Acoustical Near-Field Imaging
Exploitation of Cantilever Contact-Resonances in AFM for Ultrasonic Imaging
- Fig. 1: Mechanical model of the AFM cantilever as a rectangular elastic beam. The contact itself is described as two parallel linear springs with spring constants or contact stiffnesses kr and kr,lat, and two damping dashpots γ and γlat for both vertical and lateral contact forces. For details see reference .
- Fig. 2: Modulus and damping maps of the Ti-6Al-4V alloy heat treated at (a and b) 1223 K, (c and d) 1123 K, and (e and f) 923 K for 1 hour, followed by water quenching, respectively. The three phases α, β, and α’ possess different indentation moduli as well as damping. A detailed discussion of the underlying material science can be found in “M. Kalyan Phani, A. Kumar, W. Arnold, and K. Samwer: Elastic stiffness and damping measurements in titanium alloys using atomic force acoustic microscopy, Journal of Alloys and Compounds 676, 397 (2016) doi 10.1016/j.jallcom.2016.03.155” (reproduced with permission).
Ultrasound is combined with atomic force microscopy to achieve the lateral resolution of scanning probe techniques for ultrasonic imaging and quantitative local elasticity and damping measurements by exploiting the contact resonances of the AFM cantilevers. They are in the range from 10 kHz to several MHz. Images can be obtained with the contrast depending on the local indentation modulus or local damping. The lateral resolution is defined by the tip-sample contact radius ac which can be a few nanometers.
In Scanning Acoustic Microscopy (SAM) a focusing lens is used for imaging. Its resolution is directly related to the used wavelength . The practical limit due to attenuation in the coupling water determines the shortest wavelength employable at room temperature, λ ≈ 0.75 μm, corresponding to a frequency of 2 GHz. The resolution limit may be overcome by using the concept of “super-resolution” or near-field imaging. As the name indicates, the contrast originates in the near-field of the antenna employed.
One of the first near-field acoustical systems for non-destructive testing in the aeronautics is the so-called Fokker bond tester or mechanical impedance spectrometer . Such a system possesses an exciting oscillator and a receiving oscillator. When contacting a component, one measures with a bond tester the shift in resonance frequencies of the oscillator which depend on the changes of the elastic and adhesive properties of layers within the stress-field of the contactor, usually a sphere attached to the exciter . The quantity that is measured is the contact stiffness. Combined with a scanning system, the Fokker bond tester becomes a near-field acoustic imaging system whose spatial resolution is given by the contact radius of the contacting sphere. The exciter oscillates in the kHz range and the corresponding wavelengths are in the decimeter range. Contact radii are in the millimeter range hence we have a resolution of λ/100, that is we have “super-resolution”.
AFM Contact-Resonance Imaging
This principle can be applied to atomic force microscopy as well.
In AFM contact-resonance imaging, one measures the resonances of atomic force cantilevers with the tip contacting the specimen surface [4, 5, 6]. From such measurements one can derive the local indentation modulus M, using a suitable mechanical model relating the resonance frequencies of the oscillating cantilever to the tip-sample contact stiffness kr, which in turn is related to the local indentation modulus M. The indentation modulus is an elastic constant that accounts for the compressive and the shear deformations in the contact zone between isotropic or anisotropic materials . Cantilever contact-resonances can be excited either by transmitting ultrasound through the sample (AFAM technique) or the component under test, or by exciting the cantilever base (UAFM technique), of by exciting the cantilever itself. All these techniques have advantages and disadvantages and are discussed in detail elsewhere .
The equation of motion for flexural vibrations of a cantilever beam is a fourth-order differential equation . The interaction forces normal to the sample surface are represented as a linear spring with a characteristic spring constant kr, which is the negative derivative of the tip-sample force in the equilibrium position:
where z is the tip-sample distance, F(z) is the tip-sample interaction force, and ze is the equilibrium position. The lateral contact stiffness klat is defined analogously. The ensuing mechanical model of a vibrating cantilever is presented in figure 1. The superposition of vertical and lateral tip-sample interaction forces are represented by a set of springs and dashpots. Taking into account the boundary conditions for a free cantilever, one obtains the characteristic equation for which the solutions κnL (n = 1, 2, 3….) yield the wave numbers κn of an infinite set of flexural modes. The dispersion relations for the cantilever motion as a distributed mass system are used to calculate the resonance frequency fn of the nth mode. The boundary conditions change when the tip is in contact with the sample surface . When approaching the sample surface, the tip first senses long-range attractive forces before contact and in contact repulsive tip-sample forces. If the static load Fo = kr×dc, applied to the tip by a cantilever deflection dc, is large enough, attractive forces can be neglected.
The contact stiffness k for the case of an elastic contact between a spherical tip and a flat surface is given by :
kr = 2acE , (2)
where ac is the contact radius. For isotropic solids holds. Here, Es,tip are the Young’s modulus of the sample and of the tip, respectively, and νs,tip are the corresponding Poisson ratios. In the case of anisotropic solids an indentation modulus M is introduced. In this case where MS and Mtip are the indentation moduli of the sample and the tip, respectively. Because tip radii of AFM cantilever tips are of the order of some 10 nm, the contact radius can be as small as a few nanometers. This is the base of super-resolution for in AFM CR-imaging. The spatial resolution is given by the contact radius ac which depends on the tip radius R, the applied static force F0 acting on the tip, including the forces of adhesion F0, and the reduced elastic modulus E of the contact. In the case of a Hertzian contact with a spherical tip, this value is given by:
In order to obtain absolute CR- data, one has to go through several steps to invert the resonance frequency data into indentation modulus data, which entails corresponding inaccuracies as in all inverse problems .
Contact-resonance imaging has been used to measure the stiffness distribution on the metallic glass PdCuSi caused by its energy landscape related to the structural disorder . The contact resonances showed a much wider distribution of their frequencies than in the crystallized PdCuSi material or in a crystalline material. An analog distribution of anelastic properties in a ZrCuNiAl metallic glass film was observed by using the tapping mode AFM . Previously, CR-AFM was used for the measurement of elastic properties of thin-film nano-crystalline ferrites , piezoelectric ceramic materials , clay , precipitates in polycrystalline metals , elastic properties of carbon nanotubes , and size-dependent effects in tellurium nanowires . Contact-resonance imaging has also been applied to study the contribution of grain boundaries to the overall elasticity of nano-crystalline materials , and there are many other applications which the reader may find by cross-referencing. An example is shown in figure 2.
It is of much interest to know not only the local contact stiffness and indentation modulus, but also the local damping, which might be caused by various physical mechanisms. Ultrasonic attenuation studies and internal friction studies have a long tradition and were used in order to gain information on elementary excitations in solids such as phonons, electrons and their interaction with a time-varying stress field, on dislocation-phonon interactions, on the diffusion of precipitates, on the gap function in superconductors, on tunneling sites in disordered materials and many other phenomena. But until now, it has not been possible to probe these interactions locally on the scale of the microstructure or the nanostructure of a material using ultrasound. CR imaging renders this possible. The damping in the tip-sample contact influences the motion of the cantilever from which the contact damping γ can be extracted. In a formal way this entails a complex contact stiffness k*:
k* = kr + iωγ . (4)
The complex contact stiffness can be related to a contact damping factor Q-1 = ωγ/kr = E’’/E’ , where E’’ is the loss modulus and E’ is the storage modulus, see figures 1 and 2 b-f. Experimentally, the Q-factors of the cantilever motion in contact, in air and/or relative to a known material is measured and evaluated . CR-AFM contact damping in nano-crystalline Ni as a function of grain size was used, in order to monitor the influence of homogeneous dislocation generation on the local Q-1 . Local damping using CR-AFM on polymers has been studied as well . By controlling the load and contact radius, one can obtain an appreciable amount of micro-sliding in the tip-sample contact allowing to observe frictional phenomena due to the electronic system in a metal-metal transition .
Quite a few efforts have been made to achieve subsurface imaging using various schemes of CR-imaging. Subsurface imaging is possible if the physical quantity measured exhibits a gradient into the material within the penetration of the stress-field emanating from the tip-sample contact. For example, the stiffness of the material changes at the surface when there are defects within a certain depth. This has been exploited for detecting dislocations generated below the surface , to probe the thickness of thin-solid films , or to detect subsurface defects and cavities [25, 26, 27]. The depth range is roughly given by the depth range of the stress field that the tip generates at the surface, the stress field of the defect or their joint action. Therefore, the depth range depends on the static load, the tip radius and the tip shape, as well as the cantilever mode employed. As a rule of thumb, one can assume that the depth range z is about at least three times the contact radius if the Hertzian contact mechanics between tip and surface prevails, depending on the parameters of the tip-sample contact as discussed above.
1Department of Material Science and Materials Engineering, Saarland University, Saarbrücken, Germany
2I. Phys. Institut, Georg-August Universität Göttingen, Göttingen, Germany
 G.A.D. Briggs: An Introduction to Scanning Acoustic Microscopy. (Oxford University Press, Oxford, 1985).
 V.V. Lange: The Mechanical Impedance Analysis Method of Non-Destructive Testing: A Review, Non-Destructive Testing and Evaluation 11, 177 (1994) doi 10.1080/10589759408952830
 C.C.H. Guyott, P. Cawley, and R.D. Adams: The Non-Destructive Testing of Adhesively Bonded Structure-A Review, Journal of Adhesion 20, 129 (1986) doi 10.1080/00218468608074943
 U. Rabe, K. Janser, and W. Arnold: Vibrations of free and surface‐coupled atomic force microscope cantilevers: Theory and experiment, Review of Scientific Instruments 67 (9), 3281 (1996) doi 10.1063/1.1147409
 K. Yamanaka, T. Tsuji, A. Noguchi, T. Koike, and T. Mihara: Nanoscale elasticity measurement with in situ tip shape estimation in atomic force microscopy, Review of Scientific Instruments 71 (6), 2403 (2000) doi 10.1063/1.1150627
 D. C. Hurley, M. Kopycinska-Muller, A. B. Kos, and R. H. Geiss: Nanoscale elastic-property measurements and mapping using atomic force acoustic microscopy methods, Measurement Science and Technology 16 (11), 2167 (2005) doi 10.1088/0957-0233/16/11/006
 W.C. Oliver and G.M. Pharr: Measurement of hardness and elastic modulus by instrumented indentation: Advances in understanding and refinements to methodology, Journal of Materials Research 19, 1 (2004) doi 10.1557/jmr.2004.19.1.3
 U. Rabe: Atomic Force Acoustic Microscopy, in Applied Scanning Probe Microscopy, edited by B. Bhushan and H. Fuchs (Springer, Berlin, 2006), Vol. II, pp. 37.
 H. Wagner, D. Bedorf, S. Küchemann, M. Schwabe, B. Zhang, W. Arnold, and K. Samwer: Local elastic properties of a metallic glass, Nature Materials 10 (6), 439 (2011) doi 10.1038/nmat3024.
 Y. H. Liu, D. Wang, K. Nakajima, W. Zhang, A. Hirata, T. Nishi, A. Inoue, and M. W. Chen: Characterization of Nanoscale Mechanical Heterogeneity in a Metallic Glass by Dynamic Force Microscopy, Physical Review Letters 106 (12) (2011) doi 10.1103/PhysRevLett.106.125504
 E. Kester, U. Rabe, L. Presmanes, P. Tailhades, and W. Arnold: Measurement of Young's modulus of nanocrystalline ferrites with spinel structures by atomic force acoustic microscopy, Journal of Physics and Chemistry of Solids 61 (8), 1275 (2000) doi 10.1016/S0022-3697(99)00412-6
 U. Rabe, M. Kopycinska, S. Hirsekorn, J.M. Saldana, G.A. Schneider, and W. Arnold: High-resolution characterization of piezoelectric ceramics by ultrasonic scanning force microscopy techniques, Journal of Physics D-Applied Physics 35 (20), 2621 (2002).
 M. Prasad, M. Kopycinska, U. Rabe, and W. Arnold: Measurement of Young’s modulus of clay minerals using atomic force acoustic microscopy, Geophysical Research Letters 29 (8) (2002) doi 10.1029/2001GL014054
 A. Kumar, U. Rabe, S. Hirsekorn, and W. Arnold: Elasticity mapping of precipitates in polycrystalline materials using atomic force acoustic microscopy, Applied Physics Letters 92 (18) (2008) doi 10.1063/1.2919730
 D. Passeri, M. Rossi, A. Alippi, A. Bettucci, M. L. Terranova, E. Tamburri, and F. Toschi: Characterization of epoxy/single-walled carbon nanotubes composite samples via atomic force acoustic microscopy, Physica E-Low-Dimensional Systems & Nanostructures 40 (7), 2419 (2008) doi 10.1016/j.physe.2007.07.012
 G. Stan, S. Krylyuk, A. V. Davydov, M. D. Vaudin, L. A. Bendersky, and R. F. Cook: Contact-resonance atomic force microscopy for nanoscale elastic property measurements: Spectroscopy and imaging, Ultramicroscopy 109 (8), 929 (2009) doi 10.1016/j.ultramic.2009.03.025
 M. Kopycinska-Müller, A. Caron, S. Hirsekorn, U. Rabe, H. Natter, R. Hempelmann, R. Birringer, and W. Arnold: Quantitative Evaluation of Elastic Properties of Nano-Crystalline Nickel Using Atomic Force Acoustic Microscopy, Zeitschrift für Physikalische Chemie - International Journal of Research in Physical Chemistry & Chemical Physics 222 (2-3), 471 (2008) doi 10.1524/zpch.2008.222.2-3.471
 P.A. Yuya, D.C. Hurley, and J.A. Turner: Contact-resonance atomic force microscopy for viscoelasticity, Journal of Applied Physics 104 (7), 074916 (2008) doi 10.1063/1.2996259
 M.K. Phani, A. Kumar, W. Arnold, and K. Samwer: Elastic stiffness and damping measurements in titanium alloys using atomic force acoustic microscopy, Journal of Alloys and Compounds 676, 397-406 (2016) doi 10.1016/j.jallcom.2016.03.155
 V. Pfahl, A.K. Phani, M. Büchsenschütz-Göbeler, A. Kumar, V. Moshnyaga, W. Arnold, and K. Samwer: Conduction electrons as dissipation channel in friction experiments at the metal-metal transition of LSMO measured by contact-resonance atomic force microscopy, Applied Physics Letters 110, 053102 (2017) doi 10.1063/1.4975072
 D.C. Hurley, S.E. Campbell, J.P. Killgore, L.M. Cox, and Y.F. Ding: Measurement of Viscoelastic Loss Tangent with Contact Resonance Modes of Atomic Force Microscopy, Macromolecules 46 (23), 9396 (2013) doi 10.1021/ma401988h
 V. Pfahl, M. K. Phani, M. Büchsenschütz-Göbeler, A. Kumar, V. Moshnyaga, W. Arnold, and K. Samwer: Conduction Electrons as Dissipation Channel in Friction Experiments at the Metal-Metal Transition of LSMO Measured by Contact-Resonance Atomic Force Microscopy, Applied Physics Letters 110 (5) (2017) doi 10.1063/1.4975072
 K. Yamanaka, K. Kobari, and T. Tsuji: Evaluation of Functional Materials and Devices Using Atomic Force Microscopy with Ultrasonic Measurements, Japanese Journal of Applied Physics 47 (7), 6070 (2008) doi 10.1143/JJAP.47.6070
 K. B. Crozier, G. G. Yaralioglu, F. L. Degertekin, J. D. Adams, S. C. Minne, and C. F. Quate: Thin film characterization by atomic force microscopy at ultrasonic frequencies, Applied Physics Letters 76 (14), 1950 (2000) doi 10.1063/1.126222
 A. Striegler, B. Köhler, B. Bendjus, M. Röllig, M. Kopycinska-Müller, and N. Meyendorf: Detection of buried reference structures by use of atomic force acoustic microscopy, Ultramicroscopy 111 (8), 1405 (2011) doi 10.1016/j.ultramic.2011.05.009
 Z. Parlak and F. Levent Degertekin: Contact stiffness of finite size subsurface defects for atomic force microscopy: Three-dimensional finite element modeling and experimental verification, Journal of Applied Physics 103 (11) (2008) doi 10.1063/1.2936881
 C. F. Ma, Y. H. Chen, W. Arnold, and J. R. Chu: Detection of subsurface cavity structures using contact-resonance atomic force microscopy, Journal of Applied Physics 121 (15), 9 (2017) doi:10.1063/1.4981537