NANO LETTERS
Critical Size for Fracture during Solid−Solid Phase Transformations
2004 Vol. 4, No. 5 943-946
David Zaziski,† Stephen Prilliman,† Erik C. Scher,‡ Maria Casula,† Juanita Wickham,§ Simon M. Clark,| and A. Paul Alivisatos*,† Department of Chemistry, UniVersity of California, Berkeley, California 94720, Materials Sciences DiVision, Lawrence Berkeley National Laboratory, Berkeley, California 94720, Nanosys, Inc., 2625 HanoVer Street, Palo Alto, California 94304, and AdVanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720 Received March 24, 2004
ABSTRACT The study of nanoscale materials with well-controlled size and shape can be used to learn more about critical length scales for numerous physical and chemical phenomena in solids and extended systems.1,2 Small nanocrystals (below 5-nm diameter) have been shown to exhibit fully reversible single-domain structural phase transformations with large volume changes over multiple cycles. The same transformations in extended solids are accompanied by irreversible domain formation.3-5 Here we investigate the crossover between these regimes by studying a pressure-induced structural transformation in 4-nm-diameter nanorods varying in aspect ratio from 1 to 10. We find that above a critical length the nanorods fracture at the moment of the structural transformation. This work demonstrates the use of simple, well-defined nanoscale systems to examine fundamental structural phenomena found in extended solids.
The transformation of a solid from one crystal structure to another, accompanied by a change in volume, results in the break up of an extended solid into nano- or micrometersized domains by processes such as twinning or fracture.6 Such transformations occur frequently in the preparation of materials, most famously in the rapid cooling of steel.6 They are also ubiquitous in nature, occurring at high pressures and temperatures in the interior of rocky planets.7 In a simplified model, we can consider a growing spherical nucleus of a daughter phase embedded within the parent (Figure 1A). Because there is a volume discontinuity at the interface, fracture occurs during growth when the elastic energy exceeds a critical value, as in the case of brittle fracture.8 The goal of this work is to observe this critical size by studying structural transitions in nanocrystals of various sizes and shapes. The most promising materials for the study of these length scales are nanorods in which two dimensions are held fixed and are smaller than the single-domain limit while the third dimension is tuned over a wide range. Although spherical or low-aspect-ratio nanocrystals larger than the single-domain limit will fracture into aggregate domains that are difficult to observe separately, nanorods can break up into indepen* Corresponding author. E-mail:
[email protected]. † University of California and Materials Sciences Division, Lawrence Berkeley National Laboratory. ‡ Nanosys, Inc. § Present address: Intel Corporation, 5200 NE Elam Young Parkway, Hillsboro, Oregon 97124. | Advanced Light Source, Lawrence Berkeley National Laboratory. 10.1021/nl049537r CCC: $27.50 Published on Web 04/14/2004
© 2004 American Chemical Society
Figure 1. Pictorial illustrations of structural transformations in bulk and nanocrystalline systems. (A) Inclusion of a nucleus of a daughter phase P2 surrounded by an infinite medium of the parent phase P1. (B) Representation of multiple domains by fracture in a sphere and a rod-shaped crystal. (C) Schematic of a nanorod during the growth of the 6-coordinate phase (red stripes) in the 4-coordinate structure yielding conditions for fracture. The nucleus has a shape change and a volume reduction of 18%.5 Arrows indicate the force opposing the deformation by the surrounding pressure medium. (D) Illustration of a nanorod after multiple transformation cycles. The initial nanorod shown is larger than the critical length and fractures into multiple subcritical-sized nanocrystals that can then reversibly transform as a single domain.
dently observable nanocrystals (Figure 1B). The recently developed ability to tune the length of CdSe nanorods widely
Figure 3. Transmission electron micrograph of representative CdSe nanorods in the 4-coordinate phase (A) before and (B) after one pressure cycle. (C) Length distribution of the particles before (blue) and after (red) pressure cycling. The after distribution consists of no one single specific characteristic size, which is comparatively different than before pressurization.
Figure 2. Powder XRD diffraction patterns of CdSe crystals of different lengths in the 4-coordinate structure before and after pressure cycling. (A) 40 nm × 5 nm; (B) 16 nm × 4.6 nm; (C) 10 nm × 4.4 nm; (D) 4.5-nm-diameter sphere. Spectra before and after one or two pressure cycles are shaded blue, red, and green, respectively. The sphere pattern is displayed for the comparison of single-domain behavior and is from ref 5. The line widths are determined by Debye-Scherrer broadening. The (002) peak (Q ≈ 1.75 Å-1) corresponds to the long axis of the crystal. Two stacking sequences in the 4-coordinate phase are possible: zinc blende (ABCABC-) and wurtzite (ABAB-). The loss of intensity of the (103) peak (Q ≈ 3.25 Å-1) is from an increase in zinc blende stacking faults that are formed on recovery of the 4-coordinate phase and has no influence on the transformation.5
thus creates an excellent opportunity to observe the critical length scale for domain fracture accompanying a solid-solid structural transition.9 The pressure-induced 4- to 6-coordinate bonding change in CdSe has been investigated in nanorods with diameters of 4 to 5 nm and lengths ranging from 4 to 40 nm.9-11 The nanorods have been studied under hydrostatic compression by X-ray diffraction, electronic spectroscopy, and transmission electron microscopy over multiple cycles of the structural transformation. Pressure gradients never exceed 10% across the entire cell area of 0.1 mm and thus are 944
negligible over the length of a nanorod. X-ray diffraction patterns from the nanorods most clearly demonstrate the transition from single-domain behavior in short nanorods (4.4 nm × 10 nm) to multiple-domain behavior in longer ones (4.6 nm ×16 nm and 5 nm × 40 nm) (Figure 2B-D). The widths of the X-ray diffraction peaks are inversely proportional to the length scale over which the crystal is coherently ordered.12 In the smaller crystals, the diffraction line widths do not change over multiple cycles. In longer crystals, the diffraction line corresponding to the long axis clearly broadens significantly on the first upstroke transition but remains the same through all subsequent cycles (Figure 2D). Transmission electron microscopy on nanorods before and after the transition demonstrates that the broadening in the diffraction pattern is in fact due to the fracture of the nanorods into smaller nanocrystals (Figure 3A and B). Fracture occurs perpendicular to the long axis of the crystal. It is particularly interesting to examine the length distribution before and after fracture for 40-nm-long particles (Figure 3C). After the structural transformation, the resulting fragments are on average less than half the length of the initial nanorod sample. This shows that each nanorod fragments into at least three pieces per original nanorod on average (Figure 1D). It is important to note that the fracture occurs under conditions of hydrostatic compression of the nanorods and that the fracture coincides with the transition to the more dense phase. To understand better the microscopic mechanism leading to fracture, we have investigated the kinetics of the structural transformation. The rate at which an ensemble of nanorods transform following an abrupt increase in the pressure can Nano Lett., Vol. 4, No. 5, 2004
Figure 4. Images of 40 nm × 5 nm nanorods dissolved in ethylcyclohexane loaded in the diamond anvil cell taken at (A) 1.2, (B) 8.0, and (C) 0.5 GPa in the 4-, 6- and (recovered) 4-coordinate phases, respectively. The color change is from the loss of the electronic absorption peak in the transformation from a 4- (redorange) to a 6-coordinate (yellow) structure on going from a direct to indirect band gap semiconductor.13 Ruby is visible at the top of the images. A 25-µm-diameter Pt wire loop (shown in bottom half) facilitated the recovery of the nanorods after pressure cycling. (D) Representative transformation time dependence shown for 37 nm × 4 nm CdSe nanorods at 423 K in the forward direction at (0) 7.0 and (]) 6.7 GPa and (E) in the reverse direction at (O) 1.2 and (4) 1.4 GPa. The abscissa is the change in intensity of the absorption peak in the 4-coordinate phase and reflects the number of nanorods in the ensemble that have transformed. Fits are to a single-exponential decay. Kinetic effects arise from the presence of a large energy barrier that hinders the transformation between both structural phases.
be monitored by changes in the electronic absorption spectrum, which differs characteristically between the two structures (Figure 4A-C).13 Two features of the transition kinetics are noteworthy. First, the kinetics follow a singleexponential decay over several orders of magnitude (Figure 4D and E). This is similar to the decay kinetics observed in small nanocrystals where there is a single rate-determining step in the nucleation that governs the transformation.13,14 This strongly differs from the relaxation behavior in bulk material, which follows the multiexponential Avrami expression arising from multiple nucleation and growth occurring at disparate points throughout the crystal.7 Second, the kinetics show that the time scale for the ensemble of nanocrystals to transform between structures is on the order of hours to seconds under the conditions of our experiments. This is in sharp contrast to the estimated time for a single nanocrystal to transform from one structure to another once nucleation has taken place. At present, there are no direct observations of this time scale, but we can provide a very crude estimate by noting that it will be approximately what is required for a shear acoustic wave to move across the crystal, perhaps a few tens of picoseconds. Even if this estimate is off by 1 or 2 orders of magnitude, the fact is that nucleation events are extremely rare and each rod experiences only a single event. This allows us to discount models for fracture that may involve, for instance, double or triple Nano Lett., Vol. 4, No. 5, 2004
nucleation events giving rise to intragranular fracture where two nucleation fronts collide. Another feature in this system is the time scale over which the matrix that the rods are embedded in can react to local changes in volume. The viscosity at the transition pressure (∼7 GPa) is estimated to be at least 106 cP (which is assumed to be similar to the viscosity of methanol but it could be as high as 1015 cP, marking the solvent glass-transition point).15 The propagation of the nucleation front is significantly faster than the expected dynamic response of the matrix. It is most reasonable then to consider that strain builds up when the rod is held nearly rigid by the matrix as the nucleation front grows. We can therefore focus our attention on the stress and strain that develop when a single nucleation event occurs and the nucleation front starts to propagate across the crystal. Prior studies of the X-ray powder diffraction patterns before and after the transformation of the nanocrystals suggest that the most probable mechanism involves the sequential sliding of planes that are perpendicular to the rod axis.5 Most likely, these sliding-plane events occur “coherently”; namely, the sliding direction is always the same so that a straight cylinder is transformed into a tilted one as the nucleation front propagates (Figure 1C). The sideways displacement results in strain centered primarily at the interface boundary and builds as the nucleus grows. Typical interfacial energies can be up to 6 eV/nm2.16 The total energy required to cleave the nanocrystal can be estimated as approximately that which is required to break the bonds in one plane of the nanocrystal (∼120 Cd-Se bonds for a width of 5 nm), which is about 120 eV assuming a 1-eV bond energy.17 Thus, we can estimate qualitatively that at roughly 10 nm the strain at the interface between the nucleus and the parent phase will exceed the fracture threshold (Figure 2). For longer crystals, the growing nucleus is less likely to encounter the ends of the crystal, thus giving rise to two distinct interfaces. Cracking followed by fracture may then occur at both locations, leading to multiple nanocrystal fragments.18,19 Although we can qualitatively understand the emergence of a critical size for transformation-induced fracture under compression, the observations presented here also should provide an opportunity to learn more about the behavior of fracture in nanoscale materials. Recently, simulations of fracture in nanoscale materials showed a transition from the classical Griffith criteria to failure near the theoretical strength.20 It is therefore possible that fracture in nanorods occurs at such a limit. In the absence of a crack tip, defects present in a crystal usually control the fracture of a material.17 The nanorods, given their small size, are initially free of defects such as dislocations and vacancies because they are annealed out during the high-temperature synthesis process. Thus the emergence of a critical length scale for transformation-induced fracture presents an opportunity for us to learn more about the intrinsic mechanisms of structural transformations in solids through theory and simulation. Acknowledgment. We thank C. Erdonmez, D. Chrzan, and E. Ertekin for helpful discussions. This work was 945
supported by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering of the U.S. Department of Energy under contract no. DE-AC03-76SF00098 and the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant no. F49620-01-1-0033. Supporting Information Available: Nanomaterial synthesis and transformations, synchrotron X-ray diffraction data, kinetic measurement data, length distribution of spaghetti noodles, and spectroscopic proof of fracture. This material is available free of charge via the Internet at http:// pubs.acs.org. References (1) Bi, H. J.; Cai, W. P. Kan, C. X.; Zhang, L. D.; Martin, D.; Trager, F. J. Appl. Phys. 2002, 92, 7491-7497. (2) Jacobs, K.; Wickham, J.; Alivisatos, A. P. J. Phys. Chem. B 2002, 106, 3759-3762. (3) Tolbert, S. H.; Alivisatos, A. P. Science 1994, 265, 373-376. (4) Jacobs, K.; Zaziski, D.; Scher, E. C.; Herhold, A. B.; Alivisatos, A. P. Science 2001, 293, 1803-1806. (5) Wickham, J. N.; Herhold, A. B.; Alivisatos, A. P. Phys. ReV. Lett. 2000, 84, 923-926. (6) Christian, J. W. The Theory of Transformations in Metals and Alloys; Pergamon Press: Oxford, England, 1965. (7) Putnis, A. Introduction to Mineral Sciences; Cambridge University Press: Cambridge, England, 1992. (8) Griffith, A. A. Philos. Trans. R. Soc. London, Ser. A 1920, 221, 163.
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(9) Peng, X.; Manna, L.; Yang, W.; Wickham, J.; Scher, E.; Kadavanich, A.; Alivisatos, A. P. Nature 2000, 404, 59-61. (10) Li, L. S.; Walda, J.; Manna, L.; Alivisatos, A. P. Nano Lett. 2002, 2, 557-560. (11) Barnett, J. D.; Block, S.; Piermari, G. J. Optical Fluorescence System for QuantitatiVe Pressure Measurement in Diamond-AnVil Cell. ReV. Sci. Instrum. 1973, 44, 1-9. (12) Guinier, A. X-ray Diffraction: In Crystals, Imperfect Crystals, and Amorphous Bodies; Dover Publications: New York, 1994. (13) Chen, C. C.; Herhold, A. B.; Johnson, C. S.; Alivisatos, A. P.. Science 1997, 276, 398-401. (14) Jacobs, K.; Zaziski, D.; Scher, E. C.; Herhold, A. B.; Alivisatos, A. P. Science 2001, 293, 1803-1806. (15) Cook, R. L.; Herbst, C. A.; King, H. E. J. Phys. Chem. 1993, 97, 2355-2361. (16) Porter, D. A.; Easterling, K. E. Phase Transformations in Metals and Alloys; Chapman and Hall: London, 1992. (17) Lyndenbell, R. M. J. Phys.: Condens. Matter 1992, 4, 2127-2138. (18) A compelling comparison can be made with three-piece fracture observed following the axial compression of macroscopic rods of very high aspect ratio. Most famously, this was repeatedly investigated in spaghetti noodles by Feynman and co-workers.19 Under the applied compression, the rod structure undergoes a bow-shaped deformation leading to a first fracture that is quickly followed by whiplash recoil generating a second fracture, with the motion of this third piece conserving momentum.19 Whereas the distributions are both consistent with three-piece fracture in which two pieces are larger than the third (Figure S1, Figure 3C), the mechanism in the nanorods is still not known in detail. (19) Feynman, R. P. No Ordinary Genius: The Illustrated Richard Feynman; W. W. Norton and Co.: New York, 1994. Nickalls, O.; Nickalls, R. New Sci. 1995, 52. (20) Gao, H. J.; Ji, B. H. Eng. Fract. Mech. 2003, 70, 1777-1791.
NL049537R
Nano Lett., Vol. 4, No. 5, 2004
