River Rapids Conference Proceedings of the Society for Experimental Mechanics Series Advancement of Optical Methods in Experimental Mechanics, Volume 3 Helena Jin Cesar Sciammarella Sanichiro Yoshida Luciano Lamberti Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics River Publishers
Conference Proceedings of the Society for Experimental Mechanics Series Series Editor Tom Proulx Society for Experimental Mechanics, Inc. Bethel, CT, USA
River Publishers Helena Jin • Cesar Sciammarella • Sanichiro Yoshida • Luciano Lamberti Editors Advancement of Optical Methods in Experimental Mechanics, Volume 3 Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics
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Preface Advancement of Optical Methods in Experimental Mechanics, Volume 3: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics represents one of eight volumes of technical papers presented at the SEM 2014 SEM Annual Conference & Exposition on Experimental and Applied Mechanics organized by the Society for Experimental Mechanics held in Greenville, SC, June 2–5, 2014. The complete Proceedings also includes volumes on: Dynamic Behavior of Materials; Challenges In Mechanics of Time-Dependent Materials; Mechanics of Biological Systems and Materials; MEMS and Nanotechnology; Composite, Hybrid, and Multifunctional Materials; Fracture, Fatigue, Failure and Damage Evolution; and Experimental and Applied Mechanics. Each collection presents early findings from experimental and computational investigations on an important area within Experimental Mechanics, Optical Methods being one of these areas. With the advancement in imaging instrumentation, lighting resources, computational power, and data storage, optical methods have gained wide applications across the experimental mechanics society during the past decades. These methods have been applied for measurements over a wide range of spatial domain and temporal resolution. Optical methods have utilized a full range of wavelengths from X-Ray to visible lights and infrared. They have been developed not only to make two-dimensional and three-dimensional deformation measurements on the surface but also to make volumetric measurements throughout the interior of a material body. Livermore, CA, USA Helena Jin Chicago, IL, USA Cesar Sciammarella Hammond, LA, USA Sanichiro Yoshida Bari, Italy Luciano Lamberti v
Contents 1 The Kinematics of Crystalline Arrays at the Subnanometric Level ............................ 1 C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti 2 Comprehensive Theory of Deformation................................................. 15 Sanichiro Yoshida 3 Topography of Rough Dielectric Surfaces Utilizing Evanescent Illumination..................... 21 F.M. Sciammarella, C.A. Sciammarella, and L. Lamberti 4 Strain Assessment in Cracked Sheet Metals by Optical Grid Method........................... 39 M. Sasso, G. Chiappini, M. Rossi, and D. Amodio 5 A Preliminary Investigation on the Mechanical Behavior of Umbilical Cord With Moire´ Techniques ............................................................. 47 A. Boccaccio, R. Brunelli, L. Lamberti, M. Papi, T. Parasassi, M. De Spirito, and C. Pappalettere 6 Study on the Visco-Hyperelastic Behavior of the Zona Pellucida.............................. 53 A. Boccaccio, L. Lamberti, M. Papi, C. Douet, G. Goudet, M. De Spirito, and C. Pappalettere 7 Data Processing Techniques to Analyze Large 3-D Deformations of Cardiac Cycles ............... 63 C.A. Sciammarella, L. Lamberti, and A. Boccaccio 8 Bi-Directional Displacement Measurement by Speckle Interferometry Immune to Random Vibration........................................................ 89 Shuichi Arikawa and Satoru Yoneyama 9 Design of a Double-Illumination ESPI System for the Measurement of Very Slow Motions .......... 97 C. Casavola, G. Pappalettera, and C. Pappalettere 10 Multiplexed Holography for Single-Shot Three-Dimensional Shape and Displacement Measurements ...................................................... 103 Morteza Khaleghi, Cosme Furlong, Jeremie Guignard, Ivo Dobrev, Jeffrey Tao Cheng, and John J. Rosowski 11 Observation of Grain-Size Effect in Serration of Aluminum Alloy............................. 109 Tomohiro Sasaki, Tatsuya Nakamura, and Sanichiro Yoshida 12 Opto-Acoustic Technique to Investigate Interface of Thin-Film Systems ........................ 117 Sanichiro Yoshida, David Didie, Daniel Didie, Sushovit Adhikari, and Ik-Keun Park 13 Analysis of Fatigue of Metals by Electronic Speckle Pattern Interferometry..................... 127 Shun Hasegawa, Tomohiro Sasaki, Sanichiro Yoshida, and Seth L. Hebert 14 Simultaneous Application of Acoustic and Optical Techniques to Nondestructive Evaluation. . . . . . . . 135 Ik-Keun Park, Sanichiro Yoshida, David Didie, Haesung Park, Daniel Didie, and Saugat Ghimire vii
15 Stress Analysis on Welded Specimen with Multiple Methods ................................. 143 Sanichiro Yoshida, Tomohiro Sasaki, Sean Craft, Masaru Usui, Jeremy Haase, Tyler Becker, and Ik-Keun Park 16 Sound Attenuation for Dogs Barking Using of Transfer Function Method....................... 153 Shuichi Sakamoto, Takatsune Narumi, Yuichi Toyoshima, Nobuaki Murayama, Toru Miyairi, and Akira Hoshino 17 On the Use of Regularized DVC to Analyze Strain Localization............................... 161 Thibault Taillandier-Thomas, Thilo Morgeneyer, Ste´phane Roux, and Franc¸ois Hild 18 Determination of Surface Bi-Axial Stresses Using Raman Spectroscopy......................... 167 M. Shafiq and G. Subhash 19 Visualization and Quantification of Quasi-Static and Dynamic Surface Slopes Using a Reflection-Mode Digital Gradient Sensor.......................................... 175 Amith Jain, Chandru Periasamy, and Hareesh Tippur 20 Analysis of Linear Anisotropic Parameters by Using Hybrid Model in Mueller Optical Coherence Tomography.............................................. 183 Chia-Chi Liao and Yu-Lung Lo 21 Characterization of Time-Dependent Mechanical Behaviors of Dental Composites by DIC.......... 191 T.Y. Chen, C.L. Hsu, and S.F. Chuang 22 Deformation Distribution Measurement from Oblique Direction Using Sampling Moire Method. . . . . 197 Daiki Tomita, Motoharu Fujigaki, and Yorinobu Murata 23 Automatic Stress Measurement by Integrating Photoelasticity and Spectrometry................. 205 Po-Chi Sung, Yu-An Chiang, Wei-Chung Wang, and Te-Heng Hung 24 Observation of Fiber-Matrix Interfacial Stresses Using Phase-Stepping Photoelasticity............. 215 Takenobu Sakai, Yasunori Iihara, and Satoru Yoneyama 25 Stabilizing Heteroscedastic Noise with the Generalized Anscombe Transform: Application to Accurate Prediction of the Resolution in Displacement and Strain Maps Obtained with the Grid Method......................................... 225 M. Gre´diac and F. Sur 26 Experimental Evaluation of the Warping Deformation in Thin-Walled Open Section Profiles . . . . . . . . 231 Sandro Cammarano, Giuseppe Lacidogna, Bartolomeo Montrucchio, and Alberto Carpinteri 27 In Situ Study of Plastic Flow at Sliding Metal Surfaces ..................................... 243 A. Mahato, Y. Guo, N. Sundaram, and S. Chandrasekar 28 Stiffness Investigation of Synthetic Flapping Wings for Hovering Flight ........................ 249 Kelvin Chang, Anirban Chaudhuri, Jayson Tang, Jordan R. Van Hall, Peter Ifju, Raphael Haftka, Christopher Tyler, and Tony Schmitz 29 A Generic, Time-Resolved, Integrated Digital Image Correlation, Identification Approach.......... 257 J.P.M. Hoefnagels, J. Neggers, Benoˆıt Blaysat, Franc¸ois Hild, and M.G.D. Geers 30 Multiscale FE-Based DIC for Enhanced Measurements and Constitutive Parameter Identification. . . . 265 Laurent Robert, Jean-Charles Passieux, Florian Bugarin, Christoph David, and Jean-Noe¨l Pe´rie´ 31 Uncertainties of Digital Image Correlation Near Strain Localizations .......................... 277 Mark A. Iadicola and Adam A. Creuziger 32 Pre-qualifying DIC Performance Based on Image MTF Correlation Coefficient .................. 287 Chi-Hung Hwang, Wei-Chung Wang, Yung-Hsiang Chen, Jia-He Chen, Yan-Ting Wu, Jheng-Yong Lyu, and Ya Hsi-Chiao 33 Analysis of E-Beam Microlithography and SEM Imaging Distortions .......................... 297 A. Guery, F. Latourte, F. Hild, and S. Roux viii Contents
34 Displacement Measurements Using CAD-Based Stereo-DIC.................................. 303 J.-E. Dufour, B. Beaubier, F. Hild, and S. Roux 35 Single-Camera-Based 3D DIC for Fast-Speed Measurement ................................. 309 Hien Kieu, Zhaoyang Wang, Hieu Nguyen, and Minh Le 36 Three-Dimensional Digital Image Correlation Using a Single Color-Camera..................... 315 Wade Gubbels and Gary S. Schajer 37 Multi-Camera DIC Offers New Dimensions in Material Testing.............................. 325 Thorsten Siebert and Vinh Tran 38 Using Sampling Moire´ to Extract Displacement Information from X-Ray Images of Molten Salt Batteries ............................................................. 331 Phillip L. Reu, Enrico Quintana, and Kevin Long 39 High-Speed Digital Holography for Transient Response of the Human Tympanic Membrane. . . . . . . . 337 I. Dobrev, C. Furlong, J.J. Rosowski, and J.T. Cheng 40 Displacement and Strain Measurement with Multiple Imaging Head Using PSDHI ................ 343 Motoharu Fujigaki, Hiroki Minamino, and Yorinobu Murata 41 Simultaneous ESPI Measurements Using Multiple Wavelengths and a Color Camera.............. 349 Guillaume Richoz and Gary S. Schajer 42 Some Practical Considerations in High-Speed 3D Shape and Deformation Measurement Using Single-Shot Fringe Projection Technique........................................... 357 Minh Le, Zhaoyang Wang, and Hieu T. Nguyen 43 Fast-Speed, High-Accuracy and Real-Time 3D Imaging with Fringe Projection Technique.......... 363 Hieu Nguyen, Zhaoyang Wang, Hien Kieu, and Minh Le 44 DIC Strain Analysis of Pipeline Test Specimens Containing Metal Loss ......................... 371 Leonardo D. Rodrigues, Jose´ L.F. Freire, and Ronaldo D. Vieira 45 Experimental Inference of Inter-Particle Forces in Granular Systems Using Digital Image Correlation....................................................... 379 Nikhil Karanjgaokar and Guruswami Ravichandran 46 High Pressure Burst Testing of SiCf-SiCmComposite Nuclear Fuel Cladding.................... 387 Luis H. Alva, Xinyu Huang, George M. Jacobsen, and Christina A. Back 47 Low Cost Digital Image Correlation (DIC) for Monitoring Components Undergoing Fatigue Loading.......................................................... 395 R.K. Fruehmann and J.M. Dulieu-Barton 48 Tensile Response and the Associated Post: Yield Heating of Polycarbonate...................... 399 C. Allan Gunnarsson, Bryan Love, Paul Moy, and Tusit Weerasooriya 49 Passive 3D Face Reconstruction with 3D Digital Image Correlation............................ 403 Hien Kieu, Zhaoyang Wang, Minh Le, and Hieu Nguyen 50 On the Meso-Macro Scale Deformation of Low Carbon Steel ................................. 409 Suraj Ravindran, Behrad Koohbor, and Addis Kidane 51 Feasibility of Non-Contacting Measurement of Wind-Induced Full-Field Displacements on Asphalt Shingles ..................................................... 415 Rahim Ghorbani, Xing Zhao, Fabio Matta, Michael A. Sutton, Addis Kidane, Zhuzhao Liu, Anne Cope, and Timothy Reinhold Contents ix
Chapter 1 The Kinematics of Crystalline Arrays at the Subnanometric Level C.A. Sciammarella, F.M. Sciammarella, and L. Lamberti Abstract This article presents additional developments in the analysis of displacement fields and strains around edge dislocations via electron microscopy. The goal of this work, which extends earlier work of the authors, is to provide additional information about the connection between the Continuum Mechanics model of the events taking place in a nanometric size region of a crystal and experimentally observed geometrical changes of the crystalline array. The implications of the understanding of the connection of Continuum Mechanics and actual distortions of the crystalline array are vast. The development of new semiconductor electronic devices based on nanotubes and other semiconductors in circuits that operate at high temperature, high-power or high radiation will benefit by this input. The new technique of manufacturing, 3D printing, can also benefit from the better understanding of the process of formation of crystalline arrays and the defects that this process generates. Keywords Kinematics of crystalline arrays • Sub-nanometer observations • Edge dislocations • High resolution transmission electron microscopy • Image analysis 1.1 Introduction Properties of materials are observable variables that are determined in the macro world. These properties are a consequence of interactions that occur at the atomic level and hence are ruled by Quantum Mechanics. Between these two descriptions of matter there is a big gap and one of the endeavors of the science of Mechanics of Materials is to connect these two worlds. Human thought supported by experimental tools has provided the path way to provide solutions of this extremely difficult scientific question. The search of the solution between structure of matter and macroscopic properties can be traced back to early 19th with Cauchy and his attempt to relate Continuum Mechanics with the atomic structure of matter that resulted in the Cauchy-Born rule [1, 2]. A fundamental help in untangling this complex puzzle has been the advances in the experimental technology of observation of the behavior of matter at the atomic and subatomic levels. The developments of X-rays diffraction, electron microscopy, neutron diffraction, and other methods to observe condensed matter have provided the needed information to understand the structure of solid matter. Crystallography is the science that provides the understanding of the atomic arrangements in solid matter. X-rays give information by analyzing the interaction of electromagnetic waves (photons) with the valence electrons of atoms and ions; electron microscopy provides information on C.A. Sciammarella (*) Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, 10 SW 32nd Street, 60616 Chicago, IL, USA Department of Mechanical Engineering, College of Engineering and Engineering Technology, Northern Illinois University, 590 Garden Road, 60115 DeKalb, IL, USA e-mail: sciammarella@iit.edu F.M. Sciammarella Department of Mechanical Engineering, College of Engineering and Engineering Technology, Northern Illinois University, 590 Garden Road, 60115 DeKalb, IL, USA L. Lamberti Dipartimento Meccanica, Matematica e Management, Politecnico di Bari, Viale Japigia 182, 70126 Bari, Italy H. Jin et al. (eds.), Advancement of Optical Methods in Experimental Mechanics, Volume 3: Proceedings of the 2014 Annual Conference on Experimental and Applied Mechanics, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-06986-9_1, #The Society for Experimental Mechanics, Inc. 2015 1
the spatial arrangement of matter through the interaction of electrons, charged particles, with atoms and ions, describing the total charge distributions. Neutrons interact with atomic nuclei through the phenomenon of diffraction. By combining X-rays, electron microscopy and neutron diffraction, details of the structure of the organization of matter have been investigated and the arrangement of atoms and ions in complex structures revealed. These breakthroughs in technology provided the necessary tools to describe the structure of matter from the geometric point of view. These experimental observations have also enabled the mechanics of deformations to be the subject of theoretical and experimental investigations at different scales. As a result, the theory of Continuum Mechanics has been extended from the field of macroscopic bodies to the structure of crystalline matter. A large amount of work has been done to relate the internal local forces in crystalline structures expressed by a suitable stress tensor definition to the deformations expressed by a compatible strain tensor definition. These efforts have resulted in those aspects of the mathematical theory of dislocations closely related to the theory of elasticity. A far more difficult task has been the effort to connect the mathematical theory of plasticity with events observed at the matter structure level. A key aspect in understanding the plastic behavior of matter was the introduction of the concept of defects in crystalline materials, dislocations and point defects. Completely independent of the analysis of defects in crystalline structures, in the 1920s, the concept of a dislocation was introduced by Volterra [3], and later analyzed by Somigliana [4] and Love [5]. Volterra defined the basic type of dislocations that today are utilized in fracture mechanics. These abstract concepts originated in the analysis of the uniqueness of displacements fields came back to crystallography as a result of the need to explain discrepancies between theoretical and experimental prediction of crystal strength (see, for example, Taylor [6], Orowan [7], Polanyi [8] and Nabarro [9]). Starting with these initial developments a new discipline came into existence, the mathematical theory of dislocations [10]. The first direct experimental observation of the presence of dislocations in crystalline structures took place in the 1960s with the use of X-rays. This visualization came about through X-ray photogrammetry. In the 1970s the use of the electron microscope provided another tool to look at the structure of crystalline matter. This article presents additional developments in the analysis of displacements fields and strains around edge dislocations via electron microscopy extending earlier work of the authors [11–13]. As the kinematics of crystal structures based on finite deformations is a basic step in the mathematical theory of dislocations, this study investigates the connection between the Continuum Mechanics model of the events taking place in a nanometric size region of a crystal and experimentally observed geometrical changes of the crystalline array. The implications of the understanding of the connection of Continuum Mechanics and actual distortions of the crystalline array are vast. The development of new semiconductor electronic devices based on nanotubes and other semiconductors in circuits that operate at high temperatures, high-power or high radiation will benefit by this input. In the new manufacturing technology, 3D printing, laser beams are used to melt powders of the material that is being formed into a given geometrical shape. The energy balance between laser beam power and energy required to form a given geometrical shape is a very important subject of research. This energy balance must produce a material structure with an optimum arrangement of atoms and ions for the particular purpose that the part is fabricated for. Hence, advances in the understanding of the mathematical theory of dislocations will be of great importance in the technology of 3D printing. 1.2 High Resolution Transmission Electron Microscope Images For the full understanding of the observed images it is necessary to go over a brief explanation of the workings of an electron microscope (see the schematic shown in Fig. 1.1). The images obtained in high resolution transmission electron microscope (HRTEM) have a process of formation that is similar to that of an interference microscope. There is an illuminating beam of coherent electrons (polarized) that is focused on an object, an aperture diaphragm and a focusing lens. Using Fourier optics terminology, the focusing lens creates the diffraction pattern of the object in its back focal plane, and then through the diaphragm it performs an inverse FT. The direct beams through the specimen interfere with the diffracted beams and produce an image of the object. The incident wave front of the electron beam, ideally a plane wave front, interacts with the structure of the object. The resolution of the microscope from the point of view of the image formation can be expressed in terms of the classic Rayleigh criterion. The smallest distance that can be observed is given by the equation δ ¼ 0:61λ nr sinα ð 1:1Þ 2 C.A. Sciammarella et al.
where λis the wavelength of the radiation, nr the index of refraction of the medium between the imaging lens and the image plane, α the semi-angle of the imaging lens. The wave length associated with electrons can be approximately computed as λ ¼ 1:22 ffiffiffiE p g ð 1:2Þ In Eq. (1.2), the wavelength of the wave associated with the electron propagation is given in nm and Eg is the energy imparted to the wave expressed in eV. For a HRTEM the power of the electron beams could be for example 400 KeV. Applying Eq. (1.2), one obtains for example λ ¼0.00193 nm. Since atomic radii are of the order of 0.1 nm, the wavelength of the electron beam of a HRTEM is well within the resolution of the atomic radii. It should be noted that Eq. (1.1) applies to the case of incoherent illumination while the HRTEM image has a high degree of coherence. Furthermore, the resolution of the microscope is affected by lens aberrations: the quality of magnetic lenses is poor and may affect the information retrieved from the image. Information detection can be represented by the conventional mathematics of amplitude and phase modulation. The emerging wavefronts are given by a phasor (see Chap. 7.4.6 of [14]): E r; t ð Þ¼E r; t ð Þ eiϕ r;t ð Þ ð1:3Þ In Eq. (1.3), bold letters indicate vectors, E(r,t) is the amplitude of the wave associated with the electron propagation and ϕthe corresponding phase, r is the position vector and t is the time. As stated before, magnetic lenses have aberrations (e.g. astigmatism, spherical, chromatic) that are included in the information retrieved by the wave front. The information is captured as an intensity distribution detected in modern HRTEM by a raster sensor similar to electronic cameras. Image resolution is affected by the sensor similarly to what occurs in electronic cameras [15]. It is possible to correct the experimentally determined intensity via software by utilizing the concept of contrast transfer function, the equivalent of the optical transfer function used in visible optics. In order to understand the source of intensity distribution observed in electron microscopy, it is necessary to briefly analyze the interaction of matter with an electronic beam: that is, the diffraction phenomenon of electrons by matter, in this particular case by crystalline matter. This is an extremely complex subject and deals with the duality of particles and waves, it can only be rigorously explained in quantum mechanics terms. However, in the particular case being considered it can be summarized in few lines. When an electronic wave impinges in the cross-section of an atom different interactions take place. Of interest in our case is the stimulated emission of waves of the same frequency of the incident radiation. This interaction gives rise to waves of amplitudes proportional to the electronic density of the atom or ionic structure. This electronic density, since we are dealing with images captured on a plane, is a two dimensional density. That is, if we are observing an image plane the maximum observed intensity will take place in the region where the atomic nucleus is located and then the observed distribution can be utilized as a tool to define the position of the atom in 2D. Fig. 1.1 Schematic of high resolution transmission electron microscope (HRTEM) 1 The Kinematics of Crystalline Arrays at the Subnanometric Level 3
In order to locate atoms in 3D the different operation modes of the electron microscope can be used taking advantage of other types of interactions between electronic waves and observed atoms. Absorption is a very important mechanism: electronic waves can only penetrate matter in a dimensional range of few nm to 100 nm. Because the absorption coefficient increases with the energy of the electronic wave, if one uses 400 KeV the specimens must have a thickness of few nm. To complete the understanding of the observed images, it is necessary to analyze the process of image formation in some more detail. One is observing a specimen of a certain thickness, dob, and this thickness is imaged at a given plane. It is necessary to find out what it is the relationship of the observed image and dob. We can go back to basic concepts of classical geometrical optics: depth of field and depth of focus. The depth of field is measured in the object space and tells us how much of dob is in focus, that is how much the object can be displaced from a given position back and forth without loosing the focus in the image plane. The depth of focus refers to the image plane and tells us how much back and forth from the focal plane one can move the detector plane without an apparent loss of focus. The depth of field and the depth of focus depend on the illumination aperture, the inclination of the beam entering the lens (angle β in Fig. 1.1), and the inclination of the beam forming the image (angle αin Fig. 1.1). In HRTEM, paraxial beams are utilized and the angles are small enough that all the features contained in d0b are practically in focus [16]. Another important concept that will help the interpretation of the observed images is the concept of resolution. In the present case, resolution is the minimum distance that can be measured in the image plane [16]. As we are going to see by utilizing pixel interpolation in the image plane this distance can be reduced. 1.3 The 4HSiC Crystal Before we can proceed further it is necessary to provide additional information on the analyzed image. In earlier publications of the present authors [11–13], an image of a 4HSiC crystal was analyzed [17]. That particular image was selected because it is supported by relevant information. It was also of interest to study the basic unit of a hexagonal polytype of SiC whose schematic is shown in Fig. 1.2. Two inches wafers were grown by vapor deposition oriented along the plane (0001) within 0.5 of SiC crystal. KOH etching at 500 C or 510 C was done on the Si (0001) faces of the wafers. The etch pits were observed with Nomarski interferometry while X-rays were utilized to observe the orientation of each pit array. The most common polytypes require four and six Si-C bi-layers, respectively, to define the unit cell repeating distance along the c-axis [0001] directions. Figure 1.2 shows the sequence ACAB of the 4HSiC crystal; this sequence is relative because the radii of the atomic regions corresponding to each atom all have the same magnitude. Hence, relative positions result from the possible geometrical arrangements of spheres. Figure 1.2a shows the arrangement denoted by the letter A and assuming a Si atom as starting the sequence. The stacking arrangement is outlined in Fig. 1.2b and a perspective of the three initial layers is also illustrated in Fig. 1.2c. Figure 1.3a shows an optical micrograph of the etch pit bands—(0001) face—of a 4HSiC wafer. The bands run along directions 1120 and reveal the presence of threading edge dislocations. The theoretical structure of the hexagonal polytype of silicon carbide with a1 ¼a2 ¼a3 is shown in Fig. 1.3b. Figure 1.4 illustrates the process of formation of the dislocations seen in Fig. 1.3. This type of threading edge dislocation originated from the cooling of the SiC crystal which then caused a misoriented crystal to grow (Fig. 1.3). The large amount Fig. 1.2 Staking sequence ACAB of the 4HSiC crystal. The red dots in (a) indicate the position of the following atom layer shown in (b); Perspective of the three initial layers is shown in (c) 4 C.A. Sciammarella et al.
of energy that concentrated in that region led to the formation of dislocations (slip bands) that caused the plane to slide as illustrated in Fig. 1.4. As a result of this dislocation movement two extra half planes—indicated by the black dots in Fig. 1.4—were generated to reduce the energy in the unit cell thus allowing stability of the crystal structure to be achieved. Figure 1.4 shows together the optical micrograph (see Fig. 1.4a) with a electron microscope image of an etch pitch (see Fig. 1.4b) and a HRTEM image around a threading dislocation in the array (see Fig. 1.4d). The Burgers vectors of the edge dislocations are of the type a/3 1120 (see Fig. 1.4c) where a is the 4HSiC lattice parameter. 1.4 HRTEM Image Analysis The HRTEM image encompasses areas where the basic cell of the crystal is highly distorted and thus provides information suitable to perform the analysis of deformations at different scales and relate macro continuum mechanics to deformations in the micro- and nano-scales. The first step of analysis is to interpret the images produced by HRTEM. Image formation is the Fig. 1.3 (a) Optical micrograph of etch pit bands—(0001) face—of a 4HSiC wafer with bands along directions 1120 ; (b) Theoretical structure of the hexagonal polytype of silicon carbide with a1 ¼a2 ¼a3 Fig. 1.4 Gliding of dislocations in the slip system 1120 1100 during post-growth cooling: (a) Optical micrograph; (b) electron microscope image of the red circled region; (c) Theoretical SiC structure with a1 ¼a2 ¼a3 and Burgers vector of dislocation oriented as a3; (d) HRTEM pattern of the crystal (dark dots indicate the extra-atomic planes) 1 The Kinematics of Crystalline Arrays at the Subnanometric Level 5
result of complex processes involving subject matters of great theoretical complexity and difficulty. The very thin specimen investigated in this study is the order of 10 nm and hence contains a large number of atomic planes, more than 30. In Sect. 1.2, it was pointed out that usually the depth of focus is large enough that the whole depth of the specimen is in focus [16]: that is, features at different depths are shown in the image. To evaluate the image content, we can analyze the actual image of a hexagonal elementary cell shown in Fig. 1.5a. The cell is extracted away from the region circled in yellow in Fig. 1.4c. Figure 1.5b recalls the theoretical atomic structure explained in Fig. 1.2. Different aspects of the image formation can be evaluated. It is easy to notice the similitude between the experimentally observed image and the theoretical structure of Fig. 1.2a. It is possible to conclude that the observed pattern is the image of an atomic layer. First it is necessary to explain the process followed to put together this image. From the original image (Fig. 1.4c) a small square region containing the elementary cell was cropped. A bicubical spline interpolation of the pixels was applied resulting in the scale shown in Fig. 1.5. The red circles in the figure have the radius of the spheres r ¼a/2 ¼0.3073/2 ¼0.1537 nm, that corresponds to the lattice parameter a ¼0.3073 nm at 300 K. With the above parameter the corresponding hexagon connecting the center of the spheres has been drawn. The question of the minimum distance that can be measured in the image can be now addressed. The image of the crystal was recorded with a HRTEM JEOL 4000 EX-TEM operating at 400 KeV [18]. According to Eq. (1.2) the wavelength of the electron wave is 0.00193 nm that is of the order of 0.002 nm. Since the lattice parameter a is 0.3073 nm the ratio a/λ is 159. According with the criteria of optical microscopy, with this wavelength one should be able to resolve λ/2: that is, we should be able to measure 0.001 nm. We have seen that this is not feasible in electron microscopy and the resolution defined as the fine detail of an image is much lower than the theoretical value. However, in electron microscopy another resolution concept is utilized [16]: the point to point distance, this is the concept that applies to the measurements that we perform in the analyzed image. The HRTEM JEOL 4000EX-TEM is rated [18] as having a maximum classical resolution of 0.14 nm. The original of the picture has a pixel value S ¼0.0368 nm/pixel. The sub-image has a pixel value S ¼0.0011 nm/pixel that is a gain in resolution of 33 times. The bicubical spline interpolation has yielded a pixel resolution of the order of 0.001 nm which implies a 140 increase of sensitivity to measure distances with respect to the rated minimum classical resolution of 0.14 nm. 1.5 Analysis of the Kinematics of the Distorted Region Figure 1.6a shows the positions of crystallographic axes in the electron microscope patterns. The red dots show the extra atomic rows. Figure 1.6b shows an enlargement of the dislocation region showing the position of two edge dislocations generated by the added extra atomic planes. The analysis of the kinematics of the elementary cells is based on the frame work and limitations that came from the idealization of 3D array of atoms as a superposition of identical 2D array of atoms stacked in the vertical direction. Since this stack is not identical in depth due to the sequence shown in Fig. 1.2 based in the covalent bonding of Si atoms and C atoms, the analysis is restricted to a two-dimensional array. Figure 1.7 represents the FFT of the HRTEM pattern of the crystal shown in Fig. 1.4c. The direction fx corresponds to the family of crystallographic directions a1 [1000] (See Fig. 1.6a). The scale of Fig. 1.7 is Sf ¼(0.0111) 1/nm per pixel. Fig. 1.5 Elementary cell of the hexagonal packing identified in the HRTEM pattern resembling the theoretical structure represented in (b) 6 C.A. Sciammarella et al.
Figure 1.7 provides a statistical average of the changes of the elementary cells in the region under analysis. The hexagon represented in Fig. 1.7 corresponds to the average size of a at 300 K. The departures of the positions of the corners of the hexagon with respect to the average position indicated in Fig. 1.7 reflect the distortions of the elementary cells caused by the presence of the dislocations. Figure 1.2a (i.e. Fig. 1.5b) represent the elementary undeformed cell that is characterized for this particular crystal by the constancy of the atomic vector a. This vector reflects the spatial configuration of the atomic bonds that bind together the atoms. Since atoms are in continuous oscillatory motion caused by the temperature T, a is a statistical average in time. Figure 1.7 shows that upon the deformation caused by the presence of the dislocations the values of aare modified. In the 2D approach we can analyze the changes of the elementary cell as superposition of different changes in the dimensions and orientations of a. Figure 1.8 shows the changes of the elementary cell that can be superimposed to get the different configurations statistically observed in Fig. 1.7. All the main diagonals of the hexagon show increases and decreases of the value of the modulusof a. Increasing frequencies correspond to shortening of a1, modulus of ain the direction [1000]. Lower frequencies correspond to elongations of a1. Let a1r be the value of a1 for 300 K and a1(+f) the value of a1 for increasing frequencies: it holds a1(+f) <a1r. Conversely, if a1( f) is the value of a1 for decreasing frequencies, it holds a1( f) >a1r. The corresponding distribution of a1r is asymmetric with wider ranges for the a1( f) than for the a1(+f). This distribution reflects the asymmetry of the potential function that determines the forces binding atoms. From the position of equilibrium a1r, the repulsion forces between atoms have a much steeper gradient that the attraction forces between atoms as they move apart. This asymmetry is very important in the analysis of the kinematics of the elementary cell. In solid mechanics and for small deformations, compression and tension displacements are considered symmetric. This is not the case in the deformation of the elementary cell. It is possible to see also in Fig. 1.7 that the diameters of the hexagon experience rigid body rotations. Because the size of the region under observation is very small, rigid body rotations also are very small. Fig. 1.6 (a) Crystallographic directions 1120 a3; (b) Schematic view of the position of atoms in the neighborhood of the edge dislocations (the meaning of the yellow dots will be discussed later on the paper) Fig. 1.7 (a) FFT of the HRTEM pattern shown in Fig. 1.6a; (b) Reference system associated with the FFT 1 The Kinematics of Crystalline Arrays at the Subnanometric Level 7
Figure 1.8 shows the basic system of reference with the versors ai (i ¼1, 2, 3). The displacements of the atomic positions in the deformed configuration in the hexagon can be expressed as: u ¼ua1a1 þua2a2 þua3a3 ð1:4Þ If the displacement components along three versors are available, the displacement derivatives with respect to these directions that enter in any definition of strain tensor adopted in the analysis are related with the Cartesian components of the displacement derivatives. Looking at Fig. 1.9, the following equations can be written: ∂ua1 ∂a1 ¼ ∂u ∂x cos2θ a1 þ ∂v ∂y sin2θ a1 þ ∂u ∂yþ ∂v ∂x 2 4 3 5 cosθa1 sinθa1 ∂ua2 ∂a2 ¼ ∂u ∂x cos2θ a2 þ ∂v ∂y sin2θ a2 þ ∂u ∂yþ ∂v ∂x 2 4 3 5 cosθa2 sinθa2 ∂ua3 ∂a3 ¼ ∂u ∂x cos2θ a3 þ ∂v ∂y sin2θ a3 þ ∂u ∂yþ ∂v ∂x 2 4 3 5 cosθa3 sinθa3 8> >>>>>>< >>>>>>>: ð1:5Þ From the above equations, it is possible to determine the principal derivatives with respect to coordinate system X-Y and the orientation of the principal directions with respect to these axes: ∂u1 ∂x ¼ ∂u ∂xþ ∂v ∂y 2 þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂u ∂x ∂v ∂y 2 2 666 64 3 777 75 2 þ ∂u ∂yþ ∂v ∂x 2 4 3 5 2 vuu uuu uut ∂u2 ∂y ¼ ∂u ∂xþ ∂v ∂y 2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∂u ∂x ∂v ∂y 2 2 666 64 3 777 75 2 þ ∂u ∂yþ ∂v ∂x 2 4 3 5 2 vuu uuu uut 8> >>>>>>>>< >>>>>>>>>>: ð1:6Þ a1 a2 a3 e3 e2 e1 a b c a1 a2 a3 a1 a2 a3 Fig. 1.8 (a) Fundamental 4HSiC cell; (b) Biaxial deformation (tension) and (c) Shear deformation of the elementary cell Fig. 1.9 Relationship between derivatives of displacements measured along three different versors and Cartesian coordinates 8 C.A. Sciammarella et al.
tan2θ ¼ ∂u ∂y þ ∂v ∂x ∂u ∂x ∂v ∂y ð1:7Þ The above quantities can be determined experimentally to apply the preceding equations. It is important to realize that the above relationships provide an Eulerian description of the deformed crystal and the adopted system of reference is attached to the structure of the crystal. In the derivation of Eqs. (1.5, 1.6, and 1.7) it was assumed that, upon deformations, the directions a1, a2 and a3 change orientations with respect to the initial configuration. The above equations can be applied to the deformed shapes of the elementary cells located in different positions in Figs. 1.4c and 1.6a. There is an alternative analysis procedure based in extending Continuum Mechanics to the field under observation. Figure 1.7a shows the FFT of the crystal structure and the labeling of the corresponding axes in the physical space. These axes correspond to the undeformed crystal orientations, with axis e1 oriented along the crystallographic family of directions a1 <1000>. We can introduce the concept of digital moire´ (see Chap. 13.8 of [14]) and through this procedure provide a tool to analyze the kinematics of the hexagonal crystal at the level of the elementary cell. Symbolically, the deformation of the elementary cell can be represented as a function F(e1,e2,e3,t) and the gradient of this function represented by∇F(e1,e2,e3, t) provides the derivatives that characterize the state of deformation (see Fig. 1.8). From the inverse FFT of Fig. 1.7 we can get three systems of fringes at orientation 120 to each other. We have the equivalent of a three elements rosette with arms at 0 , 120 and 240 . The correspondence between rosette orientations and directions (orthogonal to fringe orientation) in the FFT space is also shown in the figure. Calling e1, e2 and e3 the axes of the rosette as shown in Fig. 1.8a and following a similar argument to that utilized for deriving Eqs. (1.6 and 1.7), principal derivatives can be obtained as a function of the derivatives measured along the rosette axes: ∂u1 ∂x1 , ∂u2 ∂x2p ¼ 1 3 ∂ue1 ∂e1 þ ∂ue2 ∂e2 þ ∂ue3 ∂e3 ffiffifi 2p 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifffifi ∂ue1 ∂e1 ∂ue2 ∂e2 2 þ ∂ue2 ∂e2 ∂ue3 ∂e3 2 þ ∂ue3 ∂e3 ∂ue1 ∂e1 2 2s 4 3 5 ð 1:8Þ where: x1 and x2 are the coordinate axes; e1 x1 and the x2-axis is orthogonal to x1. The principal directions of the derivatives are obtained from the following relationship: θ ¼ 1 2 arctg ffiffifi 3p ∂ue3 ∂e3 ∂ue2 ∂e2 2 ∂ue1 ∂e1 ∂ue2 ∂e2 þ ∂ue3 ∂e3 8< : 9= ; ð1:9Þ From Eqs. (1.8 and 1.9) one can compute principal derivatives and obtain all the necessary information on the elementary cell that has been adopted. Figure 1.10 shows the phase of the fringes obtained filtering the diffraction pattern around the directions defining (e1,e2,e3,to) where time to indicates the fact that what the microscope shows is a time average of the Fig. 1.10 Phases of the harmonics that modulate the structure of the crystal: (a) 240 in FFT space; (b) 120 in FFT space; (c) 0 in FFT space, respectively correspond to directions e2, e3 and e1 1 The Kinematics of Crystalline Arrays at the Subnanometric Level 9
events that took place in the observed thin slice. HRTEM patterns were processed with the Holo Moire´ Strain Analyzer image processing software developed by C.A. Sciammarella and his coworkers [20]. The phase patterns show phase dislocations (see Chap. 10.5.1 of [14]). Moire´ patterns provide the Eulerian description of the deformed elementary cell and again the versors are oriented in coincidence with crystallographic directions highlighted in Fig. 1.8a. The preceding developments in the kinematics of the elementary cell can be considered as an extension of the conceptual idea contained in the so called 1st Cauchy–Born rule. It directly connects the Continuum Mechanics kinematics of solids to the movement of atoms in the hexagonal crystal in a two-dimensional context. 1.6 Analysis of Elementary Cells To further shed light on the proposed approach to the crystal kinematics, it is interesting to survey some regions of the HRTEM pattern shown in Figs. 1.4c and 1.6a. The scale of Fig. 1.11a is Sc ¼0.000900 nm/pixel, the axis a1 is oriented as the crystallographic direction [1000]. In Fig. 1.11a, the green lines represent the limits of the region of influence of each atom. These outlines are no longer circles but ellipses, they fit better the elementary cell configuration and the elementary cell is contained in an ellipse that is very closely similar to the shapes of the green ellipses that define the atoms positions of equilibrium. Principal strains can be computed: ε1 e ¼0.0501, ε 2 e ¼ 0.0332, the principal direction ε 1 e lays along the crystallographic direction [1000] while the other principal direction ε2 e is perpendicular to it. These results are compatible with the frequencies of the diffraction pattern shown in Fig. 1.7. Figure 1.11b shows an elementary cell located in the compression region. The scale of Fig. 1.11b is Sc ¼0.0009789 nm/pixel. The principal strain ε1 e ¼ 0.02550 and coincides with the direction [1000], ε 2 e ¼ 0.05240 and is perpendicular to this direction. The above shown results are also compatible with the frequencies of the diffraction pattern shown in Fig. 1.7. Figure 1.12 shows the region of the dislocation and the positions of the atoms in this region. The red dots represent the atoms of the extra-rows. The yellow lines indicate the direction of the extra rows that are shown also as red dots in Fig. 1.6. The edge dislocations are indicated with the green symbols ┴, the vector B indicates the direction of the Burgers vector of the edge dislocations. Two basic hexagonal cells are indicated in the vicinity of the extra atomic rows (Fig. 1.12). The dislocations interrupt the basic cell arrangement corresponding to the equilibrium condition shown in Fig. 1.5b. The red circles indicate the intersections of the plane of the image with the spherical regions that correspond to the equilibrium conditions (Fig. 1.5b). The intensity distributions within these circles are functions of the electron density with the maxima in correspondence with the nuclei of the atoms. The atoms positions are outlined by the corresponding circles with the exception of the atoms Fig. 1.11 (a) Elementary cell in the tension region of Fig. 1.6a; (b) Elementary cell in the compression region 10 C.A. Sciammarella et al.
just after the extra-rows end indicated by yellow circles in Fig. 1.12a and outlined by yellow dots in Fig. 1.6b. These images show high intensity with little variation like blurred images. Just behind this atoms row there is a region of very low intensity indicating a low density of electrons. This region interrupts the sequence of the hexagonal patterns corresponding to the equilibrium positions of the atoms and is outlined with green dots. This region is the beginning of a cavity that located in the intersection of the yellow lines defining the orientation of the extra-rows. The next row of atoms, close to the yellow lines, shows a deformation in the direction of the Burger vector, 1120 , of approximately +13 %. This quantity is also compatible with the frequency pattern of Fig. 1.7. Figure 1.12b shows the packing of rigid magnetic spheres; it is possible to see that the model of rigid spheres, although an interesting visualization, does not provide the actual field of atoms seen before that can be approximated by deformable ellipsoids. In the theory of dislocations itZ is assumed that a sufficient large pile up of dislocation can transform the outlined area into an actual physical cavity, the starter of a crack. The cavity formed between the atoms has an interesting property that makes its presence detectable. Because of the low electronic density if positrons are sent to this cavity a pair formed by an electron and a positron is created, this pair is not stable and decays into twoγrays. Using aγ-rays detector it is possible to count the number of events and this number can be used to measure the number of cavities present in a specimen [21]. 1.7 Extension of Continuum Kinematics at Inter-atomic Spaces As mentioned before, three systems of fringes at orientation 120 to each other can be obtained from the FFT shown in Fig. 1.7: this is the equivalent of a three arm rosette. The correspondence between rosette orientations and directions (orthogonal to fringe orientation) in the FFT space is also shown in Fig. 1.7. From Eqs. (1.8 and 1.9) we can get the derivatives that can be input into a definition of strain tensor. For example, the components of the Almansi’s non-linear strain tensor referred to the Eulerian coordinates can be computed from these derivatives. For the coordinate system shown in Fig. 1.6a with h1000i x1 and x2 in the perpendicular direction, it can be obtained the Jacobian tensor [22]: J½ ¼ I½ F½ 1 ¼ ∂u1 ∂x1 ∂u1 ∂x2 ∂u2 ∂x1 ∂u2 ∂x2 2 666 64 3 777 75 ð1:10Þ Fig. 1.12 (a) Atoms location in the crystal region hosting the dislocation; (b) Simulation of the presence of a dislocation using magnetic spheres [19] 1 The Kinematics of Crystalline Arrays at the Subnanometric Level 11
Having known the components of the tensor [J], one can compute the components of the Almansi’s strain tensor as follows: εA 11 ¼ ∂u1 ∂x1 1 2 ∂u1 ∂x1 2 þ ∂u2 ∂x1 2 " # ð1:11Þ εA 22 ¼ ∂u2 ∂x2 1 2 ∂u1 ∂x2 2 þ ∂u2 ∂x2 2 " # ð1:12Þ εA 12 ¼ 1 2 ∂u1 ∂x2 þ ∂u2 ∂x1 ∂u1 ∂x1 ∂u1 ∂x2 þ ∂u2 ∂x1 ∂u2 ∂x2 ð 1:13Þ The above components are referred to the initial undeformed system of reference. Since the final coordinate system has a rigid body rotation, one can correct the derivatives for this rigid body rotation. In this case the corrections are negligible. In Fig. 1.6b, there are indicated the atoms 1 and 2 in whose neighborhood strain tensor components have been computed. Figure 1.13a shows again the same region of Fig. 1.6b while Fig. 1.13b shows the components of the Almansi’s strain tensor. The value of the maximum strain is e11 A ¼0.5886 and it is almost in the direction of 1120 , the rigid body rotation is 1.6 ; the minimum strain is e22 A ¼ 0.3593 and it is almost perpendicular to 1120 . The fringe analysis provides important information that is being retrieved from the 2D image produced by the electron microscope. The kinematics of the continuum has been extended to the inter-atomic space by the utilization of digital moire´. The presence of fringe dislocations in the fringe patterns in the same position as physical dislocations indicates the place where there are discontinuities in the displacement field (see Chap. 10.5.1 of [14]). The analyzed region is located inside the atomic field. Strains in this region must be compressive in one direction because in this region extra atomic planes have been introduced during the crystal growth process. Figures 1.14a, b provide the displacement derivatives in the indicated directions; these derivatives show a spatial distribution that is very similar to the pattern corresponding to the solution of the theory of elasticity. The blue lines of equal values of the derivatives correspond to the area subjected to compression while the red lines correspond to the area subjected to tension. The yellow lines correspond to the transition from tensile to compressive stresses. These two regions have two extreme values almost equidistant from the transition line. It was also observed that at point “C” of Fig. 1.13a the lattice constant is very close to the value for the perfect crystal (a¼0.3073 nm). This indicates the passage from compression to tension in the array, a neutral region. As it has been said before, this neutral region is present in the elasticity solution of an edge dislocation and separates the region where the extra planes are located (compression region) from the region that is expanded (region that does not contain extra planes), Fig. 1.14c. The moire´ method provides the Eulerian components of the strain tensor: that is, the strains at the location in the deformed configuration. Through the above analysis it is possible to have an idea of the magnitude of the deformation caused by dislocations generated during the growth stage of a 4HSiC crystal in the region where the cooling rates have altered the regularity of the crystalline array. Fig. 1.13 (a) Principal strains at points 1 and 2; (b) Magnified view of the neighborhood of point 1 showing the principal strains of the Almansi’s strain tensor and the corresponding rigid body rotation of the principal strain directions 12 C.A. Sciammarella et al.
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