Micro- and Nanomechanics

Encyclopedia of Nanoscience and Nanotechnology www.aspbs.com/enn Micro- and Nanomechanics Barton C. Prorok Auburn University, Auburn, Alabama, USA Yong Zhu, Horacio D. Espinosa,∗ Zaoyang Guo, Zdenek P. Bazant Northwestern University, Evanston, Illinois, USA Yufeng Zhao, Boris I. Yakobson Rice University, Houston, Texas, USA CONTENTS of grains in a given structure (e.g., through the thickness), and the role played by the manufacturing process (e.g., tex- 1. Mechanics of Scale ture, residual stress, and dislocation structure). This last fea- 2. Micro- and Nanoscale Measurement ture is particularly important since material surfaces and Techniques microstructures are the result of the process employed to 3. Micro- and Nanoscale Measured deposit or remove material [1]. Material Properties 4. Theoretical Modeling and Scaling 1.1. Thin Films 5. Modeling One-Dimensional Materials: As thin film dimensions begin to approach that of the films Nanotubes and Nanowires microstructural features, the material mechanical proper- Glossary ties begin to exhibit a dependence on the specimen size. In metallic thin films this translates to plastic yielding occurring References at increased stresses over their bulk counterparts. Although this phenomenon was observed as early as 1959 [2], no 1. MECHANICS OF SCALE consensus or common basic understanding of it yet exists. In addition to plastic behavior, other mechanical proper- It has been known for quite some time that materials and ties can exhibit size effects, such as fracture toughness and structures with small-scale dimensions do not behave in fatigue resistance. Each of these properties operates on the same manner as their bulk counterparts. This aspect a characteristic length scale that can be compared to the was first observed in thin films where certain defect struc- physical dimensions of microelectronics, microdevices, or tures were found to have deleterious effects on the film’s nanodevices. This is shown schematically in Figure 1, which structural integrity and reliability. This became a signifi- utilizes a logarithmic length scale map beginning at the cant concern because thin films are routinely employed as atomic scale and ending at the macro scale. On the left components in microelectronics and microelectromechanical are four categories of structures and the regime where their systems (MEMS). Their properties frequently allow essen- dimensional size fits on the length scale. On the right are tial device functions and therefore accurate identification regimes indicating where dimensional size effects begin to of these properties is key to the development of new tech- affect the material mechanical properties and theories used nologies. Unfortunately, most of our knowledge is based on to predict behavior. Elastic properties are dependent on the bulk material behavior, which many times fails to describe bonding nature of the material and only exhibit size effects material response in small-scale dimensions because of the at the atomic scale. In contrast, plastic, fatigue, and frac- dominance of surface and interface effects, finite number ture properties all exhibit size effects in the micrometer and submicrometer regime. These properties all depend on ∗ Corresponding author. defect generation and evolution, which are mechanisms that ISBN: 1-58883-061-6/$35.00 Encyclopedia of Nanoscience and Nanotechnology Copyright © 2004 by American Scientific Publishers Edited by H. S. Nalwa All rights of reproduction in any form reserved. Volume 5: Pages (555–600) 556 Micro- and Nanomechanics Macro World deformation depends strongly on the ability of dislocations Conventional Components 10 to move under an induced stress [8, 9]. The ease of their 1 –1 m Classical Plasticity movement can be hindered by any number of obstacles such 10 –2 • Continuum mechanics • No internal length-scale as grain boundaries, precipitates, interfaces, etc. Specimen MEMS Devices 10 –3 parameter size then begins to govern plastic behavior by creating geo- 10 –4 mm metrical constraints, surface effects, and the competition of deformation mechanisms (i.e., dislocation motion versus 10 Fracture –5 Strain-Gradient Plasticity 10 Sandia –6 Fatigue • Modified plasticity theory based on continuum mechanics twinning or phase transitions). Other effects that specimen 10 size can have on plastic deformation involve microstructural µm Plasticity Microelectronics and Thin Films –7 10 10 –8 Discrete Dislocation Mechanics changes. This includes grain size, morphology, and crystal- Intel 10 –9 nm lographic texture. Preferential grain orientations can result Nano-Devices 10 –10 Å Elasticity Molecular Dynamics Quantum Mechanics from a minimization of surface energies [10, 11]. The aver- age grain size is also typically on the order of the film thick- "ON" JUNCTION ELECTRODE • Limited by Current Atomic World computational power SUPPORT "OFF" JUNCTION Lieber ness, due to an effect called the “specimen thickness effect” which depends upon grain boundaries being pinned by their NANOTUBE INSULATOR Figure 1. Illustration of length-scale effects on the mechanical proper- surface grooves, occurring when the mean equivalent grain ties of materials. diameter is on the order of the film thickness [12, 13]. Several theoretical models, based on single dislocation motion, have been proposed to explain the size effect phe- operate on characteristic length scales [3]. These fundamen- nomenon [14–16]. However, each predicts strength increases tal changes in mechanical behavior occur in the size scale far below experimentally obtained values. The higher yield of MEMS and microelectronics devices, and thus, a better point of metallic thin films is likely the result of a combined understanding of inelastic mechanisms is required to better interaction between strain hardening and deformation mode predict their limits of strength and reliability. transition from dislocation motion to twinning. The right side of Figure 1 also lists the theories used Several pioneering studies have experimentally identified to predict material behavior and the length scale where the existence of size effects on the plasticity of metals. These they are applicable. These include classical plasticity, strain studies were able to obtain experimental nanoindentation gradient plasticity, discrete dislocation mechanics, molecu- data showing a strong size effect as evidenced by mate- lar dynamics, and quantum mechanics. Classical plasticity rial hardness decrease as indentation depth increases [17– is described in terms of traditional continuum mechanics, 19]. Figure 2 is a reproduction of the data from Ma and which describes the relationships between stress and strain and is applicable for predicting behavior from a size of approximately 100 micrometers and greater. Molecular dynamics (MD) is at the other end of the scale and involves the generation, mobility, and interaction between individual dislocations, twinning, stacking faults, and other defects. It is only applicable at the lower end of the scale since it is based on large scale numerical simulation. Therefore, it is subse- quently limited by current computational power (i.e., sys- tems approximately one million atoms in size). In the regime between classical plasticity and discrete dislocation mechan- ics, a theory called strain gradient plasticity has been devel- 800 oped in the last decade to describe material behavior [4]. It considers the effect of gradients in strain in the description [110] of flow behavior. A large number of theories have been pro- 700 [100] posed. However, as will be discussed in Section 4, most of Hardness H (MPa) these theories exhibit unreasonable predictions as the size 600 of the structure approaches one nanometer. For this rea- son, its applicability is limited to structures with minimum dimension of a few hundred nanometers. Between this the- 500 ory and molecular dynamic, a model based on discrete dis- location dynamics has been postulated [5–7]. These models 400 are currently in an early stage of development so they are not discussed here in detail. The mechanical response of thin films depends on many 300 0 0.5 1.0 1.5 2.0 factors. Of particular importance is the existence of film Plastic Depth h (µm) thickness effects that arise because of geometrical con- straints on dislocation motion. Size effects on mechanical Figure 2. Plot of hardness vs plastic depth illustrating how hardness properties begin to play a dominant role when one or more increases with smaller plastic depth. Reprinted with permission from of the structure’s dimensions begins to approach the scale [18], Q. Ma and D. R. Clarke, J. Mater. Res. 10, 853 (1995). © 1995, of the material microstructural features. The onset of plastic Materials Research Society. Micro- and Nanomechanics 557 Clarke [18] of nanoindentation performed on epitiaxially grown silver on sodium chloride and shows that the hardness increases by a factor of two to three as penetration depth decreases. Their results have been verified and expanded in subsequent studies [20–27]. Other pioneering work involved bending strips of metal of varying thickness around a rigid rod [28] and applying a torque load to copper rods of varying diameter [29]. The strips varied in thickness from 12, 25, and 50
m and each was bent around a rigid rod whose diameter was scaled to the film thickness to ensure identical states of strain in each strip. Figure 3 shows the results in the form of normalized bending moment versus surface strain. It is clear from this plot that as each strip was strained to the same degree, the thinner strips required a larger bending moment. The copper rod experi- Figure 4. Plot of applied torque vs twist for copper rods of varying ment used rods of varying diameter, 12–170
m. A torque diameter and stress–strain of identical rods subject to direct tension. was applied to each rod in order to twist them all to the same Reprinted with permission from [29], N. A. Fleck et al., Acta Metal. degree (i.e. to identical states of strain). A plot of torque ver- Mater. 42, 475 (1994). © 1994, Elsevier Science. sus twist per unit length is shown in Figure 4 and indicates an increase in strength by a factor of three for the smallest rod over the largest. Direct tensile tests were also performed on in this regime. The generally accepted size limit for accu- identically sized copper rods (Fig. 4). The authors concluded rate description of plasticity by the classical theory is systems that for the most part, no size effects existed when subjected with dimensions greater than approximately 100
m. As to direct tension. The results of these tests show a strength previously mentioned, molecular mechanics can accurately increase for smaller structures over larger ones when subject describe material behavior at the atomic level. However, to bending and torsional loads. due to the computational cost and limitations on perform- In these pioneering studies, the size dependence of the ing atomistic simulations for more than one million atoms, mechanical properties was considered to be a result of the maximum size regime computationally approachable is nonuniform straining [23, 29, 30]. It was shown that classical systems with dimensions <0.1
m [30]. This leaves an inter- continuum plasticity could not predict the size dependence mediate region where a continuum strain gradient plasticity theory, to describe material behavior, is highly desirable [4, 29–35]. In the aforementioned work of Fleck et al. [29], direct tensile tests were also performed on identically sized cop- per wires. The authors concluded that for the most part, no size effects existed for this case. It should be noted that the smallest rod diameter investigated by this group was 12
m. The homogeneous manner in which the uniaxial tests were conducted appears to have hindered gradients in plasticity from occurring. Can size effects then exist in the absence of strain gradients? Recent work on tensile testing of thin gold films of submicrometer thickness has shown that strong size effects do indeed exist in the absence of strain gradi- ents [36–39]. In these studies, grain size was held constant at approximately 250 nm while specimen thickness and width were varied systematically. Figure 5a is a composite scanning electron microscopy (SEM) image showing the side view of the three studied membranes with different thicknesses. For each thickness, there is a characteristic number of grains composing the film ranging from one to five. Stress–strain plots for these films are presented in Figure 5b. They show that the yield stress more than tripled when film thickness was decreased from 1 to 0.3
m, with the thinner specimens exhibiting brittle-like failure and the thicker a strain soften- ing behavior. It is believed that these size effects stem from the limited number of grains in the film thickness, which lim- its the number of dislocation sources and active slip systems. Figure 3. Plot of normalized bending moment vs surface strain illus- In such case, other deformation modes such as twining and trating how the thinner films require more bending moment for the grain boundary shearing accompanied by diffusion become same state of strain. Reprinted with permission from [28], J. S. Stolken dominant [36]. The same features were observed in other and A. G. Evans, Acta Mater. 46, 5109 (1998). © 1998, Elsevier Science. face center cubic (fcc) metals [36]. 558 Micro- and Nanomechanics a certain point, other types of defects, surface forces, and (a) intermolecular processes control mechanical behavior. Nanostructures can be described as either a bulk material with a grain structure of a nominal size in the range of 1 to 100 nanometers, or structures with one or more dimen- sions below 100 nm. A more rigorous definition is based on the functionality of the structure (i.e., when dimensions are such that new and unique properties can be achieved). An example of a nanostructure is a carbon nanotube, which is a molecular scale fibrous structure made of carbon atoms. They were discovered by Sumio Iijima in 1991 and are a subset of the family of fullerene structures [40]. The sim- plest way to describe the structure of a carbon nanotube is to imagine a flat plane of carbon graphite rolled into a tube, much the same as a piece of paper; see Figure 6. Like 400 t = 0.3 µm (b) (a) 350 t = 0.5 µm t = 1.0 µm 300 Stress (MPa) 250 Armchair Face 200 150 100 50 0 0 0.002 0.004 0.006 0.008 0.01 Strain Figure 5. SEM image highlighting the number of grains existing Zigzag Face through the film thickness (a). Note the various magnifications and 45 tilt employed during imaging. Stress–strain plot for a gold membranes 0.3, 0.5, and 1.0
m thick (b). Reprinted with permission from [37], H. D. Espinosa et al., J. Mech. Phys. Solids 51, 47 (2003). © 2003, Else- vier Science. (b) Clearly there are many things to learn about materials at this scale. Future investigations should be pursued and focus (c) on varying the material microstructure systematically to gain further insight into its effects on the size dependent plasticity phenomenon. Further progress can also be made by combin- ing experimental results with analytical and computational studies to better understand the fundamental deformation (d) mechanisms, particularly the mechanics of dislocation gen- eration, motion, interaction, and the competition between inter- and intragranular deformation processes. (e) 1.2. Nanostructures As structures move beyond the submicrometer to the nano- meter scale, description of mechanical behavior focuses on issues other than the traditional ensemble of defects. For instance, the length scale of a typical dislocation and the vol- Figure 6. Schematic drawings of a two-dimensional graphene sheet (a), ume of material required for it to have significant influence rolled-up sheet (b), and armchair (c), zigzag (d), and chiral (e) nano- on deformation are large compared to the typical volume of tubes. (c)–(e) Reprinted with permission from [41], M. S. Dresselhaus a nanosized object. Therefore, it can be argued that beyond et al., Carbon 33, 883 (1995). © 1995, Elsevier Science. Micro- and Nanomechanics 559 paper, a graphene plane can be rolled in several directions to achieve varying structures [41]. Figure 7 shows transmis- sion electron microscopy (TEM) images of carbon nano- tubes as a gathered bunch and in a rope-like form [42]. Since their discovery, many scientists have been fascinated by their unique and outstanding properties. Extensive arti- (a) (c) cles and books have been dedicated to carbon nanotubes. Among those, Qian et al. [43] contributed a comprehensive plastic flow review article, “Mechanics of Carbon Nanotube,” from the perspective of both experimentation and modeling. Dislocation theory has been used to describe relaxation and intramolecular plasticity in carbon nanotubes (CNTs) [44]. By analyzing the dynamic topology of the graphene wall of a CNT, Yakobson argued that dislocations dipoles result- ing from Stone–Wales (SW) diatomic interchanges play a key role in CNT relaxation under tension. The dislocation core is identified as a 5/7 pentagon–heptagon and the dipole as a 5/7 attached to an inverted 7/5 core; see Figure 8. When the dislocations unlock, one of two possible mechanisms occurs as a function of temperature. At low temperatures, c′ = (10,9) a mechanism of transformations 7/8/7 and then 7/8/8/7, etc., leads to the brittle failure upon formation of larger molec- ular openings such as 7/14/7. At high temperatures, the two dislocations glide away from each other in a spiral path. brittle fracture b = (0,1) When enough glide has taken place, they leave behind a (b) (d) nanotube of smaller diameter and changed electrical prop- c = (10,10) erties; see Figure 8. A dominant characteristic of nanostructures is that they possess a rather large surface area to volume ratio. As this ratio increases, interfaces and interfacial energy as well Figure 8. (a) In an armchair CNT, the first Stone–Wales rotation of an equatorially oriented bond into a vertical position creates a nucleus of as surface topography are expected to play a command- relaxation. SW rotations marked by arrows show further evolution as ing role in deformation and failure processes. The picture (b) a brittle crack or (c) a couple of dislocations gliding away from of nanoscale behavior can be viewed as the following: At each other. (d) The change of the CNT chirality and a stepwise change the larger end of the length scale, 50 to 100 nanometers, of diameter cause the corresponding variations of electrical properties. dislocation generation and motion will continue to dictate Formation of the next SW defect continues the necking process, unless material behavior. As the grain size or structural dimen- the dislocations pile up at insufficient temperature. Reprinted with per- sions fall below this range, behavioral control is transitioned mission from [44], B. I. Yakobsons, Appl. Phys. Lett. 72, 918 (1998). to surface and intermolecular mechanisms. Understanding © 1998, American Institute of Physics. the mechanics of these materials and structures, and the competition and relationship between their deformation types of concerns compose the field of nanomechanics, the mechanisms, will be essential to predicting their behavior foundation of which is currently being laid out. in applications of nanoscale electronics and devices. These 2. MICRO- AND NANOSCALE MEASUREMENT TECHNIQUES Mechanical testing at the micro- and nanoscales is quite challenging. Since the physical dimensions of specimens range from a few hundred micrometers down to as small as 1.0 nanometer, novel mechanical testing methods have been developed to successfully measure their properties. Speci- mens of such size are easily damaged through handling and it is difficult to position them to ensure uniform loading along specimen axes. They are also difficult to attach to the instrument grips. Testing has been shown to suffer from inadequate load resolution as well as having data reduction formulas that are hypersensitive to precise dimensional mea- surements. To minimize these effects, a variety of micro- and Figure 7. TEM image of a gathered collection of carbon nanotubes (a) nanoscale testing techniques have been employed to investi- and bunched in a ropelike form. Reprinted with permission from [42], gate size effects on mechanical properties. Reviews detailing V. Ivankov et al., Carbon 33, 1727 (1995). © 1995, Elsevier Science. the particulars are given in [15, 45–47]. 560 Micro- and Nanomechanics 2.1. Thin Film Measurement Techniques surround the penetrating tip with plasticity mostly occurring in the vicinity of the tip and elasticity occurring ahead of the The mechanical testing of thin films has been pursued for plastic front. The situation is best described as a complex a few decades. The methods are quite unique and diverse. interplay of elastic and plastic deformation processes. After They can be grouped into four categories: depth sensing the prescribed maximum depth is achieved the unloading indentation, bending or curvature, tensile tests, and micro- process begins. As the load on the indenter is reduced, elas- electromechanical system approaches. tic recovery of the material forces the indenter upward. The measured load-deflection signature of this unloading pro- 2.1.1. Depth Sensing Indentation cess is then governed only by the elastic properties of the Depth sensing indentation is a widely used method for esti- material. Typical loading and unloading curves are shown in mating the mechanical properties of materials whereby a Figure 9. These curves show the behavior for a hard  material’s resistance to a sharp penetrating tip is contin- and a ductile () material. In comparison, the harder mate- uously measured as a function of depth into the mate- rial requires a larger load to drive the indenter to the same rial. The result is a load-displacement signature with loading and unloading segments that describe material response. In recent years, instruments have been devel- (a) 2 Pmax oped possessing subnanometer accuracy in displacement and submicro-Newton accuracy in load [48–52]. Another major improvement in the indentation methodology is the ability Loading to continuously measure contact stiffness at any point dur- ing the test. These new tools have fueled interest in studying Load, P the mechanical properties of thin films and nanostructured S materials by nanoindentation. Unloading In indentation testing, the indenter tip geometry is gen- erally pyramidal or spherical in shape and fashioned from single crystal diamond. In micro- and nanoscale testing the most frequently used indenter is the three-sided Berkovich tip. This geometry allows the tip to be ground to a very 3 fine point, as opposed to the chisel-like point in four-sided 1 Vickers indenters, resulting in self-similar geometry over a hmax Displacement, h wide range of indentation depths. Other indenter geometries include spherical (good for defining the elastic-plastic transi- (b) 300 tion of a material; however, there is great difficulty in obtain- Fused Quartz ing high-quality spheres of sufficient hardness), cube-corner 250 (sharper than the Berkovich, but produces much higher stresses and strains in the indenter vicinity that results in 200 cracking), and conical (has self-similarity like the Berkovich Load (mN) with an additional advantage of no stress concentrations 150 Aluminum from sharp edges; however, like the spherical indenter there is difficulty in manufacturing high quality tips). Table 1 sum- 100 marizes the various indenter geometries. The process of driving the indenter into the material can be described as follows. After making contact, load 50 is applied to either maintain a constant tip displacement rate or a constant strain rate. Initially, deformation consti- 0 0 200 400 600 800 1000 1200 1400 1600 tutes only elastic displacement of the material, which quickly Displacement, (nm) evolves into permanent or plastic deformation as the load (c) is increased. Thus, zones of elastic and plastic deformation 1 3 2 Table 1. Geometries of various indenters. Parameter Berkovich Cube-corner Cone Spherical Vickers hmax Shape hf C-f angle 6535 35264 — — 68 Figure 9. Schematic of the typical load-displacement signature Projected obtained by nano-indentation (a). Comparison between hard (fused Contact area 245600d 2 25981d 2 a 2 a 2 24.5044d 2 quartz) and soft (aluminum) materials (b), and side-view schematic of C-f stands for centerline to face angle, d stands for indentation depth, and the nano-indentation process where the circled numbers correspond to a stands for tip radius for cone and spherical indenters. the points in the load-displacement curve above (c). Micro- and Nanomechanics 561 penetration depth, reflective of its higher atomic cohesion. where P is the load, h is the displacement, and A, m, and The harder material also stores more elastic energy than hf are constants determined by a least squares fitting proce- the softer material and undergoes less permanent deforma- dure. Elastic modulus is then obtained from a variation of tion. This is seen during unloading where the harder mate- Eq. (2), rial traces a path closer to its loading curve than the softer material. The softer material has a nearly vertical unload- dP 2 √ S= = √ Er A (5) ing signature. This is indicative of very little elastic recovery. dh  When comparing the residual indentation marks the softer material exhibits a larger and deeper mark. where  describes the correction of the area function due Indentation testing uses the maximum point of the load- to tip blunting, approximately 1.034 for a Berkovich inden- deflection signature to determine the hardness H  of the ter [54]. The advantage of the Oliver and Pharr method is material, defined as the ratio of applied load P  to the that the indent shape does not need to be directly measured. indenter/material contact area A: The method is based on the assumption that elastic mod- ulus is independent of indentation depth. By modeling the P H= (1) load frame and specimen as two springs in series the total A compliance, C, can be expressed as The contact area is determined using the relationship √ describing the surface area of the indenter with indenta-  1 C = Cf + Cs = Cf + √ (6) tion depth, given in Table 1 for the various tip geometries. 2Er A This value is referred to as the “universal hardness” and includes contributions from both elastic and plastic defor- where Cf is the load frame compliance and the compliance mation. With today’s instrumentation testing systems, this of the specimen, Cs , is S1 ; see Eq. (2). If the elastic modu- can be calculated on a continuous basis as a function of lus is constant with indentation depth, then a plot of C vs indentation depth. A−1/2 is a linear function whose intercept is Cf . For large Young’s modulus E is another important material prop- indentation depths, the area function for a perfect Berkovich erty that can be determined with nanoindentation testing. indenter can be used, namely, There are two methods for determining these properties based on the load displacement signature of the indent. The Ahc  = 245Hc2 (7) first involves a technique developed by Doerner and Nix [53] who took the approach that during the early stages of Equation (7) is also a good starting place to estimate the unloading, the contact area between the indenter and mate- area function of the Berkovich tip. The area function for the rial is constant. Thus the unloading stiffness, S, is related to tip is ascertained from a series of indents at periodic depths, the materials modulus via 100, 200, 400, 600, 1200, 1800 nm. For large indent depths dP 2 √ it also gives initial estimates of Cf and Er . These values can S= = √ Er A (2) then be plugged back into Eq. (6) to further estimate the dh  contact area for the successively smaller indents. This data where S is measured stiffness of the upper portion of the can be fit to the function unloading curve, dP is the change in load, dh is the change in displacement, A is the projected contact area, and Er is Ahc  = 245Hc2 + C1 h1c + C2 hc1/2 + C3 hc1/4 + · · · + C8 hc1/128 the effective modulus determined by (8) 1 1 −  2  1 − i2  = + (3) where C1 through C8 are constants. The first term is the Er E Ei perfect Berkovich tip and the others describe the blunting where E and  are the elastic modulus and Poisson’s ratio of the tip. In order to obtain the most accurate area func- for the specimen and Ei and i are the same parameters tion, this new value must be plugged into Eq. (6) again, the for the indenter. An important aspect of this method is process being iterated until convergence is attained. the accurate determination of contact area. By extrapolat- Oliver and Pharr confirmed their method by obtaining ing the linear portion of the unloading curve to zero load data for nine materials of varying properties and plotting (see Fig. 9), an extrapolated depth can be used to determine the computed contact area versus indentation depth with all contact area. A perfect pyramid can be assumed at inden- materials falling on an identical linear line defining the tip tation depths above 1
m. However, below 1
m a correc- area function. See Figures 24 and 25 in Oliver and Pharr tion factor must be used in order to account for blunting of [54]. Once the area function is known, it can be used to the tip point [53]. This was accomplished by employing the calculate E and H of other materials. TEM replica technique to determine the Berkovich indenter One inherent feature of these studies is that E and H shape. Deviations from a perfect Berkovich tip below 1
m are not directly measured and that some of the assumptions were identified. used in the data reduction may be violated. One of the key Oliver and Pharr [54] built on these solutions by realizing assumptions is the power law given in (3), which is obtained that unloading data is usually not linear but better described from a contact elasticity solution of a half space. These with a power law, assumptions become quite relevant in thin films for which substrate effects are part of the experimental signatures [55– P = Ah − hf m (4) 57]. Bückle [58] has recommended that indentation depth in 562 Micro- and Nanomechanics microindentation should not exceed 10% of the film thick- between the indenter and sample. The result is that pile-up ness in order to obtain reliable results. This is currently used tends to overestimate hardness and modulus due to higher as a guideline in nanoindentation, which limits the minimum contact areas and sink-in tends to underestimate hardness film thickness that can be reliably tested, using the stan- and modulus due to lower contact area. dard data reduction procedure given by Oliver and Pharr to Attempts to deconvolute the substrate effects were inves- approximately 1–2
m and larger. In general, this limit is tigated by Saha and Nix [56]. In this work, soft films that a function of substrate properties and film roughness. Nix were deposited on a variety of relatively harder substrates and co-workers developed a methodology to account for the were probed by nanoindentation; see Table 2. These systems substrate effect, which enables the property measurement of included cases where elastic homogeneity existed between much thinner films. the film and substrate, that is, when the expected elastic The load-depth signature is a complex quantity that is modulus of the film was equivalent to the substrate (e.g., affected by events such as cracking, delamination, plastic aluminum on glass), and cases where elastic modulus dif- deformation, strain hardening, phase transitions, etc. In par- fered significantly (aluminum on sapphire). By employing ticular, gradients in elastic and plastic strains exist and are the parameter P /S 2 , where P is the indenter load and S a function of indentation depth [22, 23, 33, 35, 59]. Thus, is the contact stiffness, and plotting it versus indentation many interpretative methodologies have been devised in an depth, a description of material behavior is obtained. This attempt to deconvolute these effects [60, 61]. parameter was first proposed by Joslin and Oliver [64], who A major complication of the nanoindentation technique, realized that both components, P and S, are directly mea- observed in some materials, is the so-called “pile-up” and sured in the test. They examined and combined their ana- “sink-in” of the material around the indenter tip [24, 62, 63]. lytical relations, Eq. (1) for P and Eq. (5) for S, to obtain Both effects are primarily a result of the plastic behavior of the material and have a significant effect on the contact area P 1  H = 2 (9) between the tip and specimen. Furthermore, the solutions S2  4 Er2 given are based on elastic contact mechanics and do not con- sider the effects of plasticity in the indentation process. The which shows that the parameter is independent of the con- inclusion of plasticity into the solution is a complex process tact area (i.e., tip calibration) and thereby is also not cor- since the constitutive equations are nonlinear. Also, mate- rupted by pile-up or sink-in effects. Since H and E remain rial properties such as yield stress and work hardening must constant with indentation depth for homogeneous materi- be included, which are heavily dependent on microstructural als, the parameter P /S 2 should be independent of depth aspects that can vary from specimen to specimen. as well. By this method, Saha and Nix have shown that Figure 10 illustrates the problems of “pile-up” and “sink- when elastic homogeneity between film and substrate is met, in.” Pile-up occurs in soft materials, such as metals, whereby the Oliver and Pharr data reduction method [54] allows the ease of deformation results in localized deformation in the decoupling between indentation size effects, at small the region near the indenter tip. The result is material piling depths, and substrate induced strain gradient effects, at large up around the indenter and effectively increasing the con- depths whereby a plateau representing true intrinsic mate- tact area between sample and indenter. By contrast, stiffer rial behavior is obtained. materials are more difficult to deform and the strain fields In the same work, Saha and Nix [56] also developed a emanating from the penetrating indenter spread further into method to account for substrate effects in cases that do not the material and effectively distribute the deformation to possess elastic homogeneity. Using the elastically homoge- more volume, further away from the indenter. The sink-in neous aluminum (Al) film on glass substrate as a starting effect arises from this feature and results in less contact area point, they first assumed that since the glass substrate is much harder than the aluminum, all plastic deformation is accommodated by the Al film with no deformation occur- ring in the substrate until the indenter makes contact with it. This allowed them to make a second assumption, basically that the film hardness measured for the Al/glass system was its intrinsic value and therefore also representative of the film hardness for the Al/sapphire system. With this knowl- edge they calculated the reduced modulus via Eq. (9) and Table 2. Film/substrate systems tested by Saha and Nix. Al/Al, Al/glass, W/sapphire are elastically homogenous systems. Films Substrates E (GPa) H (GPa) Aluminum (Al) aluminum 75 1 E = 75 GPa H = 1 GPa glass 73 6 Figure 10. Schematic representation illustrating how the contact area Tungsten (W) silicon 172 13 changes during pile-up and sink-in effects. Reprinted with permission E = 410 GPa from [24], K. W. McElhaney et al., J. Mater. Res. 13, 1300 (1998). © 1998, H = 15 GPa sapphire 440 30 Materials Research Society. Micro- and Nanomechanics 563 corresponding true contact area from Eq. (5). These values (a) provided a means to compare experimental work with an existing model developed by King [65]. The King model was actually a modification of a treatment developed by Doerner and Nix [53] who, in the specific case they studied, were able to include a term in the reduced modulus, Er , to account for the substrate effect. The King treatment built on this by modeling the system with a flat triangular indenter geometry resulting in a reduced modulus given by 1 1 − i2 1 − f2 1 − s2 −t/a = + 1 − e−t/a  + e  (10) Er Ei Ef Es where Es and s are the elastic modulus and Poisson’s ratio (b) of the substrate respectively, a is the square root of the projected contact area, t is the thickness of the film below the indenter, and  is a scaling parameter that depends on a/t, which varies with indenter geometry. Saha and Nix have modified this analysis through incorporating a Berkovich indenter geometry by replacing t in Eq. (10) with t − h, where h is the indenter displacement into the film. Thus, 1 1 − i2 1 − f2 1 − s2 −t−h/a = + 1 − e−t−h/a  + e  Er Ei Ef Es (11) Figure 11. Comparison between modified King model and Saha and Substituting constants for E and  of each component, Nix experimental data for reduced modulus (a) and film modulus (b) of with Ef taken from the elastically homogeneous case, yields aluminum film on the substrates indicated. Reprinted with permission the reduced modulus as a function of indentation depth from [56], R. Saha and W. D. Nix, Acta. Mater. 50, 23 (2002). © 2002, that can be compared with the experimental Er calculated Materials Research Society. through Eq. (9). Sara and Nix found that these two meth- ods are in good agreement for up to 50% of the film thick- ness, after which the experimentally obtained Er begins 2.1.2. Bending and Curvature to decrease. Finally, by incorporating the experimentally Bending and curvature testing consists of microbeam bend- obtained Er values into the modified King model, a plot ing, wafer curvature, and the bulge test. Bending tests of the film modulus versus indentation depth is obtained; on micromachined beams were first performed by Weihs see Figure 11. Here Ef is seen to agree well up to 50% et al. [77] and repeated by others [78–85]. The method of the film thickness before beginning to decrease indicat- involves deflecting a freestanding cantilever-like beam fixed ing that the King model is overpredicting the effect of the at one end to the substrate. Building such structures on the substrate. These data show that the modified analysis based micrometer scale is achieved with standard microfabrication on Eq. (11) is an effective method for estimating thin film procedures used in the microelectronics field. Dimensions modulus wherever substrate effects become relevant. are on the order of a few micrometers to submicrom- Hardness and elastic modulus are the properties most eter thickness, tens of micrometers wide and hundreds routinely measured by nanoindentation. However, it has also of micrometers in length. Structures of this size have an been shown to be an effective method for measuring other extremely low stiffness and therefore high-resolution load material properties such as fracture toughness in small vol- cells are required to perceive the response of the beam. umes [66, 67], strain rate sensitivity and internal friction Nanoindenters have been shown to provide such load res- [68], and thermally activated plastic flow [69]. The frontier olution and are routinely used to deflect such structures. of nanoindentation revolves around combining current mea- A schematic of a typical microcantilever structure being surement and analysis with finite-element techniques [61, deflected by a nanoindenter is shown in Figure 12. By this 70–73]. This combination will further aid the study of mate- method, simple elastic beam theory can be applied. Namely, rial mechanical behavior, especially those with nonlinear
3 features. Finally, the so-called “Holy Grail” of nanoindenta- Eb t k= (12) tion will be the generation of material stress-strain behavior 41 −  2  l from indentation data such that mechanical properties, such as yield and postyield properties, can be estimated. Further where k is the stiffness, E is the elastic modulus, b is the information on recent progress in nanoindentation can be cantilever width,  is Poisson’s ratio, t is the thickness, and found in the literature [74–76]. l is the length of the cantilever at the point of contact. 564 Micro- and Nanomechanics 35.0 l = 60 µm 30.0 Film to be 25.0 –3 Load (mN) x10 tested l = 80 µm 20.0 15.0 l l = 100 µm 10.0 5.0 l = 120 µm b t 0.0 0 2000 4000 6000 8000 10000 12000 Displacement (nm) Si Substrate Figure 14. Load-deflection curves comparing load-displacement signa- tures for cantilevers lengths of 60, 80, 100, and 120
m. Reprinted with Figure 12. Schematic three-dimensional (3D) view of a freestanding permission from [84], B. C. Prorok et al., Exp. Mech. (2003). © 2003, cantilever structure. Parameters are defined in the text. Society for Experimental Mechanics. An example of results obtained from this method are of the cantilever length. Defining an equivalent cantilever shown in Figures 13 and 14. Figure 13 is an optical image of length circumvented this problem [84]. The technique also a common single crystal Si atomic force microscopy (AFM) suffers from boundary bending effects and inhomogeneous tapping-mode tip and the corresponding load-deflection sig- distribution of the strain since the bending moment is not nature obtained after deflecting it with a nanoindenter [84]. constant along the length of the beam. Florando et al. [83] In this structure the Si is oriented such that the length of have put forth a solution to this issue by using a beam with a the cantilever is parallel to the [110] direction. The stiffness, triangular width. Stress-strain behavior for the material can k, was found to vary between 258 × 10−4 to 261 × 10−4 then be more accurately obtained. mN/nm which corresponds to a modulus of 166 to 168 GPa Another bending based test is the wafer curvature when using Eq. (12), close to that of the [110] direction method. Typically when a thin film is deposited on a sub- for Si, 170 GPa [86]. Another example of microcantilever strate at elevated temperatures and then cooled to room bending results is shown in Figure 14 for thin film ultra- temperature, the difference in the thermal expansion coeffi- nanocrystalline diamond (UNCD). The figure shows load- cient between the two materials will cause curvature in the deflection results on UNCD freestanding cantilevers at var- structure to accommodate the strain. Whether this curvature ious cantilever lengths [84, 85]. As expected, the stiffness is convex or concave is determined by which material has the decreased as cantilever length increased. Using Eq. (12), greater coefficient of thermal expansion and if the film has the modulus was found to be between 945 and 963 GPa. tensile or compressive residual stress. This technique deter- The microcantilever bending test has shown to be a viable mines film properties by taking advantage of the fact that method to measure elastic properties; however, the tech- stress in the film is proportional to the radius of curvature nique exhibits some unique features. For example, the ana- of the substrate [87], lytical solution is very sensitive to the measured thickness, Es ts2 as seen in Eq. (12) where its influence is to the third = K (13) 1 − s  6tf f power warranting that extra care must be taken to accu- rately measure the specimen thickness. Undercutting during where tf is the film thickness, ts if the substrate thickness, the release step introduces uncertainties in the measurement Es /1 − s  is the biaxial modulus of the substrate, and Kf is the change in curvature. This equation relies on the assump- (a) (b) 2.0 tion that the film completely accommodates lattice mismatch Single Crystal Si-AFM Tip with the much thicker substrate [88]. A further constraint limits its application when the maximum bending deflec- 1.5 tion exceeds more than half the thickness of the substrate, ts /2. Elastic and plastic properties can also be examined Load (mN) 1.0 by varying the temperature. The technique has been used to examine a variety of thin films [15, 88–99]. There are some complexities of the wafer curvature method that hin- 0.5 der accurate property measurement such as nonuniformity Slope = 2.58–2.62 ×10–4 mN/nm of substrate-material adhesion and temperature. All in all, 0.0 the method yields information on the material properties 0 2000 4000 6000 8000 when confined by the substrate. Displacement (nm) The third major bending based test is the bulge test, Figure 13. Optical images of (a) a silicon AFM tapping-mode tip and developed by Beams in 1959 [100], where a freestanding (b) corresponding load-displacement curve. Reprinted with permission film is deflected by applying pressure with a compressed from [84], B. C. Prorok et al., Exp. Mech. (2003). © 2003, Society for gas or liquid. These specimens also take advantage of stan- Experimental Mechanics. dard microfabrication procedures to define their structures. Micro- and Nanomechanics 565 The geometry of the typical specimen consists of a thin (a) film membrane that spans a cylindrical or rectangular cham- ber beneath. The film is fixed at the edges of the chamber such that the remainder of its structure is freestanding. The chamber is pressurized in a controlled manner that results in the freestanding film bulging upward. The resulting “bulge” height can then be measured by interferometry and other techniques. The test is designed to determine the in-plane mechanical properties of the film by eliminating specimen edge effects. Moreover, it also avoids the complexities of substrate material adhesion problems. The technique has evolved over the years to settle on high aspect ratio rectangular chambers due to the “wrinkling effect” caused by biaxial states of stress that develop near the corners. This geometry confines the effect to the vicinity of the rectangle’s short ends and allows uniform deforma- tion to occur in the middle of the structure. Figure 15 is a cross-section of this region. Here, H is the bulge height, a is the membrane half-width, and P is the applied gas pressure. Pressure is related to bulge height through the relation (b) 20 t 4Et P= H+ 4 H3 (14) a 2 3a 1 −  2  where 0 is the residual stress, t is the film thickness, E is the material elastic modulus, and  is the Poisson’s ratio. The result of a test is a pressure-deflection plot describ- ing the membrane behavior. A typical membrane response with several loading and unloading cycles is shown in Figure 16a for a freestanding Au film 1.8
m thick. A comparison between the bulge and tensile tests was made for Si3 N4 by Edwards et al. [101], whereby the elastic modulus of each technique was found to vary by as little a 1 GPa, 257 ± 5 GPa for tensile and 258 ± 1 GPa for bulge, and validating the bulge test as a viable wafer-level technique. The method is able to examine both elastic and plastic properties. As in the case of nanoindentation, the stress- strain state in the film is not measured directly and requires a data reduction procedure that accounts for boundary Figure 16. Pressure–height plot (a) and stress–strain plot (b) for bulge effects. However, in the case the high aspect ratio mem- testing of an evaporated Au membrane 1.8
m thick. Reprinted with brane, stress and strain are nearly uniform in the short direc- permission from [46], O. Kraft and C. A. Volkert, Adv. Eng. Mater. 3, tion and can be approximated by 99 (2001). © 2001, Wiley-VCH. 2 != H2 (15) Using these equations, stress and strain can be extracted 3a2 from the data. A plot of the stress-strain response of the E a2 P  = 0 + ! = (16) data in Figure 16a is shown in Figure 16b. Several studies 1 −  2  2t H have utilized this technique to test thin films [102–107]. The apparatus required to perform a bulge test is simple and the method is an easy way of evaluating the in-plane mechanical properties. However, sample preparation is involved and is Film restricted to thin films with tensile residual stresses. Films Material with compressive stresses can buckle; in such a case the ini- H tial dome height must be determined as accurately as pos- sible to avoid large errors in the experimental results. The a experimental values are more accurate when the membrane is flatter. P Substrate 2.1.3. Tensile Testing The previously mentioned techniques can all be character- Figure 15. Cross-section of the middle of a high aspect ratio rectangu- ized as methods that subject the specimen to gradients of lar membrane; parameters are defined in the text. strain, which at the micro- and nanoscales can complicate 566 Micro- and Nanomechanics extraction of material property data. Also, their flexibility with a large pad at the other end. Schematics of the archi- for testing specimens of varying geometry is limited. Thus, tecture and gripping process are shown in Figure 18. After the equivalent of a tensile test, customarily to that per- fabrication and release of the cantilever specimen, a probe formed on bulk samples, is desirable in this regard. Tensile is aligned and brought into contact with the specimen free testing is the most direct method for obtaining a mate- end to be gripped. An electrostatic attractive force is gener- rial’s mechanical properties. Loads and strains are measured ated between the two surfaces with an applied voltage. Up directly and independently, and no mathematical assump- to certain specimen dimensions, this force is rather large tions are needed to identify quantities describing the mate- compared to the force required to deform the specimen in rial response. Many researchers have been involved in an tension; therefore, the two remain rigidly fixed together as effort to establish a scale-equivalent test [36–39, 83–85, 108– long as the voltage is applied. Tensile testing is then achieved 125]. However, tensile testing at the micro- and nanoscales through piezoelectric actuation of the probe along the axis has been difficult to achieve. Difficulties arise from load of the specimen with displacements measured by a strain resolution, specimen fabrication, handling and mounting, gauge at the probe. Specimen dimensions in the gauged uniformity of geometry from specimen to specimen, and region are on the order of: length 30–300
m; width 2–5
m; independent measurement of stress and strain. The most and thickness 0.1–2.0
m. attractive features of direct tensile testing are that data Chasiotis and Knauss [112, 113] developed a testing pro- reduction is straightforward and the tests are much less cedure similar to that of Tsuchiya et al. Their test employs susceptible to geometrically induced errors. Nonetheless, electrostatic forces to pull the specimen pad to the substrate results reported in the many referenced studies vary some- while an ultraviolet curing adhesive then fixes a probe to what, reinforcing the need for an easy to use tensile test that it. This process ensures that the specimen experiences min- minimizes sample preparation, handling, and mounting that imum handling during attachment and improves alignment can produce numerous specimens of identical features. between the specimen and probe since the specimen is tem- Several noteworthy tensile testing schemes are worth porarily fixed to the substrate. The electrostatic force is then detailing here. Sharpe et al. [114–118] have developed a reversed through identical poling of each side to release the micromachined frame containing the specimen. The fabrica- tion process involves patterning the dog-bone specimen on Tensile force a silicon wafer and then etching a window underneath. The (a) Grip final structure is shown in Figure 17. The finished tensile specimen is then mounted in the testing rig and grips are attached at either end. The two narrow sides are then cut Electrostatic with a rotary tool to free the specimen from the frame sup- force port. A piezoelectric actuator is employed to displace the Free specimen and subject it to uniaxial tension. Load is mea- sured with a load cell possessing a resolution of 0.01 g and a load range of ±100 g, and strain with an interferometric strain displacement gauge. The typical width of specimens is 600
m and the gauge length is 400 mm. Two sets of orthog- onal strips are patterned on the surface of the specimen to reflect the laser beam used in the interferometric strain dis- Specimen thickness: placement gauge setup. When illuminated with the laser, the Fixed to substrate 0.05 – 0.1 µm strips produce interference fringe patterns that are used to measure strain with a resolution of ±5 microstrain. Another tensile testing technique, developed by Tsuchiya Power Supply et al. [122, 124, 125], employs an electrostatic force grip- (b) Load Probe ping system to load the film. The specimen is fabricated as a freestanding thin-film cantilever fixed at one end and Insulating Film Substrate Specimen Etching Hole Tested part Free End Probe Probe Electrodes Figure 17. A tensile specimen fabricated at MIT (a) and at CWRU (b). Figure 18. Schematics showing the architecture of the electrostatic grip Reprinted with permission from [118], K. M. Jackson et al., Mater. Res. system. Reprinted with permission from [125], T. Tsuchiya et al., Mater. Soc. Symp. Proc. 687 (2001). © 2001, Materials Research Society. Res. Soc. Symp. Proc. 687 (2002). © 2002, Materials Research Society. Micro- and Nanomechanics 567 combined pair from the substrate. Tensile testing proceeds Nanoindenter Wedge Tip in a similar manner to that of Tsuchiya et al. with the main Suspended Membrane difference being that measurement of strain fields is per- formed through AFM and digital image correlation (DIC). Another variation of the microscale tensile test employs the cantilever architecture, but it possess a ring at the free end rather than a gripping pad [120, 126]; see Figure 19. A probe with a diameter just smaller than the inner diameter Mirau of the ring is inserted and then pulled in the direction of the Objective (10 specimen axis to apply direct tension. An optical encoder is 0) S iW afe r used to independently measure displacement. Results were compared for two groups of specimens of significantly vary- ing lengths to eliminate the error in stiffness due to defor- Figure 20. 3D schematic view of the membrane deflection experiment. mation of the ring. By assuming that the effective stiffness was the same for each measurement the effect of the ring was canceled out. A problem with this test is the difficulty Another noteworthy microscale tensile test, called the of eliminating friction between probe and substrate. This membrane deflection experiment (MDE), was developed feature complicates the data reduction procedure and inter- by Espinosa and co-workers [36–39, 119]. It involves the pretation of the data. stretching of freestanding, thin-film membranes in a fixed– fixed configuration with submicrometer thickness. In this technique, the membrane is attached at both ends and spans a micromachined window beneath (see Fig. 20). A (a) nanoindenter applies a line-load at the center of the span to achieve deflection. Simultaneously, an interferometer focused on the bottom side of the membrane records the deflection. The result is direct tension in the gauged regions of the membrane with load and deflection being measured independently. The geometry of the membranes is such that they contain tapered regions to eliminate boundary-bending effects and ensure failure in the gauge region (see Fig. 21). The result is direct tension, in the absence of bending and strain gradients of the specimen. The MDE test has certain advantages; for instance, the simplicity of sample microfabrication and ease of handling results in a robust on-chip testing technique. The loading procedure is straightforward and accomplished in a highly (b) sensitive manner while preserving the independent measure- ment of stress and strain. It can also test specimens of widely varying geometry, thickness from submicrometer to several micrometers, and width from one micrometer to tens of micrometers. Probe LM PV θ Wafer ∆ PM PM Gauge Section Mirau Anchored Microscope Optics Objective To Wafer Figure 19. SEM image of the cantilever-ring architecture (a) and Figure 21. Side view of the MDE test showing vertical load being schematic of the loading process with experimental results for two spec- applied by the nanoindenter, PV , the membrane in-plane load, PM , and imens of different lengths (b). Reprinted with permission from [126], the position of the Mirau microscope objective. Reprinted with permis- S. Greek et al., J. Micromech. Microeng. 9, 245 (1999). © 1999, Institute sion from [36], H. Espinosa et al., J. Mech. Phys. Solids (2003). © 2003, of Physics. Elsevier Science. 568 Micro- and Nanomechanics The data directly obtained from the MDE test must then Optical image of a Gold thin film strip be reduced to arrive at a stress-strain signature for the mem- brane. The load in the plane of the membrane is found as a component of the vertical nanoindenter load by the equa- tions No stretching Moderate stretching, material Large stretching, localized elastic-plastic transition exceeded deformation, specimen near failure % PV tan $ = and PM = (17) LM 2 sin $ where (from Fig. 21) $ is the angle of deflection, % is the Deflection = 0 µm 20 µm Deflection = 33 µm Deflection = 53 µm displacement, LM is the membrane half-length, PM is the load in the plane of the membrane, and PV is the load mea- Stress-Strain Curve sured by the nanoindenter. Once PM is obtained the nominal stress, t, can be computed from PM Figure 22. Magnified area of a thin gold membrane as it is stretched t = (18) A until it breaks. Three time points of stretching are shown that include the interferometric displacement image and the resulting stress–strain where A is the cross-sectional area of the membrane in the curve. Reprinted with permission from [36], H. Espinosa et al., J. Mech. gauge region. The cross-sectional area dimensions are typi- Phys. Solids (2003). © 2003, Elsevier Science. cally measured using AFM [37]. The interferometer yields vertical displacement infor- mation in the form of monochromatic images taken at is achieved; see Figure 23. The toughness is computed from periodic intervals. The relationship between the spacing the equations between fringes, ', is related through the wavelength of the monochromatic light used. Assuming that the membrane 
a √ is deforming uniformly along its gauge length, the relative KIC = f af (20) deflection between two points can be calculated, indepen- W


2 dently of the nanoindenter measurements, by counting the a a a total number of fringes and multiplying by (/2. Normally, f = 112 + 0429 − 478 W W W part of the membrane is out of the focal plane and thus all
3 fringes cannot be counted. By finding the average distance a + 1544 (21) between the number of fringes that are in the focal plane of W the interferometer, an overall strain, !t, for the membrane can be computed from the following relation: where f is the fracture stress, a is the length of the  crack, and W is the width of the gauge region as shown in '2 +(/22 Figure 23c. !t = −1 (19) ' This relationship is valid when deflections and angles are small. Large angles require a more comprehensive relation to account for the additional path length due to reflection off of the deflected membrane. This task and further details are given by Espinosa et al. [37]. The interferometer allows for in-situ monitoring of the test optically. Figure 22 shows combined interferomet- ric images and stress-strain plots obtained from a typical membrane deflection experiment. The figure shows three (a) instances of stress in stretching a thin gold strip obtained from the test. The first is at zero stretch and the second is a at an intermediate stretch where the elastic–plastic behav- σ σ ior transition is exceeded and the strip is being permanently deformed. The third is at a large stretch and shows large W local deformation indicating the strip is near failure. Data a from MDE tests of thin gold, copper, and aluminum mem- branes have indeed shown that strong size effects exist in the absence of strain gradients [36]. (b) 5 µm (c) Recently, the membrane deflection experiment was extended to measure fracture toughness of freestanding Figure 23. (a) A scanning electron micrograph of the geometry of the films [127]. In this method, two symmetric edge cracks are UNCD membranes, (b) a magnified area of the edge cracks, and (c) the machined using a focused ion beam. A tip radius of 100 nm schematic drawing of the two-symmetric-edge cracks model. Micro- and Nanomechanics 569 2.1.4. Testing Methods Based on Microelectromechanical System Technology Microelectromechanical systems can be advantageously employed in the testing of micro- and nanoscale specimens. These devices consist of micromachined elements such as comb-drive actuators and strain sensors that are integrated components on the wafer. They have the potential to impact Figure 25. SEM micrographs of the fatigue test structure; (a) mass, the small-scale testing field through high resolution force (b) comb-drive actuator, (c) capacitive displacement sensor, and (d) and displacement measurements. Several possibilities for notched cantilever beam specimen are shown. The nominal dimensions actuation and deformation measurement exist. The methol- of the specimen are as indicated in the schematic. Reprinted with per- ogy is based on the fabrication of numerous specimens mission from [137], C. L. Mohlstein et al., Mater. Res. Soc. Symp. Proc. of identical geometry and microstructure through standard 687 (2002). © 2002, Materials Research Society. micromachining techniques. A promising MEMS-based testing approach has been end attached to a rigid mount and the other to a large perfo- developed by Saif et al. [128–132]. A single crystal micro- rated plate, which sweeps in an arc-like fashion when driven machined structure is used for stressing submicrometer thin electrostatically by a comb-drive actuator (see Fig. 25). The films. In-situ SEM or TEM can be performed using this resulting motion of the structure is recorded capacitively structure; see Figure 24. One end of the structure is attached by the comb-drive sensor on the opposite side. The result to a bulk piezoelectric actuator while the other end is fixed. is mode I stress concentration at the specimen notch. The Folded and supporting beams are employed to uniformly specimen is tested until failure occurs and yields fracture transfer the load to the specimen, which is attached to a and fatigue information about the material. Further details supporting fixed–fixed beam. This beam, of known spring can be found in the cited literature. constant, is then used as the load sensor. Two displacement Other MEMS techniques have also employed electrostat- elements are placed at either end of the specimen where ically driven comb-drives to perform other types of load- the magnitude of displacement is imaged directly from the ings. One such approach studied the effect of microstructure separation of beam elements. An innovative feature of the on fracture toughness through controlled crack propagation design is that the supporting beam structure is configured [140, 141]. The testing rig consists of a specimen anchored such that it can compensate and translate nonuniaxial loads to a rigid support at one end and linked perpendicularly to into direct tensile loads on the specimen. In other words, the a comb-drive actuator. The other end is attached to a beam piezoelectric actuator is not required to pull on the structure that connects to a comb-drive actuator (see Fig. 26). A notch exactly in a direction aligned with the specimen, thus solving is either micromachined into the specimen (blunt notch) or the difficult issue of loading device–specimen alignment. a crack is propagated into the specimen through a Vickers Another MEMS-based testing approach employs a comb- microindent made in close proximity to the specimen. This drive actuator to achieve time dependent stressing of the step is performed as an intermediate step during the micro- specimen through voltage modulation [133–139]. The device fabrication of the specimen. The cracks, which radiate from architecture consists of the microscale specimen with one the indent corners, travel into the specimen. Upon actua- tion of the comb-drive, the connecting beam applies mode I (a) (b) (c) (d) Figure 24. A SEM micrograph of the tensile test chip can be performed in-situ inside a SEM or TEM. The freestanding specimen is cofabri- Figure 26. SEM images a MEMS fracture device. (a) is the overall cated with force and displacement sensors by microelectronic fabrica- device architecture, (b) is a close-up of the specimen, (c) gives the tion. Reprinted with permission from [132], M. A. Hague and M. T. A. relative dimensions, and (d) shows a buckled support beam, a prob- Saif, in “Proc. of the SEM Ann. Conf. on Exp. and Appl. Mech.,” Mil- lem stemming from residual stress during fabrication that plagues many waukee, WI, (2002). © 2002, Society for Experimental Mechanics, Inc. MEMS devices. Image courtesy of R. Ballerini. 570 Micro- and Nanomechanics loading at the specimen notch or sharp crack. At a critical we review techniques and methodologies addressing these value of displacement, controlled fracture is attained. challenges. Another MEMS-based technique utilizes electrothermal actuation to load specimens in direct tension [142, 143]. 2.2.1. Manipulation and Positioning The device is designed such that slanted or axial beams of Nanotubes impose a deformation on the sample (see Fig. 27). The pro- duced Joule effect causes local heating and expansion of There are several methods used today to synthesize CNTs the beams. The thermal actuator pulls directly on the spec- including electric arc-discharge [148, 149], laser ablation imen, stressing it in uniform tension. Strain is determined [150], and catalytic chemical vapor deposition [151]. CNTs from an integrated capacitive sensor and verified through made by these methods are commercially available, although digital image correlation. The rig can also be employed still very expensive. During synthesis, nanotubes are usu- for fatigue testing by using a modulated voltage. However, ally mixed with residues including various types of carbon thermal actuation is hindered by a relatively slow response particles. For applications or tests, a purification process time. is required in most cases. In the most common approach, These MEMS techniques show great promise to test ever- nanotubes are ultrasonically dispersed in a liquid (e.g., iso- smaller specimens and are expected to have a great impact propanol) and the suspension is centrifuged to remove large on the development of nanoscale devices. When coupled particles. Other methods including dielectriphoretic separa- with finite element multiphysics modeling they should be tion are being developed to provide improved yield. able to provide an accurate description of nanoscale struc- Random Dispersion Random dispersion is the easiest tural response and associated features needed to predict method for most of the mechanical testing experiments to their behavior. These data are important for exploiting date, but it is only modestly effective. After purification, a micro- and nanoscale properties in the design of novel and small aliquot of the nanotube suspension is dropped onto reliable devices with increased functionality. a substrate. The result is CNTs randomly dispersed on the substrate. A metal layer is then uniformly deposited on top 2.2. Nanoscale Measurement Techniques of the substrate and patterned by a photolithography pro- cess, after which some of the nanotubes become pinned by The property measurement of one-dimensional nano- a grid of pads [152]. To improve the probability of nano- objects, such as nanowires (NWs) and CNTs, is extremely tube coverage, CNTs on the substrate are imaged inside a challenging because of the miniscule size. As such, early scanning electron microscope and then this image is digi- studies of their mechanical properties focused on theoret- tized and imported to the mask drawing software, where the ical analyses and numerical simulations. They allowed the mask for the subsequent electron beam lithography (EBL) is prediction of Young’s modulus, buckling and local defor- designed. In the mask layout, the pads are designed to super- mation, and tensile strength [144–147]. Owing to advances impose over the CNTs [153]. This process requires an align- in microscopy, especially scanning probe microscopy (SPM) ment capability of lithography with a resolution of 0.1
m and electron microscopy, nanoscale experiments employing or better. these tools have been developed. The main challenges in the experimental study of one-dimensional nanosize speci- Nanomanipulation SPM can be used both to image and mens are: (1) constructing appropriate tools to manipulate to manipulate carbon nanotubes [154]. Using AFM, an indi- and position specimens; (2) applying and precisely measur- vidual multiwalled carbon nanotube (MWCNT) was suc- ing forces in the nano-Newton range, and (3) measuring cessfully isolated from a group of overlapped MWCNTs. local mechanical deformation precisely. In the next sections, A “NanoManipulator” AFM system, comprising an advanced visual interface, teleoperation capabilities for manual control of the AFM tip, and tactile presentation of the AFM data, was developed at the University of North Capacitive sensor Carolina [155–157]. The NanoManipulator can take con- trol of the AFM’s probe, move it to the desired location, and manipulate atomic-scale structures. A software program integrates force feedback and AFM. A haptic interface, which is a penlike device, enables the users to remotely oper- ate the NanoManipulator. More recently, they combined AFM, SEM, and the NanoManipulator interface to produce Specimen a manipulation system with simultaneous microscopy imag- Specimen Chevron thermal actuator ing [294]. Electron microscopy provides the imaging capability for Thermal actuator manipulation of CNTs and NWs with nanometer resolu- tion. Various sophisticated nanomanipulators under either Figure 27. Micromechanical fatigue testers with: (a) an eight-beam SEM [294, 158, 159] or TEM [160–163] have been devel- chevron actuator and (b) a two-beam actuator. Reprinted with permis- oped. These manipulators are usually composed of both a sion from [142], E. E. Fischer and P. E. Labossiere, in “Proc. of the coarse micrometer-resolution translation stage and a fine SEM Ann. Conf. on Exp. and Appl. Mech.,” Milwaukee, WI, 2002. nanometer-resolution translation stage; the latter is based © 2002, Society for Experimental Mechanics, Inc. on piezo-driven mechanisms. The manipulators have the Micro- and Nanomechanics 571 capability of motion in three linear degrees of freedom, protrudes, with a tip having a radius of curvature compara- and some even have rotational capabilities. Several probes ble to that of a single-walled nanotube. are attached to the manipulator and can be operated inde- pendently. In general, the manipulation and positioning of nanotubes is accomplished in the following manner: (1) a 2.2.2. High Resolution Force source of nanotubes is positioned close to the manipulator and Displacement Measurements inside the microscope; (2) the manipulator probe is moved SEM, TEM, and SPM have been widely used in character- close to the nanotubes under visual surveillance of the izing nanotubes. These provide effective ways of measuring microscope monitor until a protruding nanotube is attracted dimension and deformation of nanotubes with nanometer to the manipulator due to either van der Waals forces or resolution. Electron microscopy uses high-energy electron electrostatic forces; (3) the free end of the attracted nano- beams for scattering (SEM) and diffraction (TEM). Field tube is positioned in contact with the probe and is “spot emission gun SEM has a resolution of about 1 nm and TEM welded” by the electron beam [164]; (4) the other end of is capable of achieving a point-to-point resolution of 0.1– the nanotube is placed at the desired location and “spot 0.2 nm. The resolution of SEM is limited by the interaction welded.” Nanodevices can be made using this approach. The volume between the electron beam and the sample surface. technique is being implemented with force feedback for hap- The resolution of TEM is limited by the spread in energy of tic control. Limitations in perception depth are so alleviated the electron beam and the quality of the microscope optics. [155, 159]. AFM has become a powerful tool in the characterization External Field Alignment Dc and ac/dc electric fields of CNTs due to its capability not only to map the surface have been used for the alignment of CNTs and nanoparticles topography with nanometer resolution but also to manipu- [165, 166, 301]. Microfabricated electrodes are typically used late CNTs. AFM can be operated in several modes: contact to create an electric field in the gap between them. A droplet mode, tapping mode (or force modulation mode), noncon- containing CNTs in suspension is dispensed into the gap tact mode, and lateral force mode [176–180]. The tapping with a micropipette. The applied electric field aligns the mode has been used to induce radial deformation of nano- nanotubes, due to the dielectrophoresis effect, which results tubes in addition to the contact mode and the lateral force in the bridging of the electrodes by a single nanotube. The mode [154, 181]. Scanning tunneling microscopy (STM) has voltage drop that arises when the circuit is closed (dc com- not been widely used in the mechanical testing of CNTs at ponent) ensures the manipulation of a single nanotube. this stage, but it shows enormous potential since it can reveal Dc/ac fields have been successfully used in the manipulation the atomic structure and the electronic properties of CNTs of nanowires [167], nanotubes [165, 295], and bioparticles [154]. The STM can be operated in two modes: constant [168–170]. current mode and constant height mode. Figure 28 provides Huang [171] demonstrated another method of aligning several typical images taken by SEM, TEM, AFM, and STM. nanotubes. A laminar flow was employed to achieve pref- Commercial force sensors usually cannot reach nano- erential orientation of nanotubes on chemically patterned Newton resolution. Therefore, AFM cantilevers have been surfaces. This method was successfully used in the alignment effectively employed as force sensors [164, 187, 188], pro- of silicon nanowires. Magnetic fields have also been used to vided that their spring constant has been accurately cali- align carbon nanotubes [172]. brated. Alternatively, MEMS technology offers the capa- bility to measure force with nano-Newton resolution. This Direct Growth Instead of manipulating and aligning CNTs point will be further discussed in Section 2.3. after their manufacturing, researchers have also exam- To date, the experimental techniques employed in the ined methods for controlled direct growth. Huang et al. mechanical testing of nanotubes can be grouped into five [173] used the microcontact printing technique to directly categories: resonance, bending, radial, tensile, and torsion grow aligned nanotubes vertically. Dai et al. [174, 298–300] loading. reported several patterned growth approaches developed in their group. The idea is to pattern the catalyst in an arrayed fashion and control the growth of CNTs from spe- 2.2.3. Measurement Techniques cific catalytic sites. The author successfully carried out pat- for Nanotubes and Nanowires terned growth of both MWCNTs and single-walled carbon Resonance Treacy et al. [184] estimated the Young’s nanotubes (SWCNTs) and exploited methods including self- modulus of MWCNTs by measuring the amplitude of their assembly and external electric field control. thermal vibrations during in-situ TEM imaging (Fig. 29). Nanomachining It is of scientific interest to open the cap The nanotubes were attached to the edge of a hole in of MWCNTs to investigate the nanotube inner structure 3-mm-diameter nickel rings for TEM observation, with one and intershell frictional behavior. Cumings and Zettl [253] end clamped and the other free. The TEM images were have implemented an electric sharpening method to open blurred at the free ends, and increasing specimen tempera- the ends of MWCNTs using TEM. The process involves ture significantly increased the blurring. This indicated that the electrically driven vaporization of successive outer lay- the vibration was of thermal origin. Blurring occurs because ers from the end of the MWCNT, leaving the nanotube core the vibration cycle is much shorter than the integration time intact and protruding from the bulk of the nanotube. This needed for capturing the TEM image. peeling and sharpening process can be applied repeatedly A nanotube can be considered as a homogeneous cylin- to the same multiwalled nanotube until the innermost tube drical cantilever of length L with outer and inner radii a 572 Micro- and Nanomechanics Figure 29. TEM micrographs showing the blurring at the tips due to thermal vibration at 300 and 600 K, respectively. Reprinted with permis- sion from [184], M. M. J. Treacy et al., Nature 381, 678 (1996). © 1996, Macmillan Publishers Ltd. Figure 28. Typical images of CNTs taken by (a) SEM (reprinted with MWCNTs. The actuation was achieved utilizing an ac elec- permission from [164], M. F. Yu et al., Science 287, 637 (2000). © 2000, trostatic field within a TEM (Fig. 30). In the experiment, American Association for the Advancement of Science.); (b) TEM the nanotubes were attached to a fine gold wire, on which a (reprinted with permission from [293], R. H. Baughman et al., Science potential was applied. In order to precisely position the wire 297, 787 (2002). © 2002, American Association for the Advancement of Science); (c) AFM (image courtesy of C. Ke); and (d) STM (reprinted with permission from [183], T. W. Odom et al., Nature 391, 62 (1998). © 1998, Macmillan Publishers Ltd.). The imaging resolution of various instruments is illustrated. and b, respectively. For such a structure, the square of the thermal vibration amplitude is given by 16L3 kT  −4 L3 kT A2 =  ≈ 04243 (22) Ea4 − b 4  j j Ea4 − b 4  where A is the amplitude at the free end, k is the Boltzmann constant, T is the temperature, E is Young’s modulus, and j is a constant for free vibration mode n. By comparing the blurred images, one can estimate the vibration amplitude and deduce the value of Young’s modulus. This method is fairly simple to implement and exploits available instrumentation, including TEM holders with heat- ing capability. As a matter of fact, this was one of the first experiments to measure Young’s modulus of carbon nanotubes. There are some drawbacks associated with this method. Its accuracy to determine the vibration amplitude by comparing the blurred images is limited, the shape of the nanotubes is not exactly identical to a cylindrical can- tilever, and the boundary conditions present some uncer- Figure 30. Dynamic responses to alternate applied potentials, (A) tainty. Krishnan et al. [185] later applied this method to absence of a potential, (B) at fundamental mode, and (C) at second SWCNTs. harmonic mode. Reprinted with permission from [160], P. Poncharal Poncharal et al. [160] measured Young’s modulus by using et al., Science 283, 1513 (1999). © 1999, American Association for the a method based on the mechanical resonance of cantilevered Advancement of Science. Micro- and Nanomechanics 573 near the grounded electrode, a special TEM holder with a recorded at various locations along the nanotube. Single piezo-driven translation stage and a micrometer-resolution crystal MoS2 was used as the substrate due to its low friction translation stage was used. Application of an ac voltage to coefficient. By modeling the nanotube as a beam, the F –d the nanotubes caused a time-dependent deflection. The res- data acquired by this method were used to estimate Young’s onant frequencies were then related to Young’s modulus, modulus. In Figure 31e, the response of a beam to a force P viz., applied at a distance a (along the x axis) from the fixed point  x = 0 is schematically illustrated. The governing equation 2j 1  E for the elastic curve is j = a2 + b 2  (23) 8 L2 . d4 y EI = −f + P 'x − a (24) dx4 where a is the outer diameter, b is the inner diameter, E is the elastic modulus, . is the density, and j is a con- where y is the deflection, I is the moment of inertia of the stant for the j harmonic. The elastic modulus can then be nanotube, and f is the friction force between nanotube and estimated from the observed resonance frequencies. This substrate. This term was considered small and omitted in the method requires the precise positioning of the nanotubes analysis. By integrating this equation and defining the spring against the counterelectrode, which can only be achieved constant at position x as kx = dP/dy, one can express it by a high-precision manipulator. The advantage is that the in terms of Young’s modulus and tube geometry, viz., resonance frequency can be much more precisely measured than the vibration amplitude. 3r 4 kx = E (25) 4x3 Bending and Curvature Falvo et al. [156] used AFM in contact mode to manipulate and bend a MWCNT resting on where r 4 /4 is the moment of inertia for a solid cylinder of a substrate with the assistance of a nanomanipulator. The radius r. AFM tip was used to apply lateral force at locations along Bending of nanotubes resting on a substrate is straight- the tube to produce translation and bending. One end of forward to implement. Nevertheless, it cannot eliminate the the nanotube was pinned to the substrate by e-beam carbon effect of adhesion and friction from the substrate. To solve deposition. After the bending, some of the deformed nano- the friction issue, Walters et al. [186] suspended the nano- tubes were fixed by the friction between the nanotubes and tube over a microfabricated trench and bent the nanotube the substrate and some returned to the undeformed config- repeatedly in lateral force mode. Salvetat et al. [187, 188] uration. Falvo et al. [157] applied this method to investigate introduced a similar method to measure Young’s modu- the rolling and sliding behaviors of nanotubes. lus of SWCNTs and MWCNTs. The nanotubes were dis- Wong et al. [152] measured Young’s modulus, strength, persed in ethanol and a droplet was deposited on a com- and toughness of MWCNTs by using AFM in lateral force mercially available alumina ultrafiltration membrane with mode (Fig. 31). In their method, nanotubes were dispersed 200 nm pores (Whatman Anodics). Some nanotubes were randomly on a flat surface and pinned to this substrate by suspended over the pores. The adhesion between the nano- means of microfabricated patches. Then AFM was used to tubes and the membrane was found sufficiently strong, so bend the cantilevered nanotubes transversely. At a certain that the nanotubes were effectively clamped. Using AFM location x along the length of each nanotube, the force in contact mode, the authors applied vertical load to the versus deflection (F –d) curve was recorded to obtain the suspended nanotubes and recorded the force and deflec- spring constant of the system. Multiple F –d curves were tion simultaneously. In this case, the nanotubes behaved like clamped beams subjected to a concentrated load. Radial Compression Shen et al. [154] performed an indentation test on MWCNTs using AFM. After separating the overlapped nanotubes using AFM in indentation/scratch mode, the authors used AFM in tapping mode to scan the tubes and selected a well-shaped tube to perform further testing. The sample stage was lifted against the AFM tip. After the tube made contact with the tip, the AFM can- tilever was bent and the tube was compressed. The can- tilever bending changed the position of the laser spot on the four-quadrant photodectector and thereby produced a volt- age signal proportional to it. When the signal reached a trig- ger value, the sample was retracted. The radial compression was obtained from the stage motion and cantilever deflec- tion, and the force was calculated using the known spring constant of the AFM cantilever and its deflection. In sum- mary, Shen et al. actually squeezed the nanotubes by moving Figure 31. Overview of the approach used to probe mechanical proper- them against the AFM cantilever, similar to nanoindenta- ties of NRs and nanotubes. Reprinted with permission from [152], E. W. tion. However, to study the same problem with the same Wong et al., Science 277, 1971 (1997). © 1997, American Association tool, Yu et al. [181] took a different strategy. They com- for the Advancement of Science. pressed the nanotubes by AFM while imaging the nanotubes 574 Micro- and Nanomechanics in tapping mode. In tapping-mode AFM, the cantilever was oscillated with amplitude A0 above the surface. When scan- ning the sample, the tip struck the sample at the bottom of each oscillation cycle. Such intermittent contacts lead to a decrease of cantilever amplitude of value A. The set point S was defined as the ratio of A/A0 . From the value of the set point, the authors deduced the contact force. Tensile Testing Tensile testing is the most widely used technique in macro- and microscale material characteriza- tion. In the testing of nano-objects, gripping and measuring force–displacement signatures is a major challenge. Direct stretch testing of nanotubes is hard to perform; however, ingenious experiments have been carried out. Pan et al. [189] used a stress–strain rig to pull a very long (∼2 mm) MWCNT rope containing tens of thou- sands of parallel tubes. They reported Young’s modulus and tensile strength for this very long MWCNT. Yu et al. [164] conducted an in-situ SEM tensile testing of MWCNTs with the aid of a SEM nanomanipulator (Fig. 32). A sin- gle nanotube was clamped to the AFM tips by localized Figure 33. Schematic representation of the intershell experiments electron beam induced deposition of carbonaceous material performed inside a TEM. Reprinted with permission from [175], J. inside the SEM chamber. The experiment setup consisted Comings and A. Zettl, Science 289, 602 (2000). © 2000, American Asso- of three parts: a soft AFM probe (force constant less than ciation for the Advancement of Science. 0.1 N/m) as a load sensor, a rigid AFM probe as an actuator, which was driven by a linear picomotor, and the nanotubes schematically in Figure 33. A MWCNT was fixed at one end mounted between two AFM tips. Following the motion of (Fig. 33a) and nanomachined at the other end to expose the rigid cantilever, the soft cantilever was bent by the ten- the inner tubes (Fig. 33b). A nanomanipulator was brought sile load, equal to the force applied on the nanotube. The into contact with the core tubes and was spot-welded to the nanotube deformation was recorded by SEM imaging, and core by means of a short, controlled electrical current pulse the force was measured by recording the deflection of the (Fig. 33c). In Figure 33d and e two deformation modes are soft cantilever. The force–displacement signature was then illustrated. In Figure 33d, the manipulator was moved right converted to stress versus strain data, allowing modulus and and left, thus telescoping the core out from or reinserting strength of the MWCNTs to be measured. Yu et al. [296, it into the outer housing of nanotube shells. The extrac- 297] applied the same method to investigate the mechanical tion and reinsertion process was repeated many times while properties of ropes of SWCNTs, and the intershell friction being viewed at high TEM resolution to examine for atomic- of MWCNTs. scale nanotube surface wear and fatigue. In Figure 33e, the Cumings and Zettl [175] accomplished an in-situ TEM manipulator first telescoped the inner core out, then fully tensile testing of MWCNTs with the configuration shown disengaged, which allowed the core to be drawn back into the outer shells by the intertube van der Waals force, conse- quently lowering the total system energy. A real-time video recording of the core bundle dynamics gave information per- taining to van der Waals and frictional forces between the tube shells. Torsional Testing Williams et al. [153] recently intro- duced a microfabricated device which offers the capability to conduct torsion tests (Fig. 34). They used an advanced fab- rication technique to make this task possible. They started with depositing metal pads by photolithography. Then, align- ment marks were deposited on a substrate by EBL and lift-off. The dispersion of MWCNTs onto the surface fol- lowed. SEM images were taken to help determine accu- rate locations of the paddles. Then the underlying silicon oxide was etched to suspend the paddles. The suspended paddles were deflected with an AFM installed inside the SEM. The AFM/SEM setup allowed direct measurements of the applied force and the paddle deflection. Assuming Figure 32. An individual MWNT mounted between two opposing AFM there is no bending, one could calculate the torque and tips and stretched uniaxially by moving one tip. Reprinted with per- the corresponding rotation of the nanotube. This is an orig- mission from [164], R. H. Baughman et al., Science 297, 787 (2002). inal method to perform torsion test of nanotubes. How- © 2000, American Association for the Advancement of Science. ever, an apparent drawback is that the applied force by the Micro- and Nanomechanics 575 the material deformation mechanisms are not imaged at the nanoscale at various loading states. Postmortem studies are conducted after unloading and further spec- imen micromachining. • In the case of in-situ HRTEM, the current loading stages can only apply a prescribed displacement and thus do not possess the capabilities for independently measuring loads with adequate resolution. Hence the observed deformations are not accurately correlated with loading history. One exception is the MEMS-based stage being developed by Haque and Saif [131, 132]. • In the case of functional or intelligent materials, the current experimental setups do not have the elec- trodes and architecture needed for investigating elec- trical properties under stress (i.e., electromechanical coupling). Single walled and multiwalled carbon nanotubes have been studied experimentally by means of AFM, SEM, and TEM. A major issue in all of these studies is the scatter of the data. For instance Young’s modulus may present a scatter of more than 100%; failure strains are much smaller than the strain predicted by means of MD calcu- lations [44]. The mechanics community has a special inter- Figure 34. (a) Photolithographically patterned leads and EBL- est in assessing possible sources of errors and limitations patterned alignment marks (two crosshairs). (b) The same area with of developed techniques. Among the most obvious, one can patterned paddles; the scale bars in (a) and (b) correspond to 40
m. mention: (c) The paddle touched the substrate due to the large curvature of the undercut metal leads. (d) Residual stress in the metal film and imper- • They do not directly and independently measure load fect adhesion caused the leads to lift off the substrate. (e) One end of and deformation, as is the case in larger scale exper- the paddle stuck to the substrate. (f) A Successfully suspended paddle. The scale bars in (c)–(f) correspond to 2
m. Reprinted with permis- iments. In fact, beam or string mechanics is used to sion from [153], P. A. Williams et al., Appl. Phys. Lett. 82, 805 (2003). infer local deformations. © 2003, American Institute of Physics. • They assume homogeneous deformations because they cannot identify CNT atomic defects and monitor their evolution. AFM introduces not only torsion but also bending of the nanotube. • Some use the same AFM tip to image and load the CNTs. As a result, imaging under loading is not possi- ble. 2.3. Frontiers in Nanoscale • They cannot sense local deformations at attachment or Experimental Techniques loading points and therefore premature failure due to local deformations is not identified nor quantified. 2.3.1. Limitations of Existing Techniques • Most experimental configurations cannot measure Thin films have been studied with a variety of tech- specimen electronic properties under well-defined load- niques including nanoindentation [15], tensile testing on ing conditions. millimeter and micrometer sized specimens [190–194], mem- brane deflection experiments [36–39, 84, 119], and in-situ New measurement tools, which can be integrated into high-resolution transmission electron microscopy (HRTEM) high-resolution imaging instruments, are necessary in order [195]. These techniques provided insight on various size to make further advances in the mechanics of CNTs, NWs, scales including grain size effects and demonstrated that, single crystal films, and polycrystalline films. MEMS tech- in the case of metals, below a characteristic grain size, a nology offers unique features such as generation of micro- transition occurs in the plastic deformation mechanism from Newton load and nanometer displacement measurements intragranular dislocation motion to grain boundary sliding with high resolution. It also provides the means to bridge accompanied by substantial grain rotation and/or diffusion size scales across several orders of magnitude as needed to of clusters of vacancies. For the case of gold films, a grain investigate nano-objects. size of 25 nm was identified as the characteristic size at which the transition is observed [195]. It should be noted that in-situ HRTEM findings are preliminary and that much 2.3.2. A Novel MEMS Approach for In-situ work lays ahead. Despite these important advances, one can Electron and Probe Microscopy highlight the following limitations: In order to overcome the aforementioned limitations, • In the case of nanoindentation and most microtensile Espinosa and co-workers [196, 197] developed a new exper- testing of small samples, the defects responsible for imental setup for the testing of thin films, nanowires, and 576 Micro- and Nanomechanics nanotubes. The setup is an integrated MEMS device consist- of the movable electrode is equal to the deformation of the ing of three well-defined components: an actuator, a spec- folded beams in the axial direction. Capacitance change is imen, and load sensor; see Figures 35. For the testing of proportional to the displacement of the movable electrode thin films, the setup consists of a thermal actuator, a spec- when such displacement is sufficiently small [198]. If a volt- imen, and a load sensor based on differential capacitance age bias V0 is applied on each fixed electrode there will be measurement. Thermal actuators have been used in the past a voltage change in the moving electrode, Vsense , given as to produce up to several milli-Newton forces. In the device
 shown in Figure 35a, a set of slanted beams connected to Vsense %d %d 3 = +o (26) a trunk provides the actuation when a current is circulated V0 d d between the fixed pads. Thermal expansion of the doped polysilicon beams results in a displacement of the trunk and, where d is the gap between movable and fixed electrodes, consequently, the loading of the sample [142]. The device %d is the movable electrode displacement in the axial direc- works in displacement control, which is very advantageous tion, and V0 is the bias voltage. If %d is much smaller than in the study of thin films. the gap d, Vsense is proportional to %d. When the spring con- Sensors based on differential capacitance measurements stant of the folded beams is characterized, the force applied are well established (e.g., Analog Devices’ and Motorola’s on the load cell is proportional to Vsense . Measurements of MEMS accelerometers). In this approach, the differential specimen deformation are achieved by in-situ microscopy. capacitor serves as a load sensor upon proper calibration. Direct measurement of local displacements and strains is A readout chip, manufactured by Microsensors Co. [198], very important. It is known that peculiarities observed in is being employed to measure differential capacitance with the elastic, plastic, fracture, and transport properties of thin femtofarad resolution. The chip suppresses parasitic capac- films are directly related to atomic structure and associated itance and provides adjustable internal capacitors to select defects. This point is addressed further later. range and resolution. Here the concept of a two-chip sys- The lumped model of this device is shown in Figure 35b. tem, a MEMS chip for sample loading and another com- Compatibility and equilibrium equations are also given in plementary metal oxide semiconductor chip for capacitance the figure. To illustrate the approach, tensile testing of measurement, is employed; see Figure 35c. The movement polysilicon film can be examined as an example for design purposes. Considering the failure strain of polysilicon is about 1% [117], the required force from the thermal actu- (a) Pad ator to break the polysilicon specimen was calculated. The Capacitive Load Sensor structure of the thermal actuator (e.g., number of slanted beams and geometry of each beam) was then designed after Specimen thermal analysis [199, 302]. These estimates were also ver- Folded Beam ified using ANSYS multiphysics (from ANSYS Inc.). They show that the polysilicon specimen can be deformed to fail- ure and that the resolution of a differential plate capacitor can be used to measure the load sensor motion. Thermal actuator For the case of CNT testing, the experimental setup is similar to that of thin film testing. One difference is that a AFM image comb-drive actuator is used (force control) instead of a ther- Pad mal actuator (Fig. 36). The force to break a CNT is only sev- eral micro-Newtons, which can be achieved by a comb-drive (b) XT XL Governing Equations: actuator. A comb-drive actuator can accommodate a sev- XS = XT – XL eral micrometers motion range, which is required to study KT F KS KL KS XS = KL XL the nanotribological behavior of MWCNTs after the outside KT XT = F – KS XS shell failure. Due to the limitation in space we do not pro- (c) vide all the details of the lumped model analysis for this device; however, we just mention that the analysis is very C1 Cf similar to one previously discussed for the thermal actuator. – output The 3D nanomanipulator shown in Figure 37 is used to + LPF mount the nanosize specimen between the comb-drive actu- C2 Vref ator and the capacitive load sensor (Fig. 36). Several imag- synchronous demodulator ing tools are employed to measure specimen deformation and to identify defect initiation and evolution. In the next Figure 35. (a) MEMS device for in-situ AFM/SEM/TEM electrome- section, we report displacement measurement performed by chanical characterization of polycrystalline nanoaggregates films. The DIC of AFM images obtained at various deformation levels. whole system can fit in a 3 mm × 3 mm area. (b) Lumped model of the device shown in (a), where XS is the deformation of the specimen, XL 2.3.3. In-situ AFM Results is the displacement of the load sensor, XT is the displacement of the thermal actuator, KS is the stiffness of the specimen, KL is the stiffness Using the MEMS device shown in Figure 35, testing of of the load sensor, KT the stiffness of the thermal actuator, and F is the polysilicon thin films with in-situ AFM displacement and total force generated by the thermal actuator. (c) Two-chip architecture strain measurements of the specimens was performed. AFM used for measuring the load. was employed to scan the specimen surface before and after Micro- and Nanomechanics 577 Carbon Nanotube Gap for specimen Leaf springs (a) In-situ TEM nanoindentation Capacitive load sensor Specimen Comb drive actuator CNT (indenter) (b) h F = Nε0 V2 dA XL XL Vsense = V0 [ + o( )3 ] dS dS Figure 36. MEMS actuator for in-situ SEM/TEM/STM electromechan- ical characterization of carbon nanotubes. Various configurations will be investigated. The dimensions of the device will be such that the chip will fit in a TEM holder. Actuation and sensing pads will be wire bonded (c) to a small breadboard and from there will be wired through the TEM holder feedthrough (see bottom image). Figure 38. (a) AFM image of the topography of the specimen surface before loading. (b) AFM image during the loading. (c) Displacement contour computed by DIC within the area shown in (a) and (b). The load was applied in the x-direction. Reprinted with permission from the loading. DIC was then used to process the AFM data [197], Y. Zhu et al., in “Proc. SEM Ann. Conf. Exper. Appl. Mech.,” to quantify the displacement/strain field. Figure 38a and b 2003. © 2003, Society for Experimental Mechanics, Inc. shows two AFM images of the specimen before and after loading [196, 197]. The first image was obtained before 2.3.4. Opportunities in Nanomechanical the application of a voltage to the thermal actuator, while the second image was obtained when a 5 V was applied. Imaging of Deformation Figure 38c shows the displacement field obtained using DIC In this section we summarize efforts underway to elucidate for a scanned area of 8
m × 2
m. The displacement con- deformation and failure mechanisms in thin films, nano- tours show that the thermal actuator symmetrically stretched wires, and nanotubes. Special emphasis is placed on tech- the specimen, in the x-direction, as expected. Moreover, the niques that are being used in conjunction with the MEMS planarity of the device was investigated with an optical sur- setup discussed in the previous section. face profiler and it turned out that the device was flat and In the investigation of thin films, extensive in-situ TEM parallel to the substrate within 40 nm. work is essential to reveal the material behavior under stress at the grain level. Diffraction contrast and convergent beam electron diffraction at and around defects, with a beam diameter of a few angstroms, can be used to obtain lattice parameters and strain under load on ultrathin specimens. Higher order Laue zone line patterns can be compared to computer simulations of the deformed grains for this pur- pose. A critical aspect is the identification of dislocation sources and their densities, which are commonly assumed in atomistic, discrete dislocation, and other models. Other TEM techniques such as weak-beam dark-field microscopy and Moire patterns for defect characterization are also possible. These techniques enhance the observa- tion of defects such as dislocations, twins, and stacking faults on real space in real time. This effort is particularly relevant to assess the effectiveness of atomistic models in capturing defect distribution, annihilation, interaction with grain boundaries, and free surfaces. Defects sources, density, speed, and its relationship to material nanostructure and composition can also be quantified through this approach. Figure 37. Klocke Nanotechnik nanomanipulator within a LEO field- Moreover, the onset of inelasticity and fracture characteri- emission SEM [159]. zation at the atomic level is also possible. 578 Micro- and Nanomechanics Stack and collaborators [161, 200, 201] have performed that forces and atomic displacement are measured indepen- in-situ nanoindentation TEM studies by employing a TEM dently and directly. holder containing a piezo-actuator and a specially microfab- ricated wedge-shaped specimen. Observations of film defor- mation and nucleation of dislocations were observed in real 3. MICRO- AND NANOSCALE time. Unfortunately, due to the hysterisis of the piezo- MEASURED MATERIAL actuator and other experimental limitations, quantitative measurements of force–displacement were not possible. The PROPERTIES MEMS device shown in Figure 36 may be one approach to Thin films have long been harvested by the microelectron- overcome this limitation. Here the specimen is integrated ics industry for their unique properties. Conventional think- to the load sensor during the microfabrication steps and a ing has usually categorized their electrical properties as the sharp indenter (e.g., a CNT) is mounted to the trunk of property of primary importance. In the past decade and a the MEMS thermal actuator using a 3D nanomanipulator half though, other nonelectronic, chemical and mechanical (Klocke Nanotechnik Co.); see Figure 37. This 3D nanoma- properties have also been found to have great significance nipulator has already been implemented for in-situ SEM [15, 93]. Mechanical properties, in particular, are critical site-specific nanowelding of carbon and other nanostructures when one is concerned with the fact that devices must have [159]. structural integrity and also be reliable throughout their life Currently, the best TEM in the world has a point-to- expectancy. point resolution of about 1 angstrom. It is anticipated that Thin film materials that are widely used in microdevices transmission electron microscopes will reach a resolution include metallic, silicon-based, and carbon-based substances. of subangstrom in the next few years by means of spheri- As diverse as these materials are in the atoms that com- cal and achromatic aberration corrections. This will make pose them so are their properties. Processing techniques and feasible the identification of interatomic potentials through parameters for films are numerous and directly affect the atomic imaging of crystal planes. Using the device shown film microstructure. It is not surprising then that films of the before, experiments performed on single crystal and bicrys- same material that are processed by different methods can tal specimens could be employed to identify the interatomic have widely varying properties. The methods employed to potentials used in atomistic and molecular dynamic simu- measure the properties listed in this section also vary signif- lations. For particular crystal orientations, selected a priori, icantly and therefore so do data values as well. the atomic displacement field around a dislocation core or This section is designed to introduce the reader to thin an interface could be mapped. HRTEM images of the crys- film properties that have been measured thus far. No con- tal structure can be interpreted using software based on fast trasts or categorizations are made to separate how process- Fourier transform formalism and other approaches [202– ing or test methods affect the measured data in most cases. 204]. A cross-correlation technique can be used to assess the The scientific community has not yet come to a consen- accuracy of the simulated images. This work would certainly sus on a uniform testing and characterization methodology constitute a milestone in materials research. by which measurements can be made and classified in a For the study of nanotubes and nanowires, TEM can uniform and reproducible manner. Much work lies ahead provide information on chirality and other structural fea- before thin film behaviors, such as size effects, are fully ture [205] but is unlikely to provide atomic images from understood and before theories capable of predicting behav- which deformations can be computed. By contrast, ultravac- ior are developed. It should be noted that a number of uum STM has been successfully employed to obtain atomic the references garnered for this section have been previ- images of CNTs [182, 183]. It remains to be determined if ously collected in a review by Sharpe [207] and readers are this imaging capability could be performed in combination directed there for additional citations. with the MEMS device presented here. Capturing atomic structure under various loading degrees would be the ulti- mate goal in these studies. 3.1. Metallic Materials Electromechanical properties of CNTs are of particular Metallic materials serve many engineering functions in interest due to their potential in NEMS. Previous work by microdevices including both electrical and mechanical com- Tombler et al. [206] has shown that the conductivity of CNTs ponents. Metals display a wide range of mechanical behavior can change by several orders of magnitude when deformed including elasticity, plasticity, creep, fatigue, and fracture. by an AFM tip. The MEMS setup shown in Figure 36 Only in recent years has the mechanical behavior of thin has embedded interconnects and pads for electromechan- film metals become of concern in the design of microdevices. ical characterization of nanotubes under loading. By con- Metal films have been found to be susceptible to electro- necting the sensing pads to a signal analyzer, the electrical migration and other diffusion driven processes, stress and conductance can be measured under stress. Likewise, for the defect formation due to thermal effects, and changes in elec- case of MWCNTs, sliding forces between outer and inner trical behavior due to straining. These processes have dele- shells can be measured upon outer shell fracture. Therefore, terious effects that often lead to device failure. on-chip nanoscale tribological properties can be identified The more popular metals used in microdevices include with subnano-Newton force resolution and subnanometer gold, aluminum, copper, and nickel. These materials are displacement resolution. The main difference between pre- mainly deposited by sputtering or electron-beam evapora- vious work and the experimental approach here discussed is tion, processes that contain numerous parameters directly Micro- and Nanomechanics 579 affecting microstructure. They are normally in polycrys- previous section. Table 4 provides a compilation of some talline form with grain size on the order of film thick- results achieved over the past several years. Most of these ness. They typically possess significant textures. Table 3 results, in terms of Young’s modulus, agree well with estab- lists properties for the metals listed. With the exception of lished benchmark values. Fracture strength varies consider- Au, Young’s modulus of thin films mostly agrees with bulk ably and is governed by the orientation of the crystal as well polycrystalline values. Several materials showed increases in as the surface features particular to each micromachining yield stress as specimen size decreased, as was previously process [217]. shown for Au in Figure 5. In general, the yielding behavior Polycrystalline silicon has become a commonly employed of thin films tends to require larger stresses than their bulk material in microdevices. It is normally employed as the counterparts. This is a topic of high interest to researchers structural part of the device due to its high melting point, and is currently a major research thrust toward the devel- ease of growth and micromachineability, and somewhat opment of theories and models that can accurately predict favorable mechanical behavior [219]. It also has the dis- their behavior. tinction of having appreciable piezoresistive behavior for the transduction of deflection or other electromechanical 3.2. Silicon-Based Materials coupled variables. Polysilicon has been the focus of more thin film micromechanical properties measurements than Most materials used in microdevices are silicon-based, which any other material thus far. Table 5 lists the results from typically exhibit linear elastic behavior followed by brittle recent studies. A more complete list of experimental reports fracture. Silicon and silicon-based materials have been the can be found in [207]. dominant materials in the microelectronics revolution of the 20th century and the precursor to the microelectrome- For the most part, it can be said that Young’s modulus chanical/nanoelectromechanical systems revolution currently varies moderately between techniques. The same can be said underway. It has been the material of choice for current for fracture strength when considering that it is a function MEMS devices, mainly because devices can be fashioned of the surface flaws present. The effect of specimen size on using standard microfabrication techniques [215, 216]. The fracture strength was extensively studied by Sharpe et al. materials discussed in this section include single crystal sil- [229] and Tsuchiya et al. [122]. These pioneering studies icon, polycrystalline silicon, silicon dioxide, silicon carbide, were able to show that as specimen size decreased, trans- and silicon nitride. lating to a reduction in surface area, the fracture strength Single crystal silicon has been a vehicle for the fabrication increased due to a lower population of surface flaws. of microelectronics for several decades. It also serves as the Silicon Oxide (SiO2 ), nitride (Si3 N4 ), and carbide (SiC), most common structural material used in MEMS. Its electri- and combinations thereof, are silicon-based materials of cal properties have been well characterized over the years; growing importance in microdevices. They are less common however, the mechanical properties, such as fatigue and frac- than single or polycrystalline silicon as structural compo- ture toughness, have only recently begun to be tackled. The nents due to residual stresses that develop during processing. mechanical properties measured thus far have been shown to Silicon dioxide and silicon nitride are commonly used as sac- be dependent on the micromachining process utilized to pre- rificial layers, etch stops, or electrical/environmental passiva- pare specimens and their resulting surface conditions. The tion layers. Silicon nitride and silicon carbide are extensively techniques employed to test single crystal specimens have used to make membranes for micropumps, pressure sen- been diverse and include many of the methods listed in the sors, support for X-ray masks, etc. Table 6 lists a summary Table 3. Summary of data on thin film metals. Young’s modulus Yield strength Tensile strength Material (GPa) (GPa) (GPa) Method Ref. Au-size effect 53–55 0.055–0.220 0.78–0.35 MDE [36, 38] Au 40–80 — 0.2–0.4 tension [2] Au 74 0.26 — indentation [77, 208] Au 57 — — bending [77, 208] Al-size effect 65–70 0.150–0.180 0.240–0.375 MDE [36] Al 24.2–30.0 0.087–0.105 0.124–0.176 tension [209] Al 69 — — tension [128] Al 69–85 — — bending [210] Cu-size effect 125–129 0.200–0.345 0.45–0.80 MDE [36] Cu-size effect 120–132 0.120–0.480 tension [211] Cu 86–173 0.12–0.24 0.33–0.38 tension [212] Cu 108–145 — — indentation [212] Ni-thick 176 ± 30 032 ± 003 0.55 tension [114] Ni-thin 231 ± 12 155 ± 05 247 ± 007 tension [213] Ni-LIGA 181 ± 36 033 ± 003 044 ± 004 tension [214] Ni 156 ± 9 044 ± 003 — tension [212] MDE = membrane deflection experiment. 580 Micro- and Nanomechanics Table 4. Summary of data on single crystal silicon. Young’s modulus Fracture strength Direction (GPa) (GPa) Method Ref. 100, 110, 111 130, 170, 185 — benchmark [86] 100 168 — indentation [218] 100 (doped Si) 60–200 — indentation [151, 219] 110 163–188 3.4 indentation, MCD [208] 110 166–168 — MCD [84, 85] 110 177 ± 18 2.0–4.3 bending [220] 110 (different fab.) — 1.0–6.8 tension [125] 111, 110 — 1.3, 2.3 tension [221] 110 — 1.2 tension [222] 110 150 0.3 tension [223] 110 147 0.26–0.82 tension [224] 100, 110, 111 125–180 1.3–2.1 tension [225] 100 142 ± 9 1.73 tension [213] 110 1692 ± 35 0.6–1.2 tension [226] 100, 110, 111 115–191 — tension [121] 110 — 8.5–20 torsion [227] 75 (shear) — torsion [228] MCD = microcantilever deflection. of properties measured thus far. Silicon nitride and carbide that include ultrananocrystalline diamond, nanograined dia- both show future potential as mechanical components for mond, micrograined diamond, and diamond-like carbon. microdevices due to their high Young’s modulus. However, Each of these materials is fabricated using varying pro- more extensive studies are required to understand how their cessing schemes to achieve their microstructures with the thin film structure affects fracture strength, etc. resulting mechanical behavior being a function of these pro- cesses, microstructures, and bonding characteristics. Dia- mond materials typically have large elastic moduli reflective 3.3. Carbon-Based Materials of their strong cohesive binding and, being brittle in nature, Carbon in its various forms may become a key material for fail by means of crack propagation initiated at flaw sites. The the manufacturing of MEMS/NEMS devices in the 21st cen- UNCD and micro- and nanograined materials all yield large tury and will most probably displace silicon-based materials elastic moduli since they are composed of crystalline grains. in devices. Carbon materials have exceptional and tailorable Diamond-like carbon is described as having an amorphous properties with the potential to meet the stringent demands structure consisting of a mixture of sp2 and sp3 bonding from that MEMS/NEMS devices and other thin film applications both graphite and diamond bonding and therefore possess- require. These include UNCD, diamond-like carbon, amor- ing a lower elastic modulus. phous diamond, and carbon nanotubes. To date, the aggre- Fracture strength has only recently been measured; results gate of testing performed on these materials is preliminary are listed in Table 7. It varies from 0.55 to 5.03 GPa depend- and much work is needed to confirm property measurements ing on the type of material and testing technique. A typ- and comprehend their meaning. ical stress–strain result for a UNCD specimen is given in Table 7 summarizes the current data on thin film dia- Figure 39. In the gauge region, the specimen was 10
m mond materials. The data are separated into four groups wide and 0.5
m thick. The membrane deflection experi- ment described in Section 4 was employed. The stress–strain response of the specimen increases in a linear fashion until Table 5. Summary of data on polysilicon. Young’s Fracture Table 6. Summary of data on silicon-based materials. modulus strength (GPa) (GPa) Method Ref. Young’s Fracture modulus strength MUMPs21 136–174 1.3–2.8 tensile [112] Material (GPa) (GPa) Method Ref. MUMPs19 132 — tensile [112] Stress concentrations — 1.3–1.5 tensile [230] SiO2 — 0.6–1.9 tension [234] Size effect 154.1–159.6 1.51–1.67 tensile [229] SiO2 83 — bending [77] Size effect—doped — 2.0–2.8 tensile [122] SiO2 64 0.6 indentation [77] Microcantilever 174 ± 20 28 ± 05 bending [231] Si3 N4 222 ± 3 — bulge [105] AFM deflection 173 ± 10 26 ± 04 bending [232] Si3 N4 216 ± 10 — indentation [105] Bulk 181–203 — indentation [219] SiC 470 ± 10 — bending [235] Doped–undoped 95–175 — indentation [233] SiC 395 — indentation [236] Micro- and Nanomechanics 581 Table 7. Summary of data on thin film carbon-based materials. Young’s modulus Fracture strength Material (GPa) (GPa) Method Ref. UNCD (nanoseeded) 945–963 3.95–5.03 MDE [84] UNCD (microseeded) 930–970 0.89–2.42 MDE [84] UNCD 916–959 — MCD [84] UNCD 960 — nanoindentation [237] Nanograined diamond 910–1150 — nanoindentation [238] Nanograined diamond 675–765 — indentation [239] Nanograined diamond 510 — laser-acoustic [240] Micrograined diamond 1250 — nanoindentation [241] Micrograined diamond 1155–1207 — nanoindentation [242] Micrograined diamond 1000 — indentation [243] Micrograined diamond 884–940 — indentation [239] Micrograined diamond 830 2.72 MCD [244] Micrograined diamond — 0.55–0.82 mode I [245, 246] Diamond-like carbon 800 0.7 nanoindentation [247, 248] Diamond-like carbon 700 — laser-acoustic [249] Diamond-like carbon — 0.8–1.0 three-point bend [250] Diamond-like carbon 260 — indentation [251] Diamond-like carbon 60–145 — nanoindentation [252] failure at 5.03 GPa. The slope of the plot represents the the Weibull modulus was 5.74. By contrast, when seeding elastic modulus and was found to be 949 GPa. was performed with nanosize diamond particles, using ultra- It should also be mentioned that Table 7 contains data sonic agitation, the stress resulting in a probability of failure for a limited number of specimens of varying size and pro- of 63% increased to 4.13 GPa and the Weibull modulus was cessing schemes and therefore effects of flaw size in relation 10.76. The investigation highlights the role of microfabrica- to specimen size and processing are not defined. Fracture tion defects on material properties and reliability, as a func- strengths of 0.7 to 5.03 GPa have been measured for UNCD tion of seeding technique, when identical MPCVD chemistry films with the nanoseeded UNCD yielding by far the highest is employed. This group is currently examining how strength tensile strength [84, 85, 127]. is affected when specimen size is reduced to a degree where Espinosa and co-workers have also interpreted the it becomes comparable to the flaw size. By employing the strength of UNCD based on Weibull statistics [127]. They new membrane deflection toughness experiment developed showed that the Weibull parameters are highly dependent by Espinosa and co-workers [127], the toughness of UNCD on the seeding process used in the growth of the films. When was measured to be approximately 7 MPam1/2 . seeding was performed with micrometer size diamond par- The results collected thus far demonstrate the significant ticles, using mechanical polishing, the stress resulting in a mechanical advantages that carbon-based films can provide probability of failure of 63% was found to be 1.74 GPa, and to MEMS/NEMS and their applications over other materi- als, particularly when extrinsic flaws such as pores and sur- face flaws can be further minimized and/or eliminated. 6000.0 Clearly, much remains to be learned about the mechani- σf = 5.03 GPa cal behavior of thin film materials. Researchers in the field 5000.0 must come to a consensus on a standard technique by which properties can be measured in an accurate and repeatable 4000.0 manner. Finally, the understanding of mechanisms involved Stress (MPa) in microstructural and specimen size effects as well as frac- 3000.0 ture and fatigue will allow the development of models capa- ble of predicting film behavior. 2000.0 σ0 ~ 100 MPa 1000.0 Slope = 949 GPa 3.3.1. Carbon Nanotubes Most of the experimental techniques to date have been 0.0 reviewed in this chapter. All have been performed to char- 0 0.001 0.002 0.003 0.004 0.005 0.006 acterize the mechanical or related properties of carbon Strain nanotubes. This section lists in tabular form the results of Figure 39. Stress–strain curve representative of a typical UNCD MDE measurements of mechanical properties of MWCNTs and sample. An elastic modulus of 949 GPa, fracture stress of 5.03 GPa, and SWCNTs with focus on Young’s modulus and strength. This an estimated initial stress of 100 MPa were identified. Reprinted with section is intended to provide the reader not only values of permission from [84], B. C. Prorok et al., Exp. Mech. (2003). © 2003, mechanical properties but also the corresponding references Society for Experimental Mechanics. and test methods employed in their identification. 582 Micro- and Nanomechanics Most of the experimental measurements were conducted on MWCNTs and ropes of SWCNTs, with only one excep- tion, that of Krishnan et al. [185] who measured Young’s modulus of SWCNTs using the thermal resonance method. There exists a wide range in the reported properties primar- ily owing not only to different synthesis methods but also to the various assumptions used in the calculations of stress and strain. Presently, there are several debatable issues related to the interpretation of the data. One such issue is whether CNTs behave more like a beam or a string. Most experiments have modeled the nanotube as a beam and obtained the Young’s A modulus based on this assumption. A typical example is the measurement done by Wong et al. [152], which correlated the distance along the nanotube and corresponding force constant. However, Walters et al. [186] argued that nano- B C tubes behave as a string rather than a beam (Fig. 40). Another contentious issue is whether there exists a rela- tionship between nanotube diameter and Young’s modulus. Poncharal et al. [160] and Salvetat et al. [187, 188] observed that Young’s modulus decreases with increase in tube diame- ter. In the experiment of Poncharal et al., increasing diameter resulted in a sharp decrease of modulus. This decrease was 10 nm explained due to a wavelike distortion or ripple on the inner arc of the bent nanotube for relatively thick nanotubes (Fig. D 41). However, other measurements did not reveal a direct relation between the Young’s modulus and tube diameter. The existence and migration of buckles during the bend- ing of nanotubes was observed by Falvo et al. [156]. The 10 nm raised points along the tube were interpreted as buckles (Fig. 42), consistent with the increase in height as shown on Figure 41. Elastic properties of nanotubes. (A) Eb as a function of the right in Figure 42a. The location of the buckles shifted diameter: the dramatic drop in Eb for D ≈ 12 nm is attributed to the dramatically, which can be seen from Figure 42b–c. The onset of a wavelike distortion, which appears to be the energetically buckles in (c) appeared in regions which had been feature- favorable bending mode for thicker nanotubes. There is no remarkable less, and the buckles of (b) mainly disappeared. Table 8 sum- change in the Lorentzian line shape of the resonance (inset) for tubes marizes measured data on multiwalled carbon nanotubes. that have large or small moduli, although the low-modulus nanotubes In NEMS applications, it is essential to understand how appear to be more damped than the high-modulus tube. (D) TEM friction, wear, and lubrication effect the relative motion image of a bent nanotube showing the characteristic wavelike distor- of objects in contact. The tribology of carbon nanotubes tion. (B) and (C) Magnified views of a portion of (D). Reprinted with includes two parts: intershell friction and nanotube/substrate permission from [160], P. Poncharal et al., Science 283, 1513 (1999). © 1999, American Association for the Advancement of Science. friction. Yu et al. [164] pulled the outermost shell and mea- sured the force due to outer shell/inner shell interaction until the outermost shell broke. Cumings et al. [253] opened the end of a MWCNT and exposed the core tubes. The repeated extension and retraction of the core tubes against the outer shell did not reveal any wear or fatigue. The static friction force was estimated to be less than 66 × 10−15 N/Å2 and the dynamic friction force was less than 43 × 10−15 N/Å2 . Falvo et al. [157] studied the frictional behavior of a nano- tube on two substrates: mica and graphite. On different substrates, the nanotubes preferred either rolling or slid- ing, depending on the pulling location. Figure 43 shows the Figure 40. Lateral force on SWCNT rope as a function of AFM tip rolling process of a CNT on a graphene substrate and the position. The four symbols represent data from four consecutive lateral force curves on the same rope, showing that this rope is straining elas- measured lateral force as a function of tip position. tically with no plastic deformation. Inset: the AFM tip moves along the The effect of mechanical deformation on the electrical trench, in the plane of the surface, and displaces the rope as shown. properties of CNTs is of particular interest owing to the Reprinted with permission from [186], D. A. Walters et al., Appl. Phys. potential of CNTs in the development of nanoelectrome- Lett. 74, 3803 (1999). © 2003, American Institute of Physics. chanical systems. Paulson et al. [254] found that changes Micro- and Nanomechanics 583 20 0.06 Table 8. Multiwalled carbon nanotube. e Young’s 0 modulus Strength Method (TPa) (GPa) Comments Ref. 0 –0.06 Thermal vibration 1.8 — [184] Electrostatic 0.1–1 — varies with [160] vibration diameter f Lateral force 128 ± 059 142 ± 80 [152] bending Contact force 081 + 041 arc-discharge [188] bending 0.81–0.16 MWCNT Curvature (nm–1) Height (nm) Radial 0.0097–0.08 >53 compressive [154] indentation modulus g and strength Tensile test 045 ± 023 172 ± 064 very long [189] MWCNT Tensile test 0.27–0.95 11–63 outermost layer [164] Torsion test 0.6a [153] a Shear modulus G. h 0 400 800 1,200 s (nm) Figure 42. Curvature and height of buckles along a bent carbon nano- tube. The white scale bar (in (a) represents 300 nm and all figures are to the same scale. A 20-nm-diameter tube was manipulated from its straight shape [(a), inset)] into several bent configurations (a)–(d). The height and curvature of the bent tubes along its centerline [indicated by the arrow in (a)] are shown in (e)–(h). The upper trace in each graph depicts the height relative to the substrate; the lower trace depicts the curvature data. Height values are relative to substrate height. The “ripple”-like buckles migrate as the tube is manipulated into different configurations. The appearance and disappearance of the ripple buck- les, as well as the severe buckle at s ≈ 500 nm (e)–(f), suggest elastic reversibility. The large buckle at s ≈ 325 nm (e), (f) retains its raised topographical features even after straightening (g), (h), suggesting that damage has occurred at this point; but the tube does not fracture. The average of the buckle interval histogram [(d), inset)] and the average of the Fourier transforms [(c), inset) for a wide range of bent config- urations establish the dominant interval as 68 nm. Reprinted with per- mission from [156], M. R. Falvo et al., Nature 389, 582 (1997). © 1997, American Association for the Advancement of Science. in nanotube resistance were small unless the nanotubes fractured or the metal–nanotube contacts were perturbed. However, Tombler et al. [206] succeeded in bending indi- vidual SWNTs with an AFM tip and measuring conduc- tance as a function of deflection. The study revealed that Figure 43. Rolling behavior of CNTs on the graphene substrate (a)–(f) the conductance of SWCNT changed dramatically under the as it is manipulated from left to right. The tube is imaged before and applied deformation. Figure 44 represents the schematic of after each of the five manipulations. The insets between each topo- their experiment and measured values. Quantum mechanical graphical image show the lateral force during each manipulation. The modeling was employed to verify and explain the experimen- tube is moving from left to right, not gradually but in sudden slips in a tal findings. The conclusion reached by the authors was that stick–slip type rolling motion. In (g), three overlapped signals from sep- arate rolling trials are shown for the lateral force as the tube is pushed interaction between the AFM tip and the SWCNT deformed through several revolutions of stick–slip rolling motion. The features of the nanotube to a point where the bonding in the nanotube the force traces are reproducible. The 85 nm periodicity in the signal, was converted from sp2 to sp3 . However, in previous studies, indicated by the dashed lines, is equal to the circumference of the tube the nanotubes were more or less uniformly bent or strained; at its ends. Reprinted with permission from [157], M. R. Falvo et al., hence, the chemical bonding evidently remained sp2 . Nature 397, 236 (1999). © 1999, Macmillan Publishers Ltd. 584 Micro- and Nanomechanics (MSG) theory [33, 35] derived under certain simplifying assumptions from the concept of geometrically necessary dislocations. Based on numerical experience, this theory was recently improved as the Taylor-base nonlocal (TNT) theory [259], and the improvement consisting in the form of strain gradient dependence of the hardening function made the theory conform to a revision proposed by Bažant [260, 261] on the basis of scaling analysis. Another noteworthy theory was the Acharya and Bassani strain gradient plasticity the- ory based on the idea of lattice incompatibility [262, 263], which represented a generalization of the incremental the- ory of plasticity. The asymptotic characters of these strain gradient theories were analyzed recently and it was found that the small-size asymptotic size effect predicted by some of the theories is excessive and unreasonable [259–261]. It might seem that the small-size asymptotic behavior of gradient plasticity is irrelevant because it is approached only at sizes below the range of validity of theory, for which the spacing of the geometrically necessary dislocations (about 10 to 100 nm) and the crystal size are not negligible, and other physical phenomena, such as surface tension, gradation of crystal size, and texture, intervene. However, knowledge of both the small-size and large-size asymptotics is very use- ful for developing asymptotic matching approximations for the intermediate range, for which the solutions are much harder to obtain. For the purpose of asymptotic matching, the asymptotic behavior must be physically reasonable even if attained outside the range of validity of the theory (this has been demonstrated in the modeling of cohesive fracture Figure 44. Bending of SWCNT by an AFM tip and the corresponding electrical conductance evolution. (a) Top view of an SWCNT partly when the small-size plastic asymptote is often approached suspended over a trench for electromechanical measurements; (b) AFM only for specimen sizes much smaller than the inhomogene- image of an SWCNT with suspended length l ≈ 605 nm; (c) Side view ity size, e.g., the aggregate size in concrete [260]). of the AFM pushing suspended SWCNT. (d)–(f) cantilever deflection The present chapter reviews and summarizes several and nanotube electrical conductance evolution during repeated cycles recent papers in which it was shown that the main theories of pushing the suspended SWCNT. $ is the tube bent angle. Initial tip– proposed in the past, including couple stress theory, stress tube distance is 65 (d), 30 (e), and 8 nm (f), and the speed of the tip and rotation gradient theory, MSG, TNT, and the Acharya motion was about 22 (d), 34 (e), and 44 nm/s (f). It is seen that the and Bassani theory, suffer from excessive asymptotic size conductance of an SWCNT is reduced by two orders of magnitude when effect and some exhibit an unrealistic shape of the load- deformed by an AFM. Reprinted with permission from [206], T. W. Tombler et al., Nature 405, 769 (2000). © 2000, Macmillan Publishers deflection curve. Simple adjustments of all these theories Ltd. suffice to achieve reasonable asymptotic behavior and thus to make asymptotic matching approximations feasible. The main strain gradient theories will be briefly intro- 4. THEORETICAL MODELING duced and their asymptotic analysis presented by Bažant and AND SCALING Guo [264] will be outlined. After that, a simple asymptotic- matching approximation, suitable for predictions of yield During the 1980s and 1990s, a host of experiments on the limit and plastic hardening on the micrometer scale, will be micrometer and submicrometer scale, including microinden- presented. tation [15], microtorsion [29, 30], and microbending [28], revealed a strong size effect on the yield strength and hard- ening of metals. Similar size effects were observed also 4.1. Strain Gradient Theories in metal matrix composites with particle diameters in the micrometer and submicrometer scale [255, 256]. The clas- First we will consider the Fleck and Hutchinson phe- sical plasticity theories cannot predict these size effects nomenological strain gradient theory [29, 30] and its succes- because they involve no material characteristic length. To sive versions. In these theories, the effect of strain gradient explain them, several strain gradient theories were devel- tensor is incorporated into the potential energy density func- oped. The first one was a phenomenological theory by Fleck tion, in a manner similar to the classical theories of Toupin and Hutchinson [30] based on the existence of a poten- [265] and Mindlin [266] in which only linear elasticity was tial. This theory was later extended and improved in several considered. A higher order stress tensor needs to be intro- versions [31, 257, 258] while retaining the same basic struc- duced in these theories to provide a work conjugate to the ture. Another strain gradient theory which received consid- strain gradient tensor, and the boundary condition of classi- erable attention was the mechanism-based strain-gradient cal solid mechanics also needs to be modified as well. The Micro- and Nanomechanics 585 classical J2 deformation theory of plasticity (i.e., Hencky- relation. There are many ways to extend it to a general non- type solid strain theory) is chosen as the basis of strain gra- linear plastic material. Fleck and Hutchinson [31] chose to dient generalization. do it by defining a new variable, a scalar called the combined The Gao and Huang MSG theory [33, 35] does not use the strain quantity, E, which involves both the strain tensor and potential energy approach (and actually, potential energy the strain gradient tensor, to replace the effective strain in even does not exist in that theory). Rather, this theory is the J2 theory. W is then assumed, for a general nonlinear based on the Taylor relation between the shear strength and plastic material, to be a nonlinear function of E. To define dislocation density. A multiscale framework is used to intro- E, the strain gradient tensor
needs to be decomposed into duce the higher order stress tensor and to establish the vir- a hydrostatic part
H and deviatoric part
 : tual work balance. Numerical simulations showed that while the higher order stress tensor is affected by the material H 8ijk = 'ik 8jpp + 'jk 8ipp /4
 =
−
H (28) length characterizing the size of the framework cell (called the mesoscale cell), the stress and strain tensors are almost Due to incompressibility, we have !ij = !ij , 8ijk  = 8ijk . Fur- unaffected. This observation triggered a reformulation in the thermore,
is decomposed into three orthogonal parts
 =  form of the TNT theory [259], in which the strain gradient is m n numerically simulated as a nonlocal variable and the higher
1 +
2 +
3 such that 8ijk 8ijk = 0 when m = n [31]; n n order stress disappears. This reformulation coincided with the three invariants 8ijk 8ijk are used to define E, a revision proposed by Bažant [260, 261] for entirely differ-  ent reasons—namely, the observation that the presence of 1 1 2 2 3 3 couple stresses, dictated by the use of a strain gradient ten- E= 2!ij !ij /3 + >21 8ijk 8ijk + >22 8ijk 8ijk + >23 8ijk 8ijk sor as an independent kinematic variable, causes an exces- (29) sive small-size asymptotic size effect, indicating that couple stresses should be removed from the formulation. where >i are three length constants which are given different The Acharya and Bassani strain gradient theory [262, 263] values in different version of the theory. differs significantly from the previous theories. It represents  a generalization of incremental plasticity rather than total For CS: >1 = 0 >2 = >CS /2 >3 = 5/24>CS (30) strain theory. The effect of a strain gradient is considered by  changing the tangential modulus in the constitutive relation, For SG: >1 = >CS >2 = >CS /2 >3 = 5/24>CS (31) while the framework of classical plasticity theory remains. Here >CS is called the material characteristic length. If the 4.1.1. Fleck and Hutchinson Theories strain gradient part is ignored, scalar E becomes identical to the effective strain ! used in the classical plasticity theories. The first phenomenological strain-gradient theory developed Now the strain energy density W can be expressed as a by Fleck and Hutchinson [29, 30] is called the couple stress function of E instead of ! as W = W E; thus the Cauchy theory (denoted by CS). The subsequent modification [31] is stress tensor  and the higher order stress tensor  can be called the stretch and rotation gradients theory (denoted by expressed as SG). Since the main idea of these two theories is the same, we will consider them jointly. To simplify the problem, only ;W dW ;E 2!ij dW incompressible materials will be considered and the elastic ij = = = (32) part will be ignored because it is negligible compared to ;!ij dE ;!ij 3E dE large plastic deformation of metals. ;W dW ;E In the classical work of Toupin [265] and Mindlin [266], <ijk = = ;8ijk dE ;8ijk and dealing only with the linear elasticity case, the strain   gradient is introduced into the strain energy density W as dW 1 2 3 2 1 ;8lmn 2 2 ;8lmn 2 3 ;8lmn = >8 + >2 8lmn + >3 8lmn E dE 1 lmn ;8ijk ;8ijk ;8ijk W = 1/2(!ii !jj +
!ij !ij + a1 8ijj 8ikk + a2 8iik 8kjj >2CS Cijklmn 8lmn dW + a3 8iik 8jjk + a4 8ijk 8ijk + a5 8ijk 8kij (27) = (33) E dE where ( and
are the usual Lamé constants, !ij = ui: j + Here Cijklmn is a six-dimensional constant dimensionless ten- uj: i /2 is the strain, 8ijk = uk: ij is the component of strain sor [264]. Since the values of >i are different in CS and SG gradient tensor
, and an is the additional elastic stiffness theories, the tensor C will also be different in these two the- constant of the material. The sum of the first two terms on ories, although for each of them C is a constant tensor, that the right-hand side is the classical strain energy density func- is, independent of !,
, and >CS . Because of the existence of tion, while the other five terms are the contributions of the higher order stress, the field equations of equilibrium must strain gradient tensor. Based on the strain energy density be generalized as defined as (27), the Cauchy stress ij can be defined as a work conjugate to !ij (i.e. ij = ;W /;!ij ). A higher order stress tensor , work conjugate to the strain gradient tensor ik: i − <ijk: ij + fk = 0 (34)
, needs to be defined as <ijk = ;W /;8ijk . The strain energy W defined by (22) represents a linear elastic constitutive where fk is the body force. 586 Micro- and Nanomechanics 4.1.2. Gao and Huang MSG Theory [261, 264]). Upon noticing this fact, the MSG theory has and TNT Theory been replaced by the TNT theory, in which the higher order stress tensor is removed. As the first strain gradient theory based on geometrically In the TNT theory, the strain gradient is not an inde- necessary dislocations, the MSG theory is a generalization pendent variable but a nonlocal variable defined by numer- of the incremental theory of plasticity [267]. In the MSG ical integration. The gradient term !ij: k can be numerically theory, the definition of strain gradient tensor 8ijk = uk: ij is approximated in a nonlocal form as [259] the same as it is in the Fleck and Hutchinson theories, but −1 the definition of higher order stress is different. It is defined !ij: k = D!ij x + G − !ij xEGm dV Gk Gm dV by virtual work balance in a multiscale framework. The final Vcell Vcell constitutive relation reads [33, 260, 261, 264] (41)
 in which Vcell is a sufficiently small representative cell sur- 2  K H 2 ik = K'ik !nn + !ik <ijk = l!2 8ijk +?ijk + Y Aijk rounding point described by x. To simplify the integration, 3! 6  (35) Vcell can be chosen as a cube centered at x, and then the strain gradient 8ijk can be expressed as where 1 8ijk = D! G + !jk Gi − !ij Gk E dV with 1 I! Vcell ik j ?ijk = B − Cijk  Aijk = f !f  !Cijk (36) ! ijk 1 5   I! = G12 dV = l (42) ! = 2!ij !ij /3 8 = 8ijk  8ijk /2 Vcell 12 !  (37) where l! is the size of the cube. Furthermore, one may intro-  = Y f 2 ! + l8 duce the volumetric part
H and the deviatoric part
 of and tensor 
, and the effective strain gradient invariant 8 =   8ijk 8ijk /2, which is identical to that defined in the MSG Bijk = D28ijk + 8kji + 8kij − 'ik 8ppj + 'jk 8ppi /4E72 theory. Because the strain gradient tensor does not function Cijk = D!ik 8jmn + !jk 8imn − 'ik !jp + 'jk !ip 8pmn /4E in (40) as an independent kinematic variable, we need not (38) define the corresponding work-conjugate higher order stress × !mn /54!2 tensor. For p = 1, q = 2, the constitutive relation is [259] H  H 8ijk = 'ik 8jpp + 'jk 8ipp /4 8ijk = 8ijk − 8ijk 2  ik = K'ik !nn + ! where K is the elastic bulk modulus. Equation (37) defines 3! ik where the new hardening rule of the material in which Y is the  yield stress; ! and  are the effective strain and stress; 8  = Y f 2 ! + l8 (43) is the effective strain gradient, which is proportional to the density of geometrically stored dislocations; Y f ! repre- Since the new higher order stress is absent, the equilibrium sents the classical plastic hardening function; l is the mate- equation of the TNT theory is the same as in the classical rial intrinsic length (similar to parameter >CS used in the theory (i.e., ij: i + fj = 0). Fleck and Hutchinson theories [29–31]); !ij = !ij − !nn /3 is H the deviatoric strain; 8ijk is the volumetric part of strain gra- 4.1.3. Acharya and Bassani Theory dient tensor; and l! is the size of the so-called “mesoscale” The Acharya and Bassani strain gradient theory is a general- cell which is expressed by Gao et al. [33] as ization of the classical incremental plasticity theory, in which the strain gradient is assumed to affect only the instanta- l! = G/Y b (39) neous modulus. The strain gradient is considered to be a measure of lattice incompatibility and is introduced only Here G is the shear modulus, b is the Burgers vector, through the second-order tensor as [262, 263] and  is an empirical factor whose value is suggested to p be between 1 to 10 [33]. The equilibrium equations are the ij = ejkl !il: k (44) same as (29). It is also interesting to consider a more general hardening relation, where ejkl is the alternating symbol and !p is the plastic strain. Introducing the invariant:  = Y Df q ! + l8p E1/q (40)   = 2ij ji (45) where p and q are positive exponents; and MSG theory cor- responds to the case p = 1, q = 2. Acharya and Bassani modified the classical J2 flow theory as When the MSG theory is used in numerical simulations, follows [262, 263]:  the results show that when the value of l! is changed, the < = ij ij /2 <˙ = <˙cr = hH p :Ḣ p (46) stress and strain do not change much, although the higher order stress does. This means that the existence of the  p p p p higher order stress offers no advantage [35] (aside from !˙ij = Ḣ p /2<ij ˙ ij = Cijkl !˙kl − !˙kl  H p = 2!ij !ij /3 the fact that they make the asymptotic scaling problematic (47) Micro- and Nanomechanics 587 Here the instantaneous hardening modulus h depends not It is now useful to define dimensionless variables: only on plastic strain invariant H p but also on plastic strain gradient invariant . An example of this function is [262, <¯ijk = <ijk /E0 >CS  ¯ ij = ij /0 = E E 263] l l (53) 8̄ijk = 8ijk D l = 1: 2: 3
  1/2 H p N −1 l2 /H0 2 hH p :  = h0 1 + 1+ (48) Then the constitutive relation can be expressed as H0 1 + cH p /H0 2
 where l is the material intrinsic length, and h0 , H0 , c, and N 2 1 1/n 1−n/n ¯ ik = E !¯ik (54) are positive material constants. 3 E0 There exist other strain gradient theories, but generally
1/n they are similar to one of the theories introduced here. For 1 >CS 1−n/n <¯ijk = E Cijklmn 8̄lmn (55) example, the Chen and Wang [268, 269] strain gradient the- E0 D ory is similar to the Fleck and Hutchinson theories. The equilibrium equation (29) can be rewritten as 4.2. Asymptotic Analysis of Strain >CS  Gradient Theories ;i ¯ ik − ; ; <¯ + N f¯k = 0 (56) D i j ijk 0 For the purpose of scaling analysis, we need to consider geo- metrically similar structures of different sizes. This means where ;i = ;/; x̄i = derivatives with respect to the dimen- that the structures are also similarly loaded. It is obvious sionless coordinates. Substituting (49) and (55) into (56), that the strain gradient theories must reduce to the clas- one obtains the dimensionless field equation of equilibrium sical plasticity theory when the structure size is very large. in the form To discuss the asymptotic cases, it is necessary to introduce


 dimensionless variables. Diverse sets of such variables could 2 1 1/n  1−n/n  >CS 2 1 1/n ;i E !¯ik − be chosen but only one is easy to interpret, 3 E0 D E0    x̄i = xi /D ūi = ui /D !¯ij = !ij  × ;i ;j Cijklmp E 1−n/n 8̄lmp = − N f¯ (57) (49) 0 k 8̄ijk = 8ijk D f¯k = fk D/N Following Bažant [260, 261] and Bažant and Guo [264], where D is the characteristic length of the structure, and N we may simplify the analysis by replacing the surface frac- is the nominal strength. For geometrically similar structures tions with body forces applied in a very thin boundary layer, the strain distribution may often be assumed to be the same, the thickness of which tends to zero. This ensures that all and then x̄i , ūi , !¯ij , and 8̄ijk will be size independent; that the boundary conditions are homogeneous. When the struc- is, they will be the same for structures of different sizes. ture is sufficiently large, >CS /D → 0, <¯ijl vanish, according Consequently, the asymptotic behavior of the strain gradient to (55), and the equilibrium equations reduce to the classi- tensor must be 8ijk ∝ 1/D. cal equilibrium equations, as required. The combined strain quantity, E, reduces to the classical effective strain because 4.2.1. Asymptotic Analysis of the Fleck the strain gradient part can be ignored compared to the strain part. Then the strain energy density function takes the and Hutchinson Theories normal form as a function of the strain only. Scaling and Size Effect The Fleck and Hutchinson strain As proposed by Bažant and Guo [264], it is interesting gradient theory can be used to generalize various partic- to look at the opposite asymptotic character of the theory ular forms of classical constitutive relations for plasticity. when the structure size tends to zero, >CS /D → . At first, A stress–strain relation in the form of a general power law the dimensionless combined strain quantity can be rewritten relation may be chosen as an example, in which the strain as energy density is [29–31]  
n+1/n  = 2!ij !ij /3 + >2 8̄1 8̄1 + >2 8̄2 8̄2 + >2 8̄3 8̄3 D2 E n E 1 ijk ijk 2 ijk ijk 3 ijk ijk W = E (50) n + 1 0 0 E0 ∝ D−1 for >CS /D →  (58) where 0 , E0 , and n are positive material constants. For hardening materials, n ≥ 1; typically n ≈ 2–5 for normal If one defines a size-independent dimensionless variable metals. According to (32) and (33), the constitutive relation  then reads = 1 1 2 2 3 3  H >21 8̄ijk 8̄ijk + >22 8̄ijk 8̄ijk + >23 8̄ijk 8̄ijk >CS (59)
1/n 2 1 ik = 0 E 1−n/n !ik (51) the asymptotic behavior is seen to be 3 E0
1/n <ijk = 0 1 >2CS E 1−n/n Cijklmn 8lmn (52)  ≈ >CS H E  for >CS /D →  (60) E0 D 588 Micro- and Nanomechanics Substituting (60) into (57), the asymptotic form of the equi- w can be regarded as the deflection magnitude. Substituting librium equation reads this relation into the dimensionless constitutive relation (54)


 and (55), one can easily get 2 >CS 1−n/n 1 1/n   1−n/n  >CS 1+n/n ;i H !¯ik − 3 D E0 D
1/n ¯ ik = w 1/n ˆ ik <¯ijk = w l/n <ˆijk (66) 1   ×  1−n/n 8̄lmp = − N f¯k ;i ;j Cijklmp H (61) E0 0 where ˆ ik and <ˆijk are both size independent profile func- After multiplying this equation by D/>CS n+1/n and tak- tions. Substituting these relations into dimensionless equilib- ing the limit of the left-hand side for >CS /D → , one gets rium equation (56), one finds that the load-deflection curve the following asymptotic form of the equilibrium equations: must have the form    1−n/n 8̄lmp = J f¯k ;i ;j Cijklmp H f¯k ∝ w 1/n (67)
n+1/n with J = E  1/n N D (62) 0 E0 >CS This relation is similar to the traditional strain–stress rela- Because the left-hand side of the foregoing equation, as tion derived from the strain energy density function (50). well as the dimensionless body force f¯k , is independent of The reason the load deflection curve begins with a vertical size D and because the boundary conditions are homoge- tangent is that the initial elastic response is assumed to be neous and thus size independent, the parameter J must be negligible. size independent. Thus, upon solving N from (62), one finds that the small-size asymptotic scaling law is Example One important example is the microtorsion of a
n+1/n thin wire, for which a strong size effect was demonstrated −1/n >CS  N = 0 J E (63) [29–31] and described by strain gradient theories. The strain 0 D energy density function W is defined as W = E N +1 /N + or 1. Compared with (50), one finds that N = 1/n. The radius of the wire, D, is chosen as the characteristic size of the N ∝ D−n+1/n (64) structure. The deformation is characterized by the twist angle per unit length, L. The nominal stress can be defined For plastic hardening materials, we have 1 < n + 1/n ≤ as N = T /D3 , where T is the torque. For different radii of 2. Although the result (64) applies only to the special case the wire, we compare the N values corresponding to the of strain energy density function given by (50), the analyt- same dimensionless twist L̄ = LD. The nominal stress can ical technique used here is general. It is even suitable to be expressed according to the CS theory as follows: the strain energy density function defined directly in terms of strain and strain gradients, rather than as the combined 

  strain quantity. For example, if the strain energy density T 6 1 >CS 2 N +3/2 >CS N +3 function is defined as (27) for the case of linear elasticity, N = 3 =  L̄N + − D N +3 0 3 D D a similar analysis can be made and it is found that the size effect law for very small sizes reads [264] (68) N ∝ D−2 (65) When >CS /D → , one has This also shows that (64) is quite general because (65) can be regarded as a special case of (64) in which the strain 
2 N +3/2 
hardening exponent n = 1. 1 >CS >CS N +3 + − 3 D D Small-Size Asymptotic Load–Deflection Response
2
 For some special cases (e.g., the pure torsion of a long thin N +3 D >CS N +3 wire or the bending of a slender beam), the symmetry con- ≈ (69) 2 3>CS D ditions require displacement distribution to remain similar during the loading process. For such cases, the dimension- less displacement ūk can be related to a single parameter, from which w, and characterized by displacement profile ûk as ūk = w ûk [260, 261, 264]. Since ûk is dimensionless, it must be inde- N ∝ D−N −1 = D−n+1/n (70) pendent of the size D. Displacements ūk evolve during the proportional loading process while the distribution profile remains constant. Thus the parameter w can be considered For the load-deflection response, we now obtain the follow- as the displacement norm, ūk . It follows that the strain, ing relation between the load T and the deformation L̄: strain gradient, and combined strain quantity are all propor- tional to w. Therefore, E  can be similarly represented as  where E  = w E, E  is a size independent profile function, and T ∝ LN = L1/n (71) Micro- and Nanomechanics 589 4.2.2. Asymptotic Analysis of the Gao When D → 0, the fourth term is generally the dominant and Huang MSG Theory one, and so we get the asymptotic form of the equilibrium and TNT Theory equation, Scaling and Size Effect The dimensionless variables ijk  = J1 f¯k ;i ;j 8̄p/q ? (79) defined in (49) also need to be used here, and further dimen- sionless variables need to be defined as follows: with
2
2+p/q !¯ = ! 8̄ = 8D ¯ ij = ij /Y l N D J1 = (80) (72) l! Y l <¯ijk = <ijk /Y l ¯ = /Y Since D is not present in the left-hand side of (80) and the H 8̄ijk H = 8ijk D ijk = Bijk D B ijk = Cijk D C boundary conditions are also homogeneous, the parameter (73) ijk = ?ijk D ? ijk = Aijk D A J must be independent of D. Thus, the general small-size asymptotic scaling law of MSG theory reads [260, 261, 264] H These definitions are meaningful because 8ijk , Bijk , and Cijk
2
2+p/q are all homogeneous functions of degree 1 of tensors 8ijk l l N = Y J1 ! and !ij . It is not difficult to obtain a dimensionless version l D of the constitutive law of the MSG theory and for pq = 21 K 2¯  N ∝ D−5/2 (81) ¯ ik = 'ik !¯nn + !¯ Y 3!¯ ik
 This asymptotic size effect is very strong [260, 261, 264]. l2 K H ijk + 1 Aijk It is much stronger than the normal linear elastic fracture <¯ijk = ! 8̄ijk + ¯ ? (74) lD 6Y ¯ mechanics size effect, which is N ∝ D−1/2 , or the typical Weibull size effect, which is around N ∝ D−01 . where l and l! are two characteristic material lengths. The There are also some special cases. For example, in the corresponding dimensionless equilibrium equation reads case of microbending, ? ijk = 0 for all i: j: k, which makes l  the fourth term on the left-hand side of (77) vanish; in ;i ¯ ik − ; ; <¯ + N f¯ = 0 (75) the case of incompressible material, 8ijkH = 0, which makes D i j ijk Y k the third term on the left-hand side of (77) zero. So Same as before, the boundary conditions can again be the general size effect law will change to N ∝ D−2 for considered as homogeneous and the applied loads replaced microbending of a compressible material, and to N ∝ by body forces f¯k applied within a thin surface layer. The D−2+p/q for microbending of an incompressible material (in asymptotic behavior for a very large structure is simple. detail, see [264]). When l/D → 0, we also have l! /D → 0. Thus <¯ijk tends to The size effect D−5/2 , as well as D−2 , is enormous and zero, according to Eq. (69), and all the equations reduce to unrealistic. This is a consequence of the last three terms the standard equations of classical plasticity theory, which on the left-hand side of (77), which represent contributions means that there is no size effect, as required by classical from the couple stresses. A detailed analysis showed that the plasticity. couple stresses are not necessary to fit the test results and The opposite asymptotic character for sufficiently small to ensure the virtual work balance [264]. Based on this anal- structures (l/D →  and l! /D → ) is more interest- ysis, Bažant [260, 261] and Bažant and Guo [264] proposed ing. The general hardening rule (40) can be rewritten with a modified version of the MSG theory in which the couple dimensionless variables as stresses are made to vanish. This led to a theory identical to the TNT theory [260, 261, 264], which was proposed on the ¯ = Df q ! + l8̄/Dp E1/q (76) basis of numerical experience with varying the “mesoscale cell size” l! . Let us now analyze the asymptotic size effect Thus, we have ¯ ≈ l8̄/Dp/q when l/D → . Substitut- of this theory. The dimensionless variables defined for the ing (74) into (75), we can express the equilibrium equation MSG theory may again be used for TNT theory. The dimen- as follows: sionless constitutive relation of the TNT theory reads 

2 K 2 l8̄ p/q  l K 2¯  ;i 'ik !¯nn + !¯ik − ! ¯ ik = 'ik !¯nn + !¯ (82) Y 3!¯ D D Y 3!¯ ik 
p/q
p/q K H l8̄  D  and the differential equation of equilibrium in terms of the × ; i ;j 8̄ + ?ijk + Aijk 6Y ijk D l8̄ dimensionless variables takes the form   = − N f¯k (77) ;i ¯ ik + N f¯k = 0 (83) Y Y When l/D → , the five terms on the left-hand side of the For large enough sizes, D/l → , the asymptotic behavior foregoing equation are, in sequence, of the order of will be identical to the classical theory of plasticity, which implies no size effect. For very small sizes, D/l → 0, we have O1 OD−p/q  OD−2  OD−2−p/q  OD2+p/q  (78) ¯ = Df q ! + l8̄/Dp E1/q ≈ l8̄/Dp/q (84) 590 Micro- and Nanomechanics and the equilibrium equation can be rewritten as follows: which means that ? ijk is independent of w. When we con- 
 sider the beginning of the load-deflection diagram at D → 0, K 2 l8̄ p/q   we also have w → 0, which is a limit not discussed during ;i 'ik !¯nn + !¯ik = − N f¯k (85) Y 3!¯ D Y previous sizeeffect analysis. When the effect of w is con- sidered, the five terms on the left-hand side of (77) are Obviously, the second term on the lefthand side domi- proportional, in sequence, to the functions as follows: nates when D/l → 0, and so the asymptotic form of the equilibrium equation is w w/Dp/q w/D2 w p/q /D2+p/q (88)

p/q w 1−p/q w 1−p/q /D2−p/q !¯ 3 N D ;i 8̄ p/q ijk = J f¯k with J = − (86) In the case of MSG theory with p = 1: q = 2, these functions !¯ 2 Y l are Same as before, J is sizeindependent, and consequently the  small-size asymptotic scaling law for the TNT theory is w w/D w/D2 w 1/2 /D5/2 w 1/2 /D3/2 (89)
p/q 2Y l N = − J and (77) can then be expressed as 3 D p 1 and for = N ¯ q 2 − f = a1 w +a2 w p/q D−p/q +a3 wD−2 +a4 w p/q D−2−p/q Y k N ∝ D−1/2 (87) +a5 w 1−p/q D−2+p/q (90) Four possible cases of small-size asymptotic scaling for the MSG theory and the TNT theory are shown in Figure 45. where parameters ai are constants independent of D and w. Since the force f¯k should decrease when w decreases, Small-Size Asymptotic Load-Deflection Response one knows that 1 − p/q > 0, which implies p < q. As we The characteristic features of the small-size asymptotic load- discussed before, if only D → 0 is considered (or, in other deflection curves will now be determined. The MSG theory words, D  w, the dominant term is a4 w p/q D−2−p/q , which will be analyzed first, and the TNT theory can be treated as means that a special limiting case of the MSG theory. Again we con- sider only the special cases where the relative displacement f¯k ∝ w p/q for w  D (91) profile does not change during the loading process. Same as before, the displacement can be characterized by parameter For the MSG theory, this gives w as ūk = w ûk , where ûk is the displacement profile, which is not only independent of D but also invariable during the f¯k ∝ w 1/2 for w  D (92) proportional loading process. Similarly, the strain and strain We need to consider another asymptotic case in which w  gradient can be expressed as !¯ij = !ij = w !ˆij , !¯ = ! = w !, ˆ D (e.g., at the beginning of the load-deflection diagram). 8̄ijk = w 8̂ijk , 8̄ = w 8̂. Since variables Bijk and Cijk are homo- The dominant term in this case is either a4 w p/q D−2−p/q geneous functions of degree 1 of both
and , their cor- or a5 w 1−p/q D−2+p/q , depending on the value of p/q. The responding dimensionless variables can also be expressed asymptotic load-deflection behavior is as products of w and a dimensionless profile function (i.e., ijk = w B B ijk and C ijk = w C ijk ). Because of the factor 1/! f¯k ∝ w r r = minNp/q: 1 − p/qO for w  D in the definition of ?ijk [see Eq. (36)], we have ? ijk , ijk = ? (93) For MSG theory, the dominant term is a4 w 1/2 D−5/2 , and so σN the load f¯k initially increases in proportion to w 1/2 . Thus one log —– σ0 has the asymptotic load-deflection relation for MSG theory 2 as 5 2 f¯k ∝ w 1/2 for all w (94) 1 3 As discussed in the preceding section, some terms in (77) may vanish in some special cases, and a similar analysis 2 can be applied to these special cases. For example, for 1 microbending of an incompressible material, the third and 2 fourth terms in (77) vanish, and as a result (92) changes as follows:   10–9 10–6 10–3 − N f¯k = a1 w + a2 w/D + a5 w 1/2 /D3/2 (95) D (m) Y Figure 45. Four possible small-size asymptotic scaling curves for the The asymptotic load-deflection curve is simple because MSG theory and the TNT theory. the last term on the right-hand side of (95) dominates when Micro- and Nanomechanics 591 D is small, regardless of the ratio of w/D, and so the asymp- bending curvature. When D → 0, the small-size asymptotic totic load-deflection relation for this special case is again form is
 f¯k ∝ w 1/2 for all w (96) 2 2 −1/2 1/2  N = Y l l ¯ !d. f !f ¯ L̄1/2 D−3/2 (102) 9 ! 0 This means the vanishing of some terms in flexure prob- lems will not change the asymptotic load-deflection behav- This verifies for this special case the asymptotic behavior ior. The TNT theory can be treated as a special case of N ∝ D−2+p/q , as well as (96). Letting l! = 0, one finds that the MSG theory. If the last three terms on the right-hand this asymptotic character also applies to the TNT theory. of the MSG equilibrium equation (77) vanish, the equation becomes identical to equilibrium equation (85) of the TNT 4.2.3. Asymptotic Analysis of Acharya theory. So (85) can be expressed as and Bassani’s Theory N ¯  Let us now give a simple analysis of the asymptotic behavior − fk = a1 w + a2 w/D (97) of the Acharya and Bassani strain gradient theory [263]. We Y define dimensionless variables ūi = ui /D, !¯ij: k = !ij: k D, and For small enough D and w, the dominant term will be the that ¯ ij = ij D. When D → 0, the asymptotic behavior of second term on the right-hand side of (97), and so we have the plastic hardening modulus (H̄ p = H p  defined by (48) is found to be f¯k ∝ w 1/2 for all w (TNT theory) (98)
 H p N −1  −1/2 ¯ l hH p :  = h0 1 + 1 + cH p /H0 2 It should be noted that the elastic part of the response has H0 H0 D been neglected, which is why the load-deflection curves in (94), (96), and (98) begin with a vertical tangent. ∝ D−1 (103) Example The experiment of microtorsion of a thin wire This shows that, at the same strain level, the plastic hard- can also be analyzed by the MSG theory [35] or TNT theory. ening modulus (slope of load deflection curve) increases as Equation (35) in [35] can be transformed to the dimension- D−1 when D → 0. If the elastic part is neglected for very less formula large plastic strain, the nominal stress must also scale asymp- 
 2  totically as D−1 . This asymptotic size effect is again quite T 2 L̄ 1 ¯ 2 l!2 ¯  ! l! f !f ¯ strong (but not as strong as in the MSG theory). This size N = 3 =  Y . + + . d. D 3 0 !¯ 12D2 12D2 ¯ effect can be reduced by modifying the hardening function (99) hH p : . For example, if the hardening modulus is rede- fined as where N is the nominal stress, T is the torque, D is the
  1/2 radius of wire (which is also chosen as the characteristic p H p N −1 l/H0 length of the structure), Y is the yield stress of macroscale hH :  = h0 1 + 1+ (104) H0 1 + cH p /H0  metal, and L̄ = 8̄ = LD is the dimensionless specific angle of twist, where L = actual specific angle of twist (i.e., the then the asymptotic scaling becomes more reasonable: rotation  angle perunit length of wire). Substituting ¯ = ¯ + l8̄/D ≈ lL̄/D into this formula, we find that, for f 2 ! hH p :  ∝ D−1/2 when D → 0 (105) D → 0, the dominant part is obtained by integration of the second term, which leads to the following small-size asymp- totic form: 4.3. Asymptotic-Matching
 Approximation Formula  2 1/2 1 .  N = Y l! l d. L̄1/2 D−5/2 (100) After determining the asymptotic behaviors of strain gradi- 18 0 !ˆ ent plasticity theories, one can obtain an asymptotic match- This result verifies the conclusions (81) and (89), where L̄ ing approximation for a smooth transition of the nominal is considered as a measure of deflection, analogous to w in strength in the intermediate size range. In [260, 261, 264], (89). The asymptotic load-deflection behavior of the TNT a smooth transition between the case of no size effect for theory can be obtained similarly. D →  and the case of power law size effect N ∝ D−s for Another special case is the application of the MSG theory D → 0 s > 0 has been described by the simple asymptotic- to the microbending of incompressible metals [35]. Equa- matching approximation tion (29) in [35] can be transformed to the dimensionless 
2s/r r/2 version D0 N = 0 1 + (106) 1/2  2 D M L̄l2 f !f ¯  ! ¯ N = 2 = 2Y √ .¯ + ! d. (101) where r is a constant to be determined by data fitting, while D 0 3 9D2 ¯ parameters 0 and D0 can be determined by either the where D is the beam depth (the characteristic dimension asymptotic size effect formula or data fitting. This formula of the structure), M is the bending moment, L̄ = 8̄ = LD was shown to fit the results for microtorsion and microbend- is the dimensionless bending curvature, and L is the actual ing (Figs. 46 and 47). 592 Micro- and Nanomechanics 2 of validity of the theoretical models (cohesive crack model, Micro-Torsion crack band model, nonlocal damage models), that is, for log D(σN/σY) (Dimensionless Stress) Asymptotic Matching specimen sizes much smaller than the inhomogeneity size Approximation (Bazant and Guo) or much larger than the largest constructable structures. Yet 2s/r r/2 σN = σ0 [1+(D0 /D) ] , s = 1/2, r = 1 the knowledge of two-sided asymptotics has been shown to be very helpful to achieving good asymptotic matching 1 Numerical Solution of approximations for the intermediate practical range. TNT Theory (Gao and huang, 2001) It is also important to emphasize that strain gradient the- ories cannot explain the size scale effects observed in fcc metals in the absence of strain gradients [36, 37]. Clearly, Test Range new continuum theories are needed to be able to predict these size effects. 0 1 2 3 log D (Size in nm) 4 5 6 5. MODELING ONE-DIMENSIONAL MATERIALS: NANOTUBES AND Figure 46. Asymptotic-matching approximation for microtorsion. NANOWIRES One-dimensional nanoscale materials, that is, nanotubes 4.4. Concluding Remarks on Strain (carbon CNT, boron nitride BNNT, etc.) and nanowires (e.g., Gradient Theories silicon SiNW), are drawing ever more attention due to their In many applications of interest (e.g., microelectronics and promising properties for a variety of future applications. MEMS), characteristic dimensions are in excess of 100 nm– Mechanical strength of CNTs and BNNTs along with their 1
m. Modeling the mechanics of such systems at the electrical (for carbon) and thermal conductivity (for both), atomistic level is beyond present computational capabilities. are appealing for composite applications. Although SiNWs Therefore, extension of continuum theories to account for have little advantage in strength, their chemical properties size scales is of high relevance. Here we have discussed many enable ease of doping and as a result they are outstand- of the existing theories. At the same time, their range of ing light-emitters due to a quantum confinement effect. applicability was examined through small-size asymptotics. These nanowires have the additional advantage that can be Even though the small-size asymptotic behavior is straightforwardly integrated in Si circuits. Here, we review obtained only below the size range of applicability of the some of the recent results concerning the mechanical yield theory (>100 nm), it is useful to pay attention to it. Sev- and failure as well as the possible coalescence or welding eral main theories show unreasonable small-size asymptotic mechanisms of C and BN nanotubes. Fundamental struc- behavior, which impairs the representation of experimen- tures and energetics of SiNWs are also discussed, in order to tally observed behavior in the practical size range and spoils contrast bulk-based materials with the uniquely built cylin- asymptotic matching approximations. Simple adjustments of drical entities. the theories suffer to obtain reasonable asymptotics and Interest in composite application of CNTs is due to their make asymptotic matching approximation meaningful. mechanical strength combined with the electrical and ther- Finally, an analogy with quasibrittle materials such as con- mal conductivity that together could lead to development a crete, rocks, sea ice, and fiber composite may be mentioned multifunctional material basis for a variety of novel purposes [270]. For them, too, the small size as well as large size ranging from textile to aerospace applications. Besides the asymptotic behaviors are attained only outside the range economical aspects, like low cost volume production of CNT, the limiting factors for their utilization in composites include their proper dispersion in matrix media, matrix-CNT adhe- 2 sion and load transfer, intertubular connectivity and shear Micro-Bending resistance, and internal strength of individual tubes. While log D(σN/σY) (Dimensionless Stress) Asymptotic Matching manufacturing and cost issues remain mostly a subject of Approximation (Bazant and Guo) experimental empirical study, the mechanics of nanotubes 1 σN = σ0 [1+(D0 /D)2s/r ]r/2, s = 1/2, r = 1 has been a topic of detailed theoretical investigations and multiscale computational modeling. Theoretical studies of CNT mechanics have been moti- Test Range vated by experimental evidence of great resilience and by expectations of extraordinary strength. These are based 0 upon the strength of the individual carbon bonds and the elegance of two-dimensional carbon network structures. Numerical Solution of CNT mechanics, including buckling, yielding, and failure TNT Theory (Gao and Huang, 2001) mechanisms, have been extensively investigated and are –1 0 1 2 3 4 5 6 summarized in recent reviews [145, 271], while the studies log D (Size in nm) of elastic behavior, both linear and nonlinear, and buck- ling have made the transition from science to engineering Figure 47. Asymptotic-matching approximation for microbending. (via the demonstrated correspondence of atomistic modeling Micro- and Nanomechanics 593 methods and continuum elasticity and finite elements [145, 5.1. Strength, Failure, and Healing 271]). Many issues of inelastic behavior still require analy- Mechanisms in Nanotubes sis at the atomic scale utilizing the solid state physics and quantum chemistry formulations. 5.1.1. Energies and Thermodynamics As an important example, we have recently undertaken of the Yield Defects a systematic series of quantum ab initio calculations of The transition from yield to tensile strength for CNTs and its structures and moduli [272, 273] for C, BN (boron nitride underlying atomic mechanism have been subjects of particu- or “white graphite,” shown in Fig. 48) and CFx (flu- lar interest. It has been proposed that two alternative yield- orinated carbon) shells. They utilize density functional failure paths in CNT are generally possible [44, 278]: (i) a based computations (within Gaussian package) with periodic brittle fracture through a nucleation and growth of a crack as boundary conditions that yielded a set of accurate elastic in Figure 49, or (ii) a dislocation relaxation (i.e. intramolec- parameters [272] that can be further employed in engineer- ular plasticity) in case of a miniscule fiber-nanotube. Indeed, ing shell models for CNT elasticity, vibrations, or postbuck- it has been observed in detailed computer simulations that ling deformations. This careful discussion is also timely since the primary yield defect, an event at the atomic scale, is somewhat controversial data have recently emerged in the represented by individual bond rotation, which leads to the literature, caused mainly by different interpretations of the formation of two pentagon–heptagon pairs in the hexagonal effective cross-section of nanostructures, referred to recently lattice, 5/7/7/5 in Figure 48. In the chemistry of fullerenes as Yakobson’s paradox [274]. and nanotubes, this corresponds to “pyracylene” or Stone– Wales transformation. A computational study of this defect SiNW is another type of 1D structure of high interest due energy at different levels of applied tension allows one mostly to the strategy of building nanoscale technology via to determine the strain ! at which the defect formation the developed infrastructure of the silicon microelectron- becomes thermodynamically favorable, whereby the nan- ics industry. Recent efforts have demonstrated that ultrathin otube can then possibly yield to the external tensile load. SiNWs indeed have properties that make them very com- Classical empirical potentials, tight binding approxima- petitive in many aspects of nanoscale electronics. First, Si is tions (TBA), and ab initio density functional theory calcu- fairly reactive, enabling it to be easily doped. Second, faster lations all identify the range of thermodynamic instability and lower energy-consuming electronic devices have been with respect to yield as ! exceeds 6–7%. More recently we suggested that are constructed of arrays of SiNWs, prelim- applied the same approach to isomorphic BN nanotubes, inary forms of which have already been built [275]. Finally, where the Stone–Wales defect (Fig. 48) again has been the quantum confinement effect opens a direct bandgap as shown to have the lowest energy, below that of the com- wide as 2.0–3.0 eV, laying a potential foundation for effective peting structure of 4/8/8/4 polygons. The determined defect visible light emitters and the so-called all-photonic technol- formation energy under strain approximately follows a linear ogy possible using silicon materials alone [276]. Additionally, relationship, namely, this direct bandgap has the potential to enable photovoltaics of ultrahigh efficiency, that is, if bulk quantities of SiNWs ESW = 55–46! eV (107) could be produced reasonably cheaply. Recently, we found the ground-state structure of ultra- thin, pristine SiNWs by utilizing theoretical analysis and (cf. ESW = 31–45! eV in case of armchair carbon). Thus, large-scale computations [277]. Surprisingly, these SiNWs density functional analysis of yield thermodynamics for car- are polycrystalline and favor a whisker growth mode (i.e., bon (C, purely covalent) and boron nitride (BN, covalent- they grow much faster along their axis than circumferen- ionic) permits a comparison of their yield strength with the tially). Their structures and energetics with be shown in conclusion that partially ionic BN can be more resistant to comparison to other single crystalline nanowires. Moreover, yield than the homoelemental C. faceting and matching of the facets at the wire’s edges were found to play a critical role in stabilizing the SiNWs. Figure 49. Quantum-mechanical relaxation of a nanocrack in H- Figure 48. Stone–Wales 5/7/7/5 defect in (5, 5) BN nanotube. Reprinted terminated CNT. Reprinted with permission from [280], T. Dumitrica with permission from [273], H. F. Bettinger et al., Phys. Rev. B 65, et al., J. Chem. Phys. 119, 1281 (2003). © 2003, American Institute of 041406 (2002). © 2002, American Physical Society. Physics. 594 Micro- and Nanomechanics 5.1.2. Kinetic Theory of Strength 5.1.3. Welding and Reversible Failure In any practical situation, a system is expected to sustain The possibility of nanotube coalescence, which is either a tension only within a finite time limit, while thermodynamic lateral or a butt-welding process of merging separate enti- equilibrium implies time-unlimited conditions. A more con- ties into one, has been investigated in detail due to its sistent approach to failure [279, 280] comes from one based importance in formation of CNT networks, and therefore on rate equations and leads to more realistic estimates of on overall electrical conductance, thermal transport, and strength. A kinetic theory approach to strength evaluation mechanical strength [282]. We have discovered a mechanism is an important step in this regard where the key point is to of welding that consists exclusively of Stone–Wales bond determine the probability, P , of defect formation (yield) as rotations. It therefore appears as a feasible physical process a function of time. It therefore can be calculated, provided at elevated temperatures or under irradiation. An example is that the computed activation barriers E ∗ are available, by shown in Figure 51 that illustrates the intermediate steps (as numbered) of (10,10)-CNT pair butt-welding, at high tem- perature or under irradiation (e-beam or laser): P = 1015 Ld%t sec−1 nm−2 expD−E ∗ !: J/kb T E ∼ 1 (108) 10: 10 + 10: 10 → 10: 10 or where L is the CNT length and d is its diameter. Follow- 15: 0 + 15: 0 → 15: 0 (109) ing extensive computational examination of the saddle-point activation barriers E ∗ (Fig. 50a), this equation enabled cal- Emerging intermediate structures must possess unique culation of the breaking strains ! for a variety of temper- and possibly useful electronic properties as they repre- atures T , duration times %t, and chiral symmetries J of sent quantum dots with already attached nanotube-wires. SWNT, as shown in Figure 50b. With more accurate ab initio Detailed calculations of the intermediate energies, E, shown calculations of the transition state barriers E ∗ , the process in Figure 52 allow us to compare the energy barriers. The can be enhanced. main data are obtained with TBA; open squares correspond Kinetic analysis shows that although thermodynamic to density functional results, and open circles are obtained requirements for yield can be relatively low (strains of 6– from high-temperature molecular dynamics simulations. The 7%), overcoming activation barriers the strain level must data also show a possible reduction of the barriers by appli- exceed 15–20% (depending on chiral symmetry of the tube cation of external mechanical forces, F , compression or ten- and the test duration). At this high strain level, nanotubes sion, in transition from energy to enthalpy criteria, H = E + can in principle undergo brittle failure through “direct” FL. This is important for engineering new nanostructures bond breaking via a series of metastable radical states through stretching [283] or welding [281], as well as for that include dangling bonds or unpaired electrons. We improvement of bulk materials due to increased connectivity have recently [281] performed quantum mechanical struc- of the tubules in CNT bundles or possibly in composites. ture relaxations which determined the energies of several of the first metastable defect states corresponding to one, 5.2. Silicon Nanowires: Structure two, or three broken bonds in the lattice. As an exam- and Energetics ple, Figure 49 shows a singular broken-bond structure that emerges as metastable at approximately 16% tensile strain. Although SiNWs possess much weaker bonds, they have From a chemistry viewpoint, such structures represent a higher rigidity (originating from the higher coordination biradical, R· + · R, that would immediately recombine under [284, 285]) making the tiny wires extremely fragile com- normal conditions. Unlike the Stone–Wales defect, these vir- pared to carbon nanotubes. Any practical construction of tual defects do not exist in free lattice structures but can Si nanowires should not be too small in order to allow only emerge beyond the bifurcation point at high tension. them to be sustainable in natural ambient conditions. Obvi- ously, extremely thin (d < 1 nm) 1D Si structures could not emulate bulk structuring because all the atoms are actually located on a surface. Therefore, reconstruction will conform (a) (b) them into clusters or a chain of clusters even at zero temper- ature. The modeling of cluster chains considered as a SiNW currently accounts for a big portion of theoretical studies [286, 287]. Recently, a chain of fullerene-like clustered Si20 Figure 50. Activation barriers at ! = 5% (a), and breaking strain values 0 8 18 33 48 68 (b) of CNT. Reprinted with permission from [280], G. G. Samsonidze and B. I. Yakobson, Phys. Rev. Lett. 88 (2002). © 2002, American Phys- Figure 51. Steps of (10,10) + (10,10) coalescence via Stone–Wales bond ical Society. rotations. Micro- and Nanomechanics 595 45 40 0.6 Γ0 (eV) (10,10)+(10,10) 35 30 0.4 Energy (eV) 25 20 15 0.2 10 C60+(10,10) 5 0 2 4 6 8 (5,5)+(5,5) d (nm) 0 Figure 53. Energy and structure (only cross-section on the right three) –5 of three types of SiNWs. On the energy curve, diamonds and squares 0 10 20 30 40 50 60 70 denote two surface reconstructions of the square SiNW; solid and open SW sequence pentagons denote two types of pentagonal SiNWs. The five radial lines in the pentagonal model (right bottom) are {111} interfaces of stack- Figure 52. Computed energies of intermediate structures in coales- ing faults. Reprinted with permission from [277], Y. F. Zhao and B. I. cence. Reprinted with permission from [282], Y. F. Zhao et al., Phys. Yakobson, Phys. Rev. Lett. 91, 035501 (2003). Rev. Lett. 88 (2002). © 2002, American Physical Society. very important in practical applications. The new crystal- was found to have a reasonably low energy [288]. If one adds lographic structure with cross-sectional fivefold symmetry more atomic layers to the cluster, chains will systematically potentially provides opportunity for novel properties. It is accumulate strain, become distorted, and rapidly increase known that the unique growth kinetics make it possible to the energy of the system. Above a particular thickness (d ∼ produce SiNWs with identical crystallographic orientation 15 nm for pristine SiNW), crystalline whiskers are more (i.e., [110] direction). favorable and the process of faceting is the decisive factor In recent experiments, bundles of pristine SiNWs with to conserve energy. diameters of 3–7 nm and lengths of >100 nm were formed in Conventionally, the equilibrium shape of a faceted crystal high vacuum [288] (see Fig. 54), which might be of the pen- is determined by minimization of Wulff’s free energy, which tagonal type according to its growth kinetics. The hexagonal neglects the interaction of facets (edge effect) and assumes a single crystalline SiNW with hydrogen-terminated facets has fixed bulk. This is a good approximation for relatively large also been recently identified in [292]. crystals but poorly applicable to ultrathin SiNWs with d < 10 nm. In [289] we generalized Wulff’s free energy as F = sHs + Ee + Eb (110) to include the energy of matching the adjacent facets, Ee , and certain changes in the bulk, Eb , including possible inter- nal granularity or elastic strain toward lowering the overall energy. Because the shape of a crystalline SiNWs is scal- able with its diameter, the energy of an isomorphic family (with the same shape but various thickness) can be evaluated without full-scale computation. Several families of pristine whiskers with four-, five-, and six-fold symmetry in cross- section have been investigated and the relative energy is shown in Figure 53. Surprisingly, a polycrystalline family has the lowest over- all energy at d < 6 nm due to the perfect matching of the facets along the edges with very little cost to the bulk energy. This type of SiNW also has an important growth kinetics property: the dimer rows along the axis make the adatoms diffuse much faster in the axis direction than along the circumference. Because the structures of the SiNW will influence their Figure 54. Bundles of the pristine SiNW, produced in high vaccum. properties [290], the creation of new structures with con- Reprinted with permission from [288], B. Marsen and K. Sattler, Phys. trolled thickness and crystallographic orientation [291] is Rev. B 60, 11593. (1999). © 1999, American Physical Society. 596 Micro- and Nanomechanics GLOSSARY 34. H. Gao, Y. Huang, and W. D. Nix, Naturwissenschaften 86, 507 (1999). 35. Y. Huang, H. Gao, W. D. Nix, and J. W. Hutchinson, J. Mech. Phys. Solids 48, 99 (2000). 36. H. D. Espinosa, B. C. Prorok, and B. Peng, J. Mech. Phys. Solids, ACKNOWLEDGMENTS in press. 37. H. D. Espinosa, B. C. Prorok, and M. Fischer, J. Mech. Phys. 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