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Dynamic instabilities in {1 1 1} silicon
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Journal of the Mechanics and Physics of Solids 56 (2008) 376–387 www.elsevier.com/locate/jmps
Dynamic instabilities in {1 1 1} silicon D. Sherman, M. Markovitz, O. Barkai Department of Materials Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 9 November 2006; received in revised form 16 May 2007; accepted 30 May 2007
Abstract The phenomena occurring during rapid crack propagation in brittle single crystals was studied by cleaving strip-like silicon specimens along the {1 1 1} low-energy cleavage plane under bending. The experiments reveal phenomena associated with rapid crack propagation in brittle single crystals not previously reported, and new crack path instabilities in particular. In contrast to amorphous materials, the observed instabilities are generated at relatively low velocity, while at high velocity the crack path remains stable. The experiments demonstrate that crack velocity in single crystals can attain the theoretical limit. No evidence for mirror, mist, and hackle instabilities, typical in amorphous materials, was found. The important role played by the atomistic symmetry of the crystals on controlling and generating the surface instabilities is explained; the importance of the velocity and orientation-dependent cleavage energy is discussed. The surface instabilities are generated to satisfy minimum energy dissipation considerations. These findings necessitate a new approach to the fundamentals of dynamic crack propagation in brittle single crystals. r 2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic fracture; Brittle single crystals; Cleavage; Dynamic instabilities; Crystal symmetry
1. Introduction Griffith (1920), in his pioneering energy balance law for ideally brittle solids, drew the relationship between the material properties, the crack length and the critical stresses for crack initiation. He considered smooth and straight crack. Stroh (1957) defined the upper limit of the crack speed as the speed at which waves propagate along a free surface, known as the Rayleigh free surface wave speed, denoted as CR. This may be correct provided a smooth and straight Griffith crack exists. Wallner (1939) identified markings appearing on the fracture surface of brittle materials (‘‘Wallner lines’’), which were generated when the propagating crack interacted with reflected stress waves. Andrews (1959) was the first to identify the surface features in fractured brittle frozen rubber as mirror, mist, and hackle, and to introduce surface features with different amplitudes. Ravichandar and Knauss (1984) showed, years later, that these surface features in brittle plastic (Homalite100) progressively grow in size. Fineberg et al. (1991) were the first to correlate the oscillations in crack velocity with surface features in brittle amorphous polymer (PMMA), and in soda lime glass (Gross et al., 1993), and to term the effect ‘instabilities in dynamic fracture’. At velocities less than one-third of the Rayleigh Corresponding author. Tel.: +972 4 8294561; fax: +972 4 8295677.
E-mail address: [email protected] (D. Sherman). 0022-5096/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2007.05.010
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surface wave speed (CR), the fracture surface is mirror like. As the crack velocity increases, the surface instabilities increase and is manifested by mist and hackle; then, at sufficiently high velocities, the crack branches into more than one tip. It is argued that the maximum crack velocity in these solids never attains CR, as the surface undulations become an additional source of energy dissipation mechanism (Ravichandar, 1998). Over the last decade, large-scale atomistic simulations of rapidly propagating cracks in brittle single-crystalline models have shown the existence of dynamic instabilities (Abraham et al., 1994; Marder and Gross, 1995; Cramer et al., 2000; Buehler and Gao, 2006). Continuum mechanics, on the other hand, is unable to incorporate these surface instabilities in the calculation; the relationship between the total potential energy, G, the crack velocity, V, and the cleavage energy, G, was defined for straight cracks with no surface features at constant velocity as (Freund, 1990) V G G0 1 G, (1) CR where G0 is a quasi-static strain energy release rate of a stationary crack. When the crack velocity approaches the limiting velocity CR, G0 increases rapidly to satisfy the total energy balance in the system, Eq. (1). Several numerical models have attempted to rationalize the physical occurrence of rapid crack propagation in solids (Slepyan, 1993). The main problems remain the exact definition of the interatomic potentials and the number of the involved atoms. Rigorous experimental results that describe the full range of crack path instabilities, the importance of the atomistic symmetry, and the limit of crack velocity occurring during rapid crack propagation in brittle single crystals are still scarce. We use silicon single crystal specimens cut from commercial silicon wafers as a model material to elucidate the phenomena occurring during rapid crack propagation in brittle single crystals, since silicon wafers well represent nearly ideally brittle and defect-free crystals. The estimated length of the process zone in silicon is about two atomic distances; this is of profound importance when surface instabilities are discussed, since in silicon, the process zone has no influence on the generation of surface features (Lawn, 1993). The low-energy cleavage planes in silicon are {1 1 1} and {1 1 0}, with small difference in their cleavage energies. While the {1 1 1} cleavage plane is the plane with the lowest cleavage energy, their exact values are still under debate, and range between 3 and 5 J/m2 (Jaccodine, 1963; Michot, 1987; Spence et al., 1993; Ebrahimi and Kalwani, 1999; Pe´rez and Gumbsch, 2000; Bernstein and Hess, 2003). In previous investigations, we have described the crack deflection phenomenon, in which a crack, propagating under three-point bending (3PB) along the {1 1 0} low-energy cleavage plane of silicon, is deflecting to the {1 1 1} cleavage plane. The deflection was explained by minimum total energy considerations and was shown to depend upon crack velocity and crystallographic orientation (Sherman, 2005; Sherman and Be’ery, 2004). In the present paper, we report on phenomena associated with crack propagation occurring during dynamic propagation along the {1 1 1} low-energy cleavage plane of silicon. 2. Experimental Cleaving of thin and wide silicon specimens was performed under 3PB. The specimens, of 20 43 mm2 lateral dimensions, were cut from commercial [1 1 0] 400 diameter, 525 mm thick silicon wafers. The specimens were notched with a 150 mm thick diamond saw to a length between 0.15 and 2 mm. The notch length determines the energy at crack initiation and during propagation and the crack velocity at steady-state. In these specimens, the crack is propagating along the ð1¯ 1 1Þ low-energy cleavage plane, which is perpendicular to the specimen-free surfaces. The 3PB Bridge with a specimen and a schematic of half the specimen is shown in Fig. 1a and b, respectively. The 3PB has the advantage of maintaining a stable crack propagation in the x, or the ½1 1¯ 2, direction, as the maximum tensile stresses, or the maximum KI plane is located underneath the line of loading of the middle pin. This is preferable over four-point bending or double torsion loading configuration. The farfield tensile/compressive stress, szz, typical to 3PB, is maximum tensile at the bottom surface and maximum compressive at the top surface of the specimen. The bending shear stress, szy, is negligible, as the thickness/ length of the specimen is 1/80. The 3PB stresses are responsible for the unique crack front shape in these specimens; the crack initiates at the bottom left point of the specimen (Fig. 1b and c), as a result of the high
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y/h Fig. 1. The 3PB specimen and the loading configuration; (a) optical photograph of the fully articulated 3PB bridge with a specimen, (b) schematic representation of half of the specimen set up, depicting the crack front and a ridge pattern and the misaligned ð1¯ 1 1Þ plane with misalignment angle, x, (c) schematic presentation of the fracture surface depicting the evolution of the crack front and of the Vn and Vx velocities and the orientation angle, y and (d) the normalized ridge velocity, Vn/Vx and orientation, y, as a function of the normalized location, y/h.
tensile stresses generated at the bottom of the notch, and starts to propagate as a quarter circular crack, as shown in Fig. 1c. This quarter circular crack experiences high tensile szz stresses at the bottom and reduced szz tensile stresses at the top portion of the circular crack front, hence, accelerating near the bottom surface of the specimens (to the ½1 1¯ 0 direction or the x-axis) to achieve the shape of a nearly quarter ellipse (Fig. 1c). Note that the crack front, in its fully aligned situation (see below), experiences tensile mode only, due to the
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symmetric bending stresses, szz, normal to the crack faces. At a steady-state of crack propagation, the crack front is approximated by a quarter ellipse with a long, shallow ‘tail’ (Be’ery et al., 2003; Sherman and Be’ery, 2003) (Fig. 1b and c) and advancing at constant velocity, Vx, to the x direction. The crack front is actually the temporal locus of the ends of all the ridges— a ridge being the path along which the bonds between atoms break— its velocity and orientation are Vn and y, respectively. A typical single ridge and the resultant crack front shape are also shown in Fig. 1c. The relationship between the crack front velocity, Vx, and the ridge velocity, Vn, is simple: Vn ¼ Vx cos y, where y is readily obtained as the angle of the normal to the ellipse describing the crack front at a certain point along the y-axis (Be’ery et al., 2003). A wide range of velocities, Vn, is encountered by a propagating ridge in our specimens, from thousands of m/s (scaled with the Rayleigh surface wave speed) at the bottom surface to several tens of m/s at the top portion of the fracture surface. At the same time, the direction of the velocity vector, via the angle y, varies in accordance with the crystallographic orientation of the ð1 1¯ 1Þ cleavage plane from ½1 1¯ 2 to [1 1 0], see Fig. 1d. This enables examination of various aspects of crack tip velocity and crystallographic orientations using a single specimen. This is in contrast to a tensile specimen, where the crack propagates along a single orientation. The normalized tip velocity, VnXVx, and the orientation, y, vs. the normalized location along the y-axis (y/h) is shown in Fig. 1d. These features enable the determination of the fundamental phenomena of surface perturbations/instabilities associated with rapid crack propagation in brittle single crystals. The quarter elliptical shape was found to closely resemble the crack front in glass and in silicon where the crack propagates along the {1 1 0} (Be’ery et al., 2003; Sherman and Be’ery, 2003) cleavage plane. The latter experiments showed that similar behavior is evident for crack propagation on the {1 1 1} plane. The parallel velocity, Vx, was measured using the ‘Potential Drop’ technique (Stalder et al., 1983). The velocity was also measured in unnotched specimens. In this technique, a thin metallic layer (20 nm Ta) is deposited on the bottom surface of the specimen, and serves as a resistor. As the crack propagates, it cuts the metallic layer, which, hence, experiences increasing resistivity. As a result, the potential in the metallic layer drops during propagation. The potential drop was measured by means of a 200 MHz scope. These measurements provide the voltage/time relationship during crack propagation. A calibration function, for the voltage/crack length relationship, was derived by scribing similar coated specimen with a diamond scriber under an optical microscope, and the potential drop was measured every 0.25 mm. Both provide, after certain manipulations, crack length vs. time. Derivative of the latter function with time yields the average crack velocity as a function of the crack length; no detail of velocity fluctuations is accessible. We estimate the velocity accuracy at the stage of steady-state velocity by 50 m/s. It is noted that Eq. (1) is fulfilled along the crack front, namely, the dynamic energy release rate, G, is equal to the dynamic fracture energy, G, at each point along the front. The quasi-static strain energy release rate, G0, however, varies with location along the front; it is nearly G0, the quasi-static cleavage energy, along the straight ‘tail’, where the velocity is low (see Fig. 1d), and increases as the bottom surface of the specimen is approached, where the velocity is high, to fulfill Eq. (1). 3. Results 3.1. Crack velocity measurements We present the calculated average parallel velocity, Vx (Fig. 1c), of two specimens; one was notched to a length of 0.15 mm, while the other was unnotched but contained microcracks along the specimen edges generated during the dicing process. Since the velocity vector along the bottom surface of the specimen points toward the x-axis (y ¼ 0 or the ½1 1¯ 2 direction; see Fig. 1c), it can be considered to be under pure tension. The voltage/time relationship during crack propagation, obtained for an unnotched specimen, is shown in Fig. 2a. The calibration function, for the voltage/crack length relationship, is shown in Fig. 2b. Note that the crack propagation duration in Fig. 2a is nearly 5 ms. The crack velocity vs. the crack length for the notched specimen is shown in Fig. 2c. The crack rapidly accelerates for the first 3 mm of propagation; the acceleration is then considerably reduced and the crack thereafter continues to propagate in a nearly constant, steady-state velocity. The maximum crack velocity measured for the notched specimen is 3620 m/s, which is 0.8CR (CR for the {1 1 1} /1 1 2S system was taken as
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4520 m/s). The fracture surface of this specimen at steady-state velocity is the one shown in Fig. 3b. This velocity is in excellent agreement with other crack velocity measurements in silicon (Cramer et al., 2000; Hauch et al., 1999). Crack velocities having a high fraction of CR were also measured for other single crystals (Field, 1971; Hull and Beardmore, 1966). The crack velocity measured for the unnotched specimen is higher and reaches 4020 m/s (Fig. 2d). This is the highest crack velocity (0.9CR) measured to date for the {1 1 1} /1 1 2S system of silicon. It is important to note that the crack in this unnotched specimen was initiated at one of the specimen edges, but 2 mm away from the loading pin. The fracture surface contained large kink instabilities (see below) at the beginning of the propagation, followed by moderate kinks, and yet, the crack achieved high velocity. Corrugated instabilities (see below) were also evident along the fracture surface of this specimen. 3.2. The terrace-like kink instabilities The basic crack path instabilities along the ð1¯ 1 1Þ cleavage plane are shown in an optical image in Fig. 3a. Crack profile and an individual ridge pattern are also shown in that figure. The instabilities are terrace-like kinks (Lawn, 1993), propagating along nearly 2/3 of the fracture surface, where the velocity Vn (Fig. 1c and d) is relatively high, and corrugated instabilities at the upper portion of the fracture surface, where that velocity is relatively low. The Atomic Force Microscope (AFM) image (Fig. 4a) taken from the dashed circled zone in Fig. 3a, shows the well-defined kink instabilities. A typical ridge pattern is also shown. A ridge is propagating on the ð1¯ 1 1Þ cleavage plane, but every few tens of micrometers it kinks sharply onto, presumably,
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Fig. 3. Optical images of the ð1¯ 1 1Þ fracture surface under 3PB. (a) The terrace-like kink instabilities at the bottom and the corrugated instabilities at the top portion of the fracture surface and (b) the fracture surface of a well-aligned specimen showing the corrugated instabilities where most of the fracture surface is free of the kink instabilities. The crack front, the single ridge pattern, the associated velocities, Vx and Vn, and the angle y are also schematically shown. The initiation points of the corrugated instabilities are well defined and their line of propagation points towards the [2 1 1] direction.
a combination of (1 1 0) and ð1¯ 1 0Þ cleavage planes and continuing its propagation further on the ð1¯ 1 1Þ cleavage plane. The height of the kinks (Fig. 4a) is of an order of a few hundreds of nanometers. A profilometer line scan, Fig. 4b, provides a larger scale description of these instabilities. It is noted that kink instabilities were shown elsewhere with only general treatment in silicon (Hauch et al., 1999; Sherman and Be’ery, 2004), in sapphire (Sherman and Be’ery, 1998), and in GaAs (Sauthoff et al., 1999). 3.3. The corrugated instabilities We eliminated the kink instabilities by aligning the ð1¯ 1 1Þ cleavage plane with the maximum KI plane, namely, by reducing the misalignment angle x (Fig. 1b) to nearly zero; this reduction was achieved by rotating the specimens around the y-axis with respect to the bending bridge; this action required several specimens. Eliminating the kink instabilities was essential in order to investigate the corrugated instabilities. Indeed, micron scale height corrugated instabilities are only evident at the upper portion of the cleavage plane, as shown in Fig. 3b, in well-aligned specimen notched to 0.15 mm notch. The loci at which the corrugated instabilities initiated are well defined and well organized in terms of the normalized height, y/h, the density of the corrugations, and the principal lines of propagation. The latter are in the [2 1 1] direction, inclined at 601 to the x-axis. An AFM image of the zone where these instabilities initiate (dashed circled zone in Fig. 3b), is shown in Fig. 4c. Note that these instabilities are shallow, while their maximum height can reach nearly a micron. A profilometer line-scan of the corrugated instability (dashed line in Fig. 3b) is shown in Fig. 4d
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Fig. 4. A closer look at the major dynamic crack path instabilities generated along the ð1¯ 1 1Þ cleavage plane. (a) AFM image of the kink instabilities (open circle in Fig. 3a). The crack propagates along the terrace, and kinks every several tens of microns into the (1 1 0) plane. The maximum KI plane is drawn. (b) A profilometer line scan along the ½1 1¯ 2 direction of the kinks. The maximum KI plane is also shown in this image. The misalignment angle is less than 11. (c) AFM image of the initiation zone of the corrugated instabilities (open circle in Fig. 3b). Note the ‘watershed’ lines in the [2 1 1] direction. The corrugated instabilities are shallow; their inclination angle is of about 21 to the ð1¯ 1 1Þ cleavage plane (see line AB). (d) Profilometer line scan along the ½1 1¯ 2 direction of the corrugated instabilities. (e) AFM image from the bottom surface (open square in Fig. 3b) of the fully aligned specimen showing ridges propagating with high normal velocity, Vn, in the ½1 1¯ 2 direction. While the crack path is stable in the ½1 1¯ 2 direction, it is wavy with 1 nm amplitude, presumably due to the formation of the ridges.
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Fig. 5. (a) Three optical images of ð1¯ 1 1Þ fracture surfaces of fully aligned specimens with increasing notch length (0.15, 0.5, and 0.7 mm) and, hence, decreased parallel velocity, Vx, and dynamic energy. The normalized height, at which the corrugated instabilities initiate, y/h, decreases as the parallel velocity decreases. The initiation velocity of the corrugated instabilities, Vnc, is about constant, and equals 1050 m/s. The ‘watershed’ line remains in the [2 1 1] direction, inclined at 601 to the x-axis (the ½1 1¯ 2 direction). (b) The ridge velocity, Vn, as a function of the normalized height, y/h, for the three specimens. Note that the corrugated instabilities propagate at low velocities (dashed lines). The error in velocity measurements is estimated to be 750 m/s. Note also that the ridge angle, y, at initiation of the corrugations is not constant and depends on the parallel crack velocity.
and demonstrates the relatively large amplitude of these instabilities. Similar instabilities were identified in Sapphire (Sherman and Be’ery, 1998). To elucidate the role of velocity in the formation of these instabilities, we performed a set of experiments involving specimens with three different notch lengths and, thus, with three different crack velocities. It was found that as the parallel velocity, Vx, increases, the normalized height, y/h, at which the corrugated instabilities initiate, increases. The notch length, a, the parallel velocity, Vx, the normalized height, y/h, and the calculated normal velocity at which the corrugated instabilities initiates, Vnc, are shown in Fig. 5a for three different specimens. Assuming the calculations of Vn are valid, the averaged critical normal crack velocity, Vnc, is about 1050 m/s, i.e., 0.23CR. Note that Vnc constitutes the upper limit for the velocity of the corrugated instabilities, as the normal velocity along the corrugated instabilities is diminishing, as shown in Fig. 5b. Note also that the ridge angle, y, at initiation of the corrugations is not constant and depends on the parallel crack velocity. The corrugations spread for the whole length of the fracture surface of the specimen, including the surrounding of the specimen’s notch.
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3.4. AFM scan of the stable fracture surface We performed AFM scans of the fully aligned specimen with a 0.15 mm notch at the lower portion of the specimen (see dashed square in Fig. 3b) where the crack tip velocity, Vn, is nearly the maximum velocity evaluated for this specimen (Fig. 2c), 3600 m/s (0.8CR). This was done in order to trace any possible dynamic instability associated with fast cracking. The result is shown in Fig. 4e. The surface constitutes 2 nm height ridges that propagate towards the ½1 1¯ 2 direction. No evidence for mirror, mist, or hackle was detected by the 35 nm radius AFM tip. Shallow ‘Wallner lines’ are also visible in specimens with relatively low parallel velocity, Vx, and become invisible when this velocity is high (see Fig. 5a). 4. Discussion In this section, we attempt to explain the generation of the above-described dynamic instabilities from points of view of the state of stresses, the crystal symmetry, and the velocity and orientation-dependent cleavage energy. Unfortunately, a full description of the latter is yet not available. 4.1. The terrace-like kink instabilities The kink instabilities were generated due to misalignment between the ð1¯ 1 1Þ cleavage plane and the maximum KI plane, see Fig. 1b. The latter is located under the loading pin, and lies on the x–y plane of the specimen; the former makes an angle x with that plane (rotation about the y-axis). Rough estimates indicate that the misalignment of the instabilities shown in Fig. 4b is less than 11. Such misalignment leads to an in-plane mode mixity: Mode I is generated by the bending stress szz normal to the ð1¯ 1 1Þ plane, and Mode II—by the in-plane shear stress, sxz (Figs. 1b and 4a), resulting from the loss of symmetry of deformation. A ridge propagates along the ð1¯ 1 1Þ plane with increasing KII, until a critical point (Point A, Fig. 4a) where it reaches a critical KII; thereafter, the crack is abruptly attracted to the maximum KI plane. This critical KII resembles the jump of the crack from the ð1¯ 1 1Þ to a combination of (1 1 0) and ð1¯ 1 0Þ cleavage planes (Fig. 4a). It is also noted that the spatial point of kinking is affected by the local ridge velocity and the crystallographic orientation; all determine the kinking conditions. These conditions vary significantly along the kinking locus. This explains the distinct shape of the terraces than that of the ridges. The kink formation can also be rationalized by energy consideration: less energy is required for a ridge to propagate along the kink than to continue to propagate along the cleavage plane (He and Hutchinson, 1989). The density of the kinks and the height of an individual kink depend upon the angle of misalignment, x. 4.2. The corrugated instabilities ¯ The corrugated instabilities constitute a complex three-dimensional architecture consisting of ð1¯ 1¯ 0Þ, ð0 1 1Þ ¯ and ð1 1 1Þ cleavage planes, and generated from energy considerations. It is readily concluded that the velocity at which the corrugated instabilities are generated (and propagated) is relatively low; this is since the normal velocity, Vn, is low at the upper portion of the fracture surface. This fact is in contrast to the unstable nature of fast cracks in amorphous materials; the instabilities in these materials increase as the crack velocity increases. Another important observation is the direction of propagation of the corrugated instabilities; they constitute disturbances where their watershed lines are always directed to the [2 1 1] direction, see Fig. 4c, which is also the line of intersection of the ð0 1¯ 1Þ cleavage plane with the ð1¯ 1 1Þ cleavage plane. Furthermore, the corrugations are always directed out of the page, for the planes and directions noted in Fig. 4c. The critical velocity of the formation of the corrugated instabilities, Vnc, demonstrates that slow cracks in single crystals are unstable; Vnc is the upper limit for the velocity at which these instabilities can be generated; above this velocity the ð1¯ 1 1Þ fracture surface is stable. This latter argument is further supported by the following observation: the {1 1 0} planes intersect the ð1¯ 1 1Þ ¯ directions (Fig. 3b). The ½1 2 1 ¯ plane along three different directions: the ½1 1¯ 2, the [2 1 1], and the ½1 2 1
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direction is not in the ridge propagation direction and, therefore, is not considered. It is not clear why the corrugated instabilities do not develop in the ½1 1¯ 2 direction at the lower portion of the fracture surface (where y is low), although it has the same atomistic arrangement as the [2 1 1] direction. We postulate that the reason for the smooth surface is the high velocity of the bond-breaking mechanisms, since Vn4Vnc. This once again indicates that fast cracks in brittle crystals are stable. Silicon is a diamond-like cubic crystal with three independent elastic moduli (E, n, and G), and three independent elastic constants (C11, C12, C44, or S11, S12, and S44). When the geometric axes of the crystal containing a crack coincide with its principal crystallographic axes ([1 0 0], [0 1 0], and [0 0 1]), the fourth-order compliance or stiffness tensors have the well-known cubic structure: 2
S11
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The main feature of the cubic compliance (or stiffness) tensor is the lack of coupling between the tensile/ compressive stresses and the shear stresses. Tensile stresses generate no shear strain, and vice versa. The compliance tensor for a crack running along the {1 1 0} cleavage plane, irrespective of its direction, maintains the cubic symmetry, similar to Eq. (2). In this case, the external tensile stresses normal to the crack plane do not generate out-of-plane shear stresses, and therefore, no driving force exists for generating out of plane surface instabilities. Indeed, it was shown recently (Sherman and Be’ery, 2004), that the fracture surface of a crack propagating along the (1 1 0) cleavage plane is smooth in all directions. The symmetric atomistic arrangement for a crack running on the (1 1 0) cleavage, shown in Fig. 6a, dominates the crack stability. However, when the geometrical axes of the silicon crystal are rotated to coincide with the orthogonal [1 1 0], ½1 1¯ 2, and ½1¯ 1 1 axes (i.e., the crack is propagating on the ð1¯ 1 1Þ ½1 1¯ 2 cleavage system, and the external load is in the ð1¯ 1 1Þ, or z direction), the cubic symmetry is transformed into a pseudo-rhombohedral
Fig. 6. The atomistic symmetry experienced by cracks running along: (a) the (1 1 0) cleavage plane in the ½1 1¯ 0 direction and (b) the ð1¯ 1 1Þ cleavage plane in the ½1 1¯ 2 direction. In both cases, the crack plane is perpendicular to the plane of the figure, and the applied stresses are normal to the crack plane. The atomistic symmetry in (a) does not allow for generating out of plane shear stresses and, therefore, the crack path is stable, while that in (b) is anti-symmetric with respect to the crack plane, enabling the generation of out of plane shear stresses, and promoting the generation of the corrugated instabilities.
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symmetry. The non-zero components in the compliance tensor are marked by stars in Eq. (3): 2
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In Cartesian orthogonal axes, the far-field tensile stress generates in-plane shear stresses, sxz, and also outof-plane shear stresses, szy, in the crystal. This behavior is further supported by the asymmetric atomistic arrangement for this system, as shown in Fig. 6b. An individual instability is generated at a preferred crystallographic orientation, and due to energy balance; the cleavage energy of the preferred cleavage planes is velocity and orientation dependent, such that G in Eq. (1) has more complicated form (Sherman and Be’ery, 2004): G ¼ GðV ; yÞ,
(4)
which results in the anisotropic, velocity-dependent phonon emission typical of a crystal containing a propagating crack (Cramer et al., 2000; Sherman and Be’ery, 2004; Sherman, 2005; Hauch et al., 1999; Zhou et al., 1996; Zhao et al., 1996). This velocity-dependent anisotropic phonon emission is an additional, basic mechanism for energy dissipation. We postulate, therefore, that at Vnc1050 m/s, the crack propagates along a combination of ð1¯ 1 1Þ and ð0 1¯ 1Þ planes, rather than on the ð1¯ 1 1Þ plane only, for energy efficiency reasons. This unstable path is more favored energetically than the stable path below the critical velocity Vnc. The validation of the postulated phonon-based energy consumption mechanisms requires either considerable experimental work, or atomistic simulations, which are not available yet. 5. Summary The dominance of the crystallographic symmetry in the formation of the crack path instabilities was demonstrated in this investigation. Corrugated instabilities were generated along well-defined crystallographic orientations, where the atomistic arrangement is such that perturbations of the crack path along a combination of the ð1¯ 1 1Þ, the (1 1 0), and ð1 1¯ 0Þ cleavage planes are permitted. Another important finding is that these instabilities were generated at relatively low velocity, below 0.23CR. At higher velocities, the crack path is stable. The corrugated instabilities, although on the micrometer scale, do not disrupt the crack from attaining high velocity, in contrast to surface instabilities in brittle amorphous solids (PMMA; soda lime glass, brittle plastics) that cause a significant reduction in crack velocity. It is important to note that the corrugated instabilities, aside from having large amplitude, are very shallow. They are generated from minimization of energy consumption: the unstable path consumes less energy than the stable path. We also identified the kink instabilities, generated due to misalignment of the cleavage plane and the maximum KI plane. No mirror, mist, and hackle instabilities were observed at the resolution of a 35 nm radius AFM pin. No branching was observed for cracks propagating at a velocity of up to 0.9CR. This is presumably due to the stresses typical of 3PB. Cracks running on the ð1¯ 1 1Þ cleavage plane in the ½1 1¯ 2 direction can attain a high fraction of the Rayleigh surface wave speed. The high measured velocity of cracks, 0.8CR for the notched specimen with a stable crack path along the bottom of the fracture surface, and 0.9CR for the unnotched one with kink instabilities, support our assumption that crack velocities in brittle solids can attain Rayleigh surface waves speed. The crack path instabilities in silicon indicate the potential crack path instabilities in other brittle crystals. However, the existence of similar instabilities in other brittle crystals depends on the particular energies of the preferred cleavage planes and the atomistic arrangement. On the other hand, kink instabilities are expected whenever misalignment exists.
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The present results show that brittle single crystals behave profoundly differently from amorphous glasses or brittle plastics, and provide a greater challenge to the numerical analysis community, and in particular, studies by molecular dynamics simulations. Acknowledgment This research was supported by the Israel Science Foundation (ISF) under Grant No. 1110/04. I. Gavish is acknowledged for her contribution to the experimental work. References Abraham, F.F., Brodbeck, D., Rafey, R.A., Rudge, W.E., 1994. Instability in dynamics fracture—a computer-simulation investigation. Phys. Rev. Lett. 73, 272–275. Andrews, E.H., 1959. Stress waves and fracture surfaces. J. Appl. Phys. 30, 740–743. Be’ery, I., Lev, U., Sherman, D., 2003. On the lower limiting velocity of a dynamic crack in brittle solids. J. Appl. Phys. 93, 2429. Bernstein, N., Hess, D.W., 2003. Lattice trapping barriers to brittle fracture. Phys. Rev. Lett. 91, 025501. Buehler, M.J., Gao, H., 2006. Dynamical fracture instabilities due to local hyperelasticity at crack tips. Nature 439, 04408. Cramer, T., Wanner, A., Gumbsch, P., 2000. Energy dissipation and path instabilities in dynamic fracture of silicon single crystals. Phys. Rev. Lett. 85, 788. Ebrahimi, F., Kalwani, L., 1999. Fracture anisotropy in silicon single crystal. Mater. Sci. Eng. A 268, 116–126. Field, J.E., 1971. Brittle fracture: its study and application. Contemp. Phys. 12, 1–31. Fineberg, J., Gross, S.P., Marder, M., Swinney, H.L., 1991. Instabilities in dynamic fracture. Phys. Rev. Lett. 67, 457–460. Freund, L.B., 1990. Dynamic Fracture Mechanics. Cambridge University Press, Cambridge. Griffith, A.A., 1920. The phenomena of rupture and flow in solids. The phenomena of rupture and flow in solids. Mech. Eng. A 221, 163–198. Gross, S.P., Fineberg, J., Marder, M., McCormick, W.D., Swinney, H.L., 1993. Acoustic emissions from rapidly moving cracks. Phys. Rev. Lett. 71, 3162–3165. Hauch, J.A., Holland, D., Marder, M., Swinney, H., 1999. Dynamic fracture in single crystal silicon. Phys. Rev. Lett. 82, 3823–3826. He, M.Y., Hutchinson, J.W., 1989. Kinking of a crack out of an interface. J. Appl. Mech. 56, 270–278. Hull, D., Beardmore, P., 1966. Velocity of propagation of cleavage cracks in tungsten. Int. J. Fract. 2, 468. Jaccodine, R.J., 1963. Surface energy of germanium and silicon. J. Electrochem. Soc. 110, 524–527. Lawn, B., 1993. Fracture of Brittle Solids. Cambridge University Press, Cambridge, UK. Marder, M., Gross, S., 1995. Origin of crack tip instability. J. Mech. Phys. Solids 43, 1–48. Michot, G., 1987. Propagation of {1 1 0} cleavage cracks in silicon. Surf. Sci. 186, L561–L567. Pe´rez, R., Gumbsch, P., 2000. An ab initio study of the cleavage anisotropy in silicon. Acta Mater. 48, 4517–4530. Ravichandar, K., 1998. Dynamic fracture of nominaly brittle materials. Int. J. Fract. 90, 83–102. Ravichandar, K., Knauss, W.G., 1984. An experimental investigation into dynamic fracture: II. Microstructural aspects. Int. J. Fract. 26, 65–80. Sauthoff, K., Wenderoth, M., Heinrich, A.J., Rosentreter, M.A., Engel, K.J., Reusch, T.C.G., Ulbrich, R.G., 1999. Nonlinear dynamic instability in brittle fracture of GaAs. Phys. Rev. B 60, 4789–4795. Sherman, D., 2005. Macroscopic and microscopic examination of the relationship between crack velocity and path and Rayleigh surface wave speed in single crystal silicon. J. Mech. Phys. Solids 53 (12), 2742–2757. Sherman, D., Be’ery, I., 1998. The non-linear dynamic rupture in sapphire. Phys. Rev. Lett. 80, 540–543. Sherman, D., Be’ery, I., 2003. The shape and the energies of a dynamically propagating crack under bending. J. Mater. Res. 18, 2379. Sherman, D., Be’ery, I., 2004. From crack deflection to lattice vibrations—macro to atomistic examination of dynamic cleavage fracture. J. Mech. Phys. Solids 52, 1743–1761. Slepyan, L.I., 1993. Principle of maximum energy dissipation rate in crack dynamics. J. Mech. Phys. Solids 41, 1019–1033. Spence, J.C.H., Huang, Y.M., Sankey, O., 1993. Lattice trapping and surface reconstruction for silicon cleavage on (1 1 1). Ab-initio quantum molecular dynamics calculation. Acta Metall. Mater. 41, 2815–2824. Stalder, B., Beguelin, P., Kausch, H.H., 1983. A simple velocity gauge for measuring crack growth. Int. J. Fract. 22, R47–R50. Stroh, A.N., 1957. Advances in physics. Philos. Mag. Suppl. 6, 418–465. Wallner, H., 1939. Linienstrukturen an bruchflachen. Z. Phys. 114, 368. Zhao, D., Haneman, D., McAlpine, N.S., 1996. Detection of temperature rise during cleavage of silicon. Surf. Sci. 369, 76–84. Zhou, S.J., Lomdahl, P.S., Thomson, R., Holian, B.L., 1996. Dynamic crack processes via molecular dynamics. Phys. Rev. Lett. 76, 2318–2321.
Journal of the Mechanics and Physics of Solids 56 (2008) 376–387 www.elsevier.com/locate/jmps
Dynamic instabilities in {1 1 1} silicon D. Sherman, M. Markovitz, O. Barkai Department of Materials Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 9 November 2006; received in revised form 16 May 2007; accepted 30 May 2007
Abstract The phenomena occurring during rapid crack propagation in brittle single crystals was studied by cleaving strip-like silicon specimens along the {1 1 1} low-energy cleavage plane under bending. The experiments reveal phenomena associated with rapid crack propagation in brittle single crystals not previously reported, and new crack path instabilities in particular. In contrast to amorphous materials, the observed instabilities are generated at relatively low velocity, while at high velocity the crack path remains stable. The experiments demonstrate that crack velocity in single crystals can attain the theoretical limit. No evidence for mirror, mist, and hackle instabilities, typical in amorphous materials, was found. The important role played by the atomistic symmetry of the crystals on controlling and generating the surface instabilities is explained; the importance of the velocity and orientation-dependent cleavage energy is discussed. The surface instabilities are generated to satisfy minimum energy dissipation considerations. These findings necessitate a new approach to the fundamentals of dynamic crack propagation in brittle single crystals. r 2007 Elsevier Ltd. All rights reserved. Keywords: Dynamic fracture; Brittle single crystals; Cleavage; Dynamic instabilities; Crystal symmetry
1. Introduction Griffith (1920), in his pioneering energy balance law for ideally brittle solids, drew the relationship between the material properties, the crack length and the critical stresses for crack initiation. He considered smooth and straight crack. Stroh (1957) defined the upper limit of the crack speed as the speed at which waves propagate along a free surface, known as the Rayleigh free surface wave speed, denoted as CR. This may be correct provided a smooth and straight Griffith crack exists. Wallner (1939) identified markings appearing on the fracture surface of brittle materials (‘‘Wallner lines’’), which were generated when the propagating crack interacted with reflected stress waves. Andrews (1959) was the first to identify the surface features in fractured brittle frozen rubber as mirror, mist, and hackle, and to introduce surface features with different amplitudes. Ravichandar and Knauss (1984) showed, years later, that these surface features in brittle plastic (Homalite100) progressively grow in size. Fineberg et al. (1991) were the first to correlate the oscillations in crack velocity with surface features in brittle amorphous polymer (PMMA), and in soda lime glass (Gross et al., 1993), and to term the effect ‘instabilities in dynamic fracture’. At velocities less than one-third of the Rayleigh Corresponding author. Tel.: +972 4 8294561; fax: +972 4 8295677.
E-mail address: [email protected] (D. Sherman). 0022-5096/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2007.05.010
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surface wave speed (CR), the fracture surface is mirror like. As the crack velocity increases, the surface instabilities increase and is manifested by mist and hackle; then, at sufficiently high velocities, the crack branches into more than one tip. It is argued that the maximum crack velocity in these solids never attains CR, as the surface undulations become an additional source of energy dissipation mechanism (Ravichandar, 1998). Over the last decade, large-scale atomistic simulations of rapidly propagating cracks in brittle single-crystalline models have shown the existence of dynamic instabilities (Abraham et al., 1994; Marder and Gross, 1995; Cramer et al., 2000; Buehler and Gao, 2006). Continuum mechanics, on the other hand, is unable to incorporate these surface instabilities in the calculation; the relationship between the total potential energy, G, the crack velocity, V, and the cleavage energy, G, was defined for straight cracks with no surface features at constant velocity as (Freund, 1990) V G G0 1 G, (1) CR where G0 is a quasi-static strain energy release rate of a stationary crack. When the crack velocity approaches the limiting velocity CR, G0 increases rapidly to satisfy the total energy balance in the system, Eq. (1). Several numerical models have attempted to rationalize the physical occurrence of rapid crack propagation in solids (Slepyan, 1993). The main problems remain the exact definition of the interatomic potentials and the number of the involved atoms. Rigorous experimental results that describe the full range of crack path instabilities, the importance of the atomistic symmetry, and the limit of crack velocity occurring during rapid crack propagation in brittle single crystals are still scarce. We use silicon single crystal specimens cut from commercial silicon wafers as a model material to elucidate the phenomena occurring during rapid crack propagation in brittle single crystals, since silicon wafers well represent nearly ideally brittle and defect-free crystals. The estimated length of the process zone in silicon is about two atomic distances; this is of profound importance when surface instabilities are discussed, since in silicon, the process zone has no influence on the generation of surface features (Lawn, 1993). The low-energy cleavage planes in silicon are {1 1 1} and {1 1 0}, with small difference in their cleavage energies. While the {1 1 1} cleavage plane is the plane with the lowest cleavage energy, their exact values are still under debate, and range between 3 and 5 J/m2 (Jaccodine, 1963; Michot, 1987; Spence et al., 1993; Ebrahimi and Kalwani, 1999; Pe´rez and Gumbsch, 2000; Bernstein and Hess, 2003). In previous investigations, we have described the crack deflection phenomenon, in which a crack, propagating under three-point bending (3PB) along the {1 1 0} low-energy cleavage plane of silicon, is deflecting to the {1 1 1} cleavage plane. The deflection was explained by minimum total energy considerations and was shown to depend upon crack velocity and crystallographic orientation (Sherman, 2005; Sherman and Be’ery, 2004). In the present paper, we report on phenomena associated with crack propagation occurring during dynamic propagation along the {1 1 1} low-energy cleavage plane of silicon. 2. Experimental Cleaving of thin and wide silicon specimens was performed under 3PB. The specimens, of 20 43 mm2 lateral dimensions, were cut from commercial [1 1 0] 400 diameter, 525 mm thick silicon wafers. The specimens were notched with a 150 mm thick diamond saw to a length between 0.15 and 2 mm. The notch length determines the energy at crack initiation and during propagation and the crack velocity at steady-state. In these specimens, the crack is propagating along the ð1¯ 1 1Þ low-energy cleavage plane, which is perpendicular to the specimen-free surfaces. The 3PB Bridge with a specimen and a schematic of half the specimen is shown in Fig. 1a and b, respectively. The 3PB has the advantage of maintaining a stable crack propagation in the x, or the ½1 1¯ 2, direction, as the maximum tensile stresses, or the maximum KI plane is located underneath the line of loading of the middle pin. This is preferable over four-point bending or double torsion loading configuration. The farfield tensile/compressive stress, szz, typical to 3PB, is maximum tensile at the bottom surface and maximum compressive at the top surface of the specimen. The bending shear stress, szy, is negligible, as the thickness/ length of the specimen is 1/80. The 3PB stresses are responsible for the unique crack front shape in these specimens; the crack initiates at the bottom left point of the specimen (Fig. 1b and c), as a result of the high
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tensile stresses generated at the bottom of the notch, and starts to propagate as a quarter circular crack, as shown in Fig. 1c. This quarter circular crack experiences high tensile szz stresses at the bottom and reduced szz tensile stresses at the top portion of the circular crack front, hence, accelerating near the bottom surface of the specimens (to the ½1 1¯ 0 direction or the x-axis) to achieve the shape of a nearly quarter ellipse (Fig. 1c). Note that the crack front, in its fully aligned situation (see below), experiences tensile mode only, due to the
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symmetric bending stresses, szz, normal to the crack faces. At a steady-state of crack propagation, the crack front is approximated by a quarter ellipse with a long, shallow ‘tail’ (Be’ery et al., 2003; Sherman and Be’ery, 2003) (Fig. 1b and c) and advancing at constant velocity, Vx, to the x direction. The crack front is actually the temporal locus of the ends of all the ridges— a ridge being the path along which the bonds between atoms break— its velocity and orientation are Vn and y, respectively. A typical single ridge and the resultant crack front shape are also shown in Fig. 1c. The relationship between the crack front velocity, Vx, and the ridge velocity, Vn, is simple: Vn ¼ Vx cos y, where y is readily obtained as the angle of the normal to the ellipse describing the crack front at a certain point along the y-axis (Be’ery et al., 2003). A wide range of velocities, Vn, is encountered by a propagating ridge in our specimens, from thousands of m/s (scaled with the Rayleigh surface wave speed) at the bottom surface to several tens of m/s at the top portion of the fracture surface. At the same time, the direction of the velocity vector, via the angle y, varies in accordance with the crystallographic orientation of the ð1 1¯ 1Þ cleavage plane from ½1 1¯ 2 to [1 1 0], see Fig. 1d. This enables examination of various aspects of crack tip velocity and crystallographic orientations using a single specimen. This is in contrast to a tensile specimen, where the crack propagates along a single orientation. The normalized tip velocity, VnXVx, and the orientation, y, vs. the normalized location along the y-axis (y/h) is shown in Fig. 1d. These features enable the determination of the fundamental phenomena of surface perturbations/instabilities associated with rapid crack propagation in brittle single crystals. The quarter elliptical shape was found to closely resemble the crack front in glass and in silicon where the crack propagates along the {1 1 0} (Be’ery et al., 2003; Sherman and Be’ery, 2003) cleavage plane. The latter experiments showed that similar behavior is evident for crack propagation on the {1 1 1} plane. The parallel velocity, Vx, was measured using the ‘Potential Drop’ technique (Stalder et al., 1983). The velocity was also measured in unnotched specimens. In this technique, a thin metallic layer (20 nm Ta) is deposited on the bottom surface of the specimen, and serves as a resistor. As the crack propagates, it cuts the metallic layer, which, hence, experiences increasing resistivity. As a result, the potential in the metallic layer drops during propagation. The potential drop was measured by means of a 200 MHz scope. These measurements provide the voltage/time relationship during crack propagation. A calibration function, for the voltage/crack length relationship, was derived by scribing similar coated specimen with a diamond scriber under an optical microscope, and the potential drop was measured every 0.25 mm. Both provide, after certain manipulations, crack length vs. time. Derivative of the latter function with time yields the average crack velocity as a function of the crack length; no detail of velocity fluctuations is accessible. We estimate the velocity accuracy at the stage of steady-state velocity by 50 m/s. It is noted that Eq. (1) is fulfilled along the crack front, namely, the dynamic energy release rate, G, is equal to the dynamic fracture energy, G, at each point along the front. The quasi-static strain energy release rate, G0, however, varies with location along the front; it is nearly G0, the quasi-static cleavage energy, along the straight ‘tail’, where the velocity is low (see Fig. 1d), and increases as the bottom surface of the specimen is approached, where the velocity is high, to fulfill Eq. (1). 3. Results 3.1. Crack velocity measurements We present the calculated average parallel velocity, Vx (Fig. 1c), of two specimens; one was notched to a length of 0.15 mm, while the other was unnotched but contained microcracks along the specimen edges generated during the dicing process. Since the velocity vector along the bottom surface of the specimen points toward the x-axis (y ¼ 0 or the ½1 1¯ 2 direction; see Fig. 1c), it can be considered to be under pure tension. The voltage/time relationship during crack propagation, obtained for an unnotched specimen, is shown in Fig. 2a. The calibration function, for the voltage/crack length relationship, is shown in Fig. 2b. Note that the crack propagation duration in Fig. 2a is nearly 5 ms. The crack velocity vs. the crack length for the notched specimen is shown in Fig. 2c. The crack rapidly accelerates for the first 3 mm of propagation; the acceleration is then considerably reduced and the crack thereafter continues to propagate in a nearly constant, steady-state velocity. The maximum crack velocity measured for the notched specimen is 3620 m/s, which is 0.8CR (CR for the {1 1 1} /1 1 2S system was taken as
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4520 m/s). The fracture surface of this specimen at steady-state velocity is the one shown in Fig. 3b. This velocity is in excellent agreement with other crack velocity measurements in silicon (Cramer et al., 2000; Hauch et al., 1999). Crack velocities having a high fraction of CR were also measured for other single crystals (Field, 1971; Hull and Beardmore, 1966). The crack velocity measured for the unnotched specimen is higher and reaches 4020 m/s (Fig. 2d). This is the highest crack velocity (0.9CR) measured to date for the {1 1 1} /1 1 2S system of silicon. It is important to note that the crack in this unnotched specimen was initiated at one of the specimen edges, but 2 mm away from the loading pin. The fracture surface contained large kink instabilities (see below) at the beginning of the propagation, followed by moderate kinks, and yet, the crack achieved high velocity. Corrugated instabilities (see below) were also evident along the fracture surface of this specimen. 3.2. The terrace-like kink instabilities The basic crack path instabilities along the ð1¯ 1 1Þ cleavage plane are shown in an optical image in Fig. 3a. Crack profile and an individual ridge pattern are also shown in that figure. The instabilities are terrace-like kinks (Lawn, 1993), propagating along nearly 2/3 of the fracture surface, where the velocity Vn (Fig. 1c and d) is relatively high, and corrugated instabilities at the upper portion of the fracture surface, where that velocity is relatively low. The Atomic Force Microscope (AFM) image (Fig. 4a) taken from the dashed circled zone in Fig. 3a, shows the well-defined kink instabilities. A typical ridge pattern is also shown. A ridge is propagating on the ð1¯ 1 1Þ cleavage plane, but every few tens of micrometers it kinks sharply onto, presumably,
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Fig. 3. Optical images of the ð1¯ 1 1Þ fracture surface under 3PB. (a) The terrace-like kink instabilities at the bottom and the corrugated instabilities at the top portion of the fracture surface and (b) the fracture surface of a well-aligned specimen showing the corrugated instabilities where most of the fracture surface is free of the kink instabilities. The crack front, the single ridge pattern, the associated velocities, Vx and Vn, and the angle y are also schematically shown. The initiation points of the corrugated instabilities are well defined and their line of propagation points towards the [2 1 1] direction.
a combination of (1 1 0) and ð1¯ 1 0Þ cleavage planes and continuing its propagation further on the ð1¯ 1 1Þ cleavage plane. The height of the kinks (Fig. 4a) is of an order of a few hundreds of nanometers. A profilometer line scan, Fig. 4b, provides a larger scale description of these instabilities. It is noted that kink instabilities were shown elsewhere with only general treatment in silicon (Hauch et al., 1999; Sherman and Be’ery, 2004), in sapphire (Sherman and Be’ery, 1998), and in GaAs (Sauthoff et al., 1999). 3.3. The corrugated instabilities We eliminated the kink instabilities by aligning the ð1¯ 1 1Þ cleavage plane with the maximum KI plane, namely, by reducing the misalignment angle x (Fig. 1b) to nearly zero; this reduction was achieved by rotating the specimens around the y-axis with respect to the bending bridge; this action required several specimens. Eliminating the kink instabilities was essential in order to investigate the corrugated instabilities. Indeed, micron scale height corrugated instabilities are only evident at the upper portion of the cleavage plane, as shown in Fig. 3b, in well-aligned specimen notched to 0.15 mm notch. The loci at which the corrugated instabilities initiated are well defined and well organized in terms of the normalized height, y/h, the density of the corrugations, and the principal lines of propagation. The latter are in the [2 1 1] direction, inclined at 601 to the x-axis. An AFM image of the zone where these instabilities initiate (dashed circled zone in Fig. 3b), is shown in Fig. 4c. Note that these instabilities are shallow, while their maximum height can reach nearly a micron. A profilometer line-scan of the corrugated instability (dashed line in Fig. 3b) is shown in Fig. 4d
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Fig. 4. A closer look at the major dynamic crack path instabilities generated along the ð1¯ 1 1Þ cleavage plane. (a) AFM image of the kink instabilities (open circle in Fig. 3a). The crack propagates along the terrace, and kinks every several tens of microns into the (1 1 0) plane. The maximum KI plane is drawn. (b) A profilometer line scan along the ½1 1¯ 2 direction of the kinks. The maximum KI plane is also shown in this image. The misalignment angle is less than 11. (c) AFM image of the initiation zone of the corrugated instabilities (open circle in Fig. 3b). Note the ‘watershed’ lines in the [2 1 1] direction. The corrugated instabilities are shallow; their inclination angle is of about 21 to the ð1¯ 1 1Þ cleavage plane (see line AB). (d) Profilometer line scan along the ½1 1¯ 2 direction of the corrugated instabilities. (e) AFM image from the bottom surface (open square in Fig. 3b) of the fully aligned specimen showing ridges propagating with high normal velocity, Vn, in the ½1 1¯ 2 direction. While the crack path is stable in the ½1 1¯ 2 direction, it is wavy with 1 nm amplitude, presumably due to the formation of the ridges.
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Fig. 5. (a) Three optical images of ð1¯ 1 1Þ fracture surfaces of fully aligned specimens with increasing notch length (0.15, 0.5, and 0.7 mm) and, hence, decreased parallel velocity, Vx, and dynamic energy. The normalized height, at which the corrugated instabilities initiate, y/h, decreases as the parallel velocity decreases. The initiation velocity of the corrugated instabilities, Vnc, is about constant, and equals 1050 m/s. The ‘watershed’ line remains in the [2 1 1] direction, inclined at 601 to the x-axis (the ½1 1¯ 2 direction). (b) The ridge velocity, Vn, as a function of the normalized height, y/h, for the three specimens. Note that the corrugated instabilities propagate at low velocities (dashed lines). The error in velocity measurements is estimated to be 750 m/s. Note also that the ridge angle, y, at initiation of the corrugations is not constant and depends on the parallel crack velocity.
and demonstrates the relatively large amplitude of these instabilities. Similar instabilities were identified in Sapphire (Sherman and Be’ery, 1998). To elucidate the role of velocity in the formation of these instabilities, we performed a set of experiments involving specimens with three different notch lengths and, thus, with three different crack velocities. It was found that as the parallel velocity, Vx, increases, the normalized height, y/h, at which the corrugated instabilities initiate, increases. The notch length, a, the parallel velocity, Vx, the normalized height, y/h, and the calculated normal velocity at which the corrugated instabilities initiates, Vnc, are shown in Fig. 5a for three different specimens. Assuming the calculations of Vn are valid, the averaged critical normal crack velocity, Vnc, is about 1050 m/s, i.e., 0.23CR. Note that Vnc constitutes the upper limit for the velocity of the corrugated instabilities, as the normal velocity along the corrugated instabilities is diminishing, as shown in Fig. 5b. Note also that the ridge angle, y, at initiation of the corrugations is not constant and depends on the parallel crack velocity. The corrugations spread for the whole length of the fracture surface of the specimen, including the surrounding of the specimen’s notch.
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3.4. AFM scan of the stable fracture surface We performed AFM scans of the fully aligned specimen with a 0.15 mm notch at the lower portion of the specimen (see dashed square in Fig. 3b) where the crack tip velocity, Vn, is nearly the maximum velocity evaluated for this specimen (Fig. 2c), 3600 m/s (0.8CR). This was done in order to trace any possible dynamic instability associated with fast cracking. The result is shown in Fig. 4e. The surface constitutes 2 nm height ridges that propagate towards the ½1 1¯ 2 direction. No evidence for mirror, mist, or hackle was detected by the 35 nm radius AFM tip. Shallow ‘Wallner lines’ are also visible in specimens with relatively low parallel velocity, Vx, and become invisible when this velocity is high (see Fig. 5a). 4. Discussion In this section, we attempt to explain the generation of the above-described dynamic instabilities from points of view of the state of stresses, the crystal symmetry, and the velocity and orientation-dependent cleavage energy. Unfortunately, a full description of the latter is yet not available. 4.1. The terrace-like kink instabilities The kink instabilities were generated due to misalignment between the ð1¯ 1 1Þ cleavage plane and the maximum KI plane, see Fig. 1b. The latter is located under the loading pin, and lies on the x–y plane of the specimen; the former makes an angle x with that plane (rotation about the y-axis). Rough estimates indicate that the misalignment of the instabilities shown in Fig. 4b is less than 11. Such misalignment leads to an in-plane mode mixity: Mode I is generated by the bending stress szz normal to the ð1¯ 1 1Þ plane, and Mode II—by the in-plane shear stress, sxz (Figs. 1b and 4a), resulting from the loss of symmetry of deformation. A ridge propagates along the ð1¯ 1 1Þ plane with increasing KII, until a critical point (Point A, Fig. 4a) where it reaches a critical KII; thereafter, the crack is abruptly attracted to the maximum KI plane. This critical KII resembles the jump of the crack from the ð1¯ 1 1Þ to a combination of (1 1 0) and ð1¯ 1 0Þ cleavage planes (Fig. 4a). It is also noted that the spatial point of kinking is affected by the local ridge velocity and the crystallographic orientation; all determine the kinking conditions. These conditions vary significantly along the kinking locus. This explains the distinct shape of the terraces than that of the ridges. The kink formation can also be rationalized by energy consideration: less energy is required for a ridge to propagate along the kink than to continue to propagate along the cleavage plane (He and Hutchinson, 1989). The density of the kinks and the height of an individual kink depend upon the angle of misalignment, x. 4.2. The corrugated instabilities ¯ The corrugated instabilities constitute a complex three-dimensional architecture consisting of ð1¯ 1¯ 0Þ, ð0 1 1Þ ¯ and ð1 1 1Þ cleavage planes, and generated from energy considerations. It is readily concluded that the velocity at which the corrugated instabilities are generated (and propagated) is relatively low; this is since the normal velocity, Vn, is low at the upper portion of the fracture surface. This fact is in contrast to the unstable nature of fast cracks in amorphous materials; the instabilities in these materials increase as the crack velocity increases. Another important observation is the direction of propagation of the corrugated instabilities; they constitute disturbances where their watershed lines are always directed to the [2 1 1] direction, see Fig. 4c, which is also the line of intersection of the ð0 1¯ 1Þ cleavage plane with the ð1¯ 1 1Þ cleavage plane. Furthermore, the corrugations are always directed out of the page, for the planes and directions noted in Fig. 4c. The critical velocity of the formation of the corrugated instabilities, Vnc, demonstrates that slow cracks in single crystals are unstable; Vnc is the upper limit for the velocity at which these instabilities can be generated; above this velocity the ð1¯ 1 1Þ fracture surface is stable. This latter argument is further supported by the following observation: the {1 1 0} planes intersect the ð1¯ 1 1Þ ¯ directions (Fig. 3b). The ½1 2 1 ¯ plane along three different directions: the ½1 1¯ 2, the [2 1 1], and the ½1 2 1
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direction is not in the ridge propagation direction and, therefore, is not considered. It is not clear why the corrugated instabilities do not develop in the ½1 1¯ 2 direction at the lower portion of the fracture surface (where y is low), although it has the same atomistic arrangement as the [2 1 1] direction. We postulate that the reason for the smooth surface is the high velocity of the bond-breaking mechanisms, since Vn4Vnc. This once again indicates that fast cracks in brittle crystals are stable. Silicon is a diamond-like cubic crystal with three independent elastic moduli (E, n, and G), and three independent elastic constants (C11, C12, C44, or S11, S12, and S44). When the geometric axes of the crystal containing a crack coincide with its principal crystallographic axes ([1 0 0], [0 1 0], and [0 0 1]), the fourth-order compliance or stiffness tensors have the well-known cubic structure: 2
S11
6 6 S12 6 6 S12 6 ½S ¼ 6 6 0 6 6 0 4 0
S 12
S12
0
0
S 11
S12
0
0
S 12 0
S11 0
0 S 44
0 0
0 0
0 0
0 0
S 44 0
0
3
7 0 7 7 0 7 7 7. 0 7 7 0 7 5 S44
(2)
The main feature of the cubic compliance (or stiffness) tensor is the lack of coupling between the tensile/ compressive stresses and the shear stresses. Tensile stresses generate no shear strain, and vice versa. The compliance tensor for a crack running along the {1 1 0} cleavage plane, irrespective of its direction, maintains the cubic symmetry, similar to Eq. (2). In this case, the external tensile stresses normal to the crack plane do not generate out-of-plane shear stresses, and therefore, no driving force exists for generating out of plane surface instabilities. Indeed, it was shown recently (Sherman and Be’ery, 2004), that the fracture surface of a crack propagating along the (1 1 0) cleavage plane is smooth in all directions. The symmetric atomistic arrangement for a crack running on the (1 1 0) cleavage, shown in Fig. 6a, dominates the crack stability. However, when the geometrical axes of the silicon crystal are rotated to coincide with the orthogonal [1 1 0], ½1 1¯ 2, and ½1¯ 1 1 axes (i.e., the crack is propagating on the ð1¯ 1 1Þ ½1 1¯ 2 cleavage system, and the external load is in the ð1¯ 1 1Þ, or z direction), the cubic symmetry is transformed into a pseudo-rhombohedral
Fig. 6. The atomistic symmetry experienced by cracks running along: (a) the (1 1 0) cleavage plane in the ½1 1¯ 0 direction and (b) the ð1¯ 1 1Þ cleavage plane in the ½1 1¯ 2 direction. In both cases, the crack plane is perpendicular to the plane of the figure, and the applied stresses are normal to the crack plane. The atomistic symmetry in (a) does not allow for generating out of plane shear stresses and, therefore, the crack path is stable, while that in (b) is anti-symmetric with respect to the crack plane, enabling the generation of out of plane shear stresses, and promoting the generation of the corrugated instabilities.
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symmetry. The non-zero components in the compliance tensor are marked by stars in Eq. (3): 2
0
6 6 6 6 S¼6 60 6 6 4 0
07 7 7 07 7. 0 7 7 7 0 05 0
0
0
0
0 0
0 0
0
0
3
(3)
In Cartesian orthogonal axes, the far-field tensile stress generates in-plane shear stresses, sxz, and also outof-plane shear stresses, szy, in the crystal. This behavior is further supported by the asymmetric atomistic arrangement for this system, as shown in Fig. 6b. An individual instability is generated at a preferred crystallographic orientation, and due to energy balance; the cleavage energy of the preferred cleavage planes is velocity and orientation dependent, such that G in Eq. (1) has more complicated form (Sherman and Be’ery, 2004): G ¼ GðV ; yÞ,
(4)
which results in the anisotropic, velocity-dependent phonon emission typical of a crystal containing a propagating crack (Cramer et al., 2000; Sherman and Be’ery, 2004; Sherman, 2005; Hauch et al., 1999; Zhou et al., 1996; Zhao et al., 1996). This velocity-dependent anisotropic phonon emission is an additional, basic mechanism for energy dissipation. We postulate, therefore, that at Vnc1050 m/s, the crack propagates along a combination of ð1¯ 1 1Þ and ð0 1¯ 1Þ planes, rather than on the ð1¯ 1 1Þ plane only, for energy efficiency reasons. This unstable path is more favored energetically than the stable path below the critical velocity Vnc. The validation of the postulated phonon-based energy consumption mechanisms requires either considerable experimental work, or atomistic simulations, which are not available yet. 5. Summary The dominance of the crystallographic symmetry in the formation of the crack path instabilities was demonstrated in this investigation. Corrugated instabilities were generated along well-defined crystallographic orientations, where the atomistic arrangement is such that perturbations of the crack path along a combination of the ð1¯ 1 1Þ, the (1 1 0), and ð1 1¯ 0Þ cleavage planes are permitted. Another important finding is that these instabilities were generated at relatively low velocity, below 0.23CR. At higher velocities, the crack path is stable. The corrugated instabilities, although on the micrometer scale, do not disrupt the crack from attaining high velocity, in contrast to surface instabilities in brittle amorphous solids (PMMA; soda lime glass, brittle plastics) that cause a significant reduction in crack velocity. It is important to note that the corrugated instabilities, aside from having large amplitude, are very shallow. They are generated from minimization of energy consumption: the unstable path consumes less energy than the stable path. We also identified the kink instabilities, generated due to misalignment of the cleavage plane and the maximum KI plane. No mirror, mist, and hackle instabilities were observed at the resolution of a 35 nm radius AFM pin. No branching was observed for cracks propagating at a velocity of up to 0.9CR. This is presumably due to the stresses typical of 3PB. Cracks running on the ð1¯ 1 1Þ cleavage plane in the ½1 1¯ 2 direction can attain a high fraction of the Rayleigh surface wave speed. The high measured velocity of cracks, 0.8CR for the notched specimen with a stable crack path along the bottom of the fracture surface, and 0.9CR for the unnotched one with kink instabilities, support our assumption that crack velocities in brittle solids can attain Rayleigh surface waves speed. The crack path instabilities in silicon indicate the potential crack path instabilities in other brittle crystals. However, the existence of similar instabilities in other brittle crystals depends on the particular energies of the preferred cleavage planes and the atomistic arrangement. On the other hand, kink instabilities are expected whenever misalignment exists.
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