1 1012 1011 1010 109 108 107 106 (% 0 )/ f 0 ( f - f ) Contact stiffness k11 (N/m) Fig. 3 Mode sensitivity to the contact stiffness k11 Free vibrations of an oriented rectangular parallelepiped of langasite fall into four groups labeled Ag, Au, Bg and Bu.10 We adopted the Ag-1 mode for the present study for two reasons: (1) We calculated the sensitivity of many modes in the four groups to the stiffnesses , , and and found that mode A 11 k g-1 shows the largest sensitivity to the normal contact stiffness (Fig.3) but negligible sensitivity to the tangential contact stiffness and k . This is because the normal displacement u k 33 1 of the Ag-1 mode shows an antinode at the center of the bottom surface, and in-plane displacements u2 and u3 show central nodal lines as shown in Fig. 4. The Ag-1 mode shows the vibration similar to a longitudinal vibration along the X axis. The biasing force F0 is applied at three points on the calculated nodal lines of the top surface. (2) The Ag-1 mode is easily excited and shows a good spectrum shape as shown in Fig. 2. Z Y (a) (b) (c) Fig. 4 Distribution of displacements of Ag-1 vibration mode at the upper surface of the probe calculated by the Rayleigh-Ritz method. (a) Displacement u1, (b) Displacement u2 , (c) Displacement u3. Discussion: To illustrate the usefulness of the proposed method, we scanned the cross-section of an Nb-Ti/Cu-polyvinyl-resin superconductive wire11, 12 with a biasing force F0=0.13 N. Figure 5 shows the cross-section observed by optical microscopy. A single wire contains 180 NbTi filaments with a diameter of 32 µm in an oxygen-free copper matrix with 0.58 by 0.68 mm2 cross-section. The NbTi filament consists of 38.7 at.pct. Nb and 61.3 at.pct Ti. Polyvinyl resin coats the wires. Ag-8 Ag-2 k 22 k k 33 22
Fig. 5 Optical microstructure of an Nb-Ti/Cu/resin-composite superconductive wire. Figure 6 shows an ERUM image. The resonance frequency was measured at 5-µm steps. As reported elsewhere,11 the averaged-a-direction Young modulus for the copper matrix is 128.7 GPa and that for the NbTi filament is 84.3 GPa. A line trace of the frequency shift is shown in Fig. 7. We compare the measurements of the resonance frequency shift with those calculated by the dynamic-contact and static-contact models in Fig. 8(Here, f0=0.31894 MHz is free-vibration resonance frequency of langasite). Here we used the averaged Young's modulus of copper and NbTi in our calculation. Scattering of the measurement of the copper matrix and NbTi filaments arises from different grain orientations. Because of its high anisotropy, copper Young's modulus can vary between 66.7 GPa and 190.8 GPa depending on the grain orientation. Also, Young's modulus of NbTi (b.c.c. phase at room temperature) varies between 63.6 GPa and 107.6 GPa.13 Our previous study with a different crystal and different vibration mode 14 revealed that α=3.6 yields a best fit with the measurements. Furthermore, our theoretical study9 with a sphere oscillator predicts α=2.4. The dynamic model in Fig. 8 agrees well with the measurement within the error limits when α=3.3. Thus, it appears that the dynamic-contact stiffness is larger than the static-contact stiffness by a factor 2.4-3.6. The static model often used in UAFM studies gives frequency shifts much smaller than measurements. Thus, Fig. 8 demonstrates the capability of the present method for a quantitative relative measurement of local elastic constants. Figures 6 and 7 show the lowest resonance frequency at the copper-resin interfaces, indicating poor bonds or debondings. Such information is unavailable by optical microscopy. Thus, the ERUM method has the further potential for nondestructive elevation of a material's deterioration. Fig. 6 ERUM image of superconductive wire. Copper matrix NbTi filaments 0.3195 R e sonance frequency f (M H z )0.3194 0.3193 0.3192 0.3191 0.3190 Dedonding 0.3189 550 450 350 250 150 50 distance(µm) Fig. 7 Line trace of the resonance frequency along the arrowheads shown in Fig. 6. 0.000 20 40 60 80 100 120 0.16 Measurements Dynamic contact model (α=3.3) Static contact model Fig. 8 Comparison of measurements with the dynamic-contact and static-contact models. CONCLUSIONS: Based on the vibration of a langasite oscillator excited by a one-point-contact acoustic-resonance technique, we developed a electromagnetic-resonance-ultrasound-microscopy method for measuring relative elastic constants in a local surface region of solids. The isolated langasite crystal provides an accurate resonance-frequency measurement. Selection of the Ag-1 resonance mode provides high-sensitivity and temperature-stable microscopic methodology. A dynamic-contact model is adopted to predict the resonance frequency of the oscillator, which is suitable for a quantitative evaluation of localized modulus. The method revealed elastic-stiffness distribution of Nb-Ti/Cu/resin superconductive wire. It also revealed the weak bonding at the wire-resin interfaces, showing a potential use for nondestructive evaluation of local damage ACKNOLEDGMENT: The first author (J.T.) acknowledges the support provided by the Japan Society for the Promotion of Science (JSPS). Cu 0.12 (% 0.08 NbTi ) 0 )/ f ( f - f 0 0.04 Young's modulus (GPa) REFERENCES: [1] J.Tian, H.Ogi., T. Tada, and M. Hirao, J. Appl. Phys. , 94, 6472 (2003). [2] A. Briggs, Acoustic Microscopy (Clarendon Press, Oxford, 1992). [3] K.Yamanaka, H.Ogiso, and O.Kolosov, Appl. Phys. Lett. 64, 178 (1994). [4] U. Rabe, J. Turner, and W. Arnold, Appl. Phys. A 66, S277 (1998). [5] H. Ogi, N. Nakamura, K. Sato, M. Hirao, and S. Uda, IEEE Trans. Ultrason. Freq. Contr. 50, 553 (2003) . [6] A.Bungo, C. Jian, K. Yamaguchi, Y. Sawada, S. Uda, and Y. Pisarevsky, Jpn. J. Appl. Phys. 87, 7491 (2000). [7] M. Hirao and H. Ogi, EMATs for Science and Industry (Klumer-Academic, Boston,2003). [8] H. Ogi, P. Heyliger, H. Ledbetter and S. Kim, J. Acoust. Soc. Am. 108, 2829 (2000). [9] J. Y. Tian, H. Ogi, and H. Hirao, "Vibration analysis for an elastic-sphere oscillator contacting semi-infinite viscoelastic solids in resonance ultrasound microscopy," J. Appl. Phys., in press. [10] I. Ohno, Phys. Chem. Minerals 17, 371 (1990). [11] H. Ledbetter, J. Moulder, and M. Austin, Wire J. 31,27 (1981). [12] S. Kim , H. Ledbetter, and H. Ogi, J. Appl. Phys. 88, 2378 (2000). [13] C. Reid, J. Routbort, and R. Manynard, J. Appl. Phys. 44,1398 (1973). [14] J. Y. Tian, H. Ogi, T. Tada, and M. Hirao, J. Acoust. Soc. Am. 115, 630 (2004). |
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