Universal Stability and Temperature Dependent Phase Transformation

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Universal Stability and Temperature Dependent Phase Transformation in Group VIIIB
IB Transition Metal FCC Nanowires Vijay Kumar Sutrakar*,†,‡,# and D. Roy Mahapatra‡ †

Mechanical Engineering Design Division, Aeronautical Development Establishment, Defence Research and Development Organization, New Thipasandara Post, Bangalore, India 560075 ‡ Department of Aerospace Engineering, Indian Institute of Science, Bangalore, India 560012 ABSTRACT: Atomistic simulation of Ag, Al, Au, Cu, Ni, Pd, and Pt FCC metallic nanowires show a universal FCC f HCP phase transformation below a critical cross-sectional size, which is reported for the first time in this paper. The newly observed HCP structure is also confirmed from previous experimental results. Above the critical cross-sectional size, initial Æ100æ/{100} FCC metallic nanowires are found to be metastable. External thermal heating shows the transformation of metastable Æ100æ/{100} FCC nanowires into Æ110æ/{111} stable configuration. Size dependent metastability/ instability is also correlated with initial residual stresses of the nanowire by use of molecular static simulation using the conjugant gradient method at a temperature of 0 K. It is found that a smaller cross-sectional dimension of an initial FCC nanowire shows instability due to higher initial residual stresses, and the nanowire is transformed into the novel HCP structure. The initial residual stress shows reduction with an increase in the cross-sectional size of the nanowires. A size dependent critical temperature is also reported for metastable FCC nanowires using molecular dynamic, to capture the Æ110æ/{111} to Æ100æ/{100} shape memory and pseudoelasticity.

1. INTRODUCTION Recently, molecular dynamics (MD) simulations have been performed extensively to obtain material properties including phase transformation stability behavior at nanoscale. For example, a surface stress induced face-centered-cubic (FCC) to bodycentered-tetragonal (BCT) phase transformation in gold nanowires,1 FCC to pentagonal structure in Cu nanowires,2,3 B2 to BCT phase transformation in NiAl nanowires,4,5 bodycentered-cubic (BCC) to FCC/hexagonal close packed (HCP) in Fe nanowire,6 and B2 to BCT phase transformation in Cu
Zr nanowire7,8 have been observed. Such metastability/instability of various nanowires can produce shape memory in nanoscale devices via thermomechanically induced phase transformation. Liang and Zhou9 have shown a shape memory effect in FCC Cu nanowire. Size dependent critical temperatures have also been established for Cu nanowire, which shows reorientation from Æ100æ/{100} to Æ110æ/{111}, which is due to the fact that Æ110æ/{111} has a lower energy state. Below a critical temperature, a metastable state of Æ100æ/{100} Cu nanowire is also a possibility.9 Cross-section-dependent metastable phase of Æ100æ/{100} Cu nanowire can be recovered via external kinetic energy (i.e., thermal heating using the Nose
Hoover thermostat;21,22 further details are provided in section 2), which drives phase transformation from metastable Æ100æ/{100} to a stable Æ110æ/{111} state. The same behavior is also reported by Liang and Zhou10 for Ni and Au nanowires with cross-sectional dimensions below 5 nm. Haftel and Gall11 have used density functional theory (DFT) and the r 2011 American Chemical Society

tight-binding (TB) method to study the relaxation of narrow Cu, Ni, Au, Pt, and Ag nanowires originally oriented in the Æ001æ direction with an FCC structure. For a small enough diameter (d < 2 nm) each nanowire, under the compressive influence of its own surface stress, spontaneously relaxes to either a Æ110æ orientation (Cu, Ni, Ag) or a BCT Æ001æ orientation (Au, Pt), both of which are characterized by a compression of the wire axis of at least 30%. Recently, Lao and Moldovan12 have also investigated the surface stress induced structural transformations and pseudoelastic behaviors in palladium nanowires. For wires with a Æ100æ initial orientation, their simulations indicate that when the cross-sectional area is less than 2.18  2.18 nm2, the nanowire undergoes spontaneous reversible phase transformation from FCC to BCT structure. Interestingly, we observed that the newly transformed structure is HCP13 instead of the BCT structure that was reported by Lao and Moldovan.12 In recent years, HCP nanoparticles have been prepared and characterized using various experimental techniques, i.e., thermogravimetric analysis (TGA), differential thermal analysis (DTA), X-ray diffraction (XRD), transmission electron microscopy (TEM), selected area electron diffraction (SAED), atomic force microscopy (AFM), X-ray diffraction (XRD), etc.14 For example, hexagonal-close-packed Ni nanoparticles were synthesized Received: January 22, 2011 Revised: April 26, 2011 Published: May 12, 2011 10394

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using a heat-treating technique with the precursors prepared by the sol
gel method.14 The phase transformation from a hexagonal-close-packed Ni to a face-centered-cubic Ni structure occurred when the heat-treating temperature was increased. Depending upon the method of preparation of the nanocrystals, dodecanethiol-stabilized gold nanoparticles with similar average size organize into different superlattice structures, as reported by Stoeva et al.15 It is reported that gold particles synthesized by the inverse micelle technique preferentially assemble into FCC structures with long-range translational and orientational ordering. Gold nanoparticles obtained by the solvated metal atom dispersion (SMAD) method behave like “hard” spheres and predominantly organize into HCP nanocrystal superlattices with long-range translational ordering. Recently, Ahuja et al.16 have performed theoretical investigations on gold under high pressure through first-principles self-consistent total-energy calculations within the local-density approximation as well as the generalized gradient approximation using the full-potential muffin-tin-orbital method and found a phase transition from FCC to HCP of the structure at 241 GPa. The stability of this phase has also been explained through the electronic density of states. The above-mentioned theoretical and experimental studies indicate the existence of HCP structure. The FCC to HCP phase transformation and the cause of such phase transformation is not well-known in transition group nanoscale materials. In the present paper, the effects of initial residual stress, which cause phase transformation from FCC to HCP, mainly in group VIIIB
IB transition metal FCC nanowire, are studied in detail. We have considered Ag, Al, Au, Cu, Ni, Pd, and Pt transition metal nanowires (Al FCC metallic nanowire is also considered) in the present study, which is based on the availability of interatomic potentials used for MD simulations. The effects of residual stress on the stability of FCC nanowires are also studied. It is observed that a nanowire of smaller crosssectional size normally is transformed from Æ100æ FCC orientation to a new HCP structure during the energy minimization process. The MD simulations details of the phase transformation are discussed, and the experimental evidence of the HCP phase is also compared with the newly reported structure. The nanowires which show metastability during the energy minimization process, with the combined effect of temperature and size on the reorientation, have also been studied.

2. ATOMISTIC MODELING AND SIMULATION DETAILS Molecular dynamics (MD) simulations of nanowire are performed using the embedded atom method (EAM).17,18 The total energy Utotal for a system of atoms is written as Utotal ¼

N

N

N

∑ Fi ðFi Þ þ 2 i∑¼ 1 j ¼∑1, j6¼i φij ðrijÞ i¼1 1

ð1Þ

where the summations in eq 1 extend over the total number of atoms N in the system. Fi is the embedded energy as a function of the host electron density Fi induced at site i by all other atoms in the system; φij is a pair potential as a function of distance rij between atom i and j. The host electron density is given by Fi ¼

N

∑ FjðrijÞ j ¼ 1, j6¼ i

ð2Þ

Table 1. Lattice Constants and Critical Cross-Sectional Sizes of Various FCC Metallic Nanowires element lattice constant (Å)

Ni 3.52

Pd 3.89

Pt 3.92

Cu 3.615

Ag 4.09

Au 4.08

Al 3.986

critical cross-sectional

7.04

15.56 27.44 10.85

4.09

16.32 15.95

size (dcritical) (Å)

where Fj(rij) is the electron density function assigned to an atom j. MD simulations of FCC metallic nanowires are performed using the interatomic potential of Foiles et al.19 for Ag, Au, Cu, Ni, Pd, and Pt, and Jacobsen et al.20 for Al. The Æ100æ/{100} FCC metallic nanowires are created by generating atomic positions as in the bulk corresponding to the FCC crystal structure with known lattice constants (taken from refs 19 and 20); the details are given in Table 1. Nanowires of cross-sectional dimensions up to ∼50  50 Å2 are considered in the present study. It is important to mention here that nanowires of the same surface configurations are considered for the generalization of results. While performing MD simulations, the FCC metallic nanowires are first relaxed to the minimum energy configuration (at T = 0 K) while keeping the ends of the nanowire free to move in the Æ100æ direction. The maximum cross-sectional dimensions of a given nanowire which shows FCC f HCP phase transformation during the energy minimization is called dcritical. While heating the nanowires, following the energy minimization process, the wires are thermally equilibrated at a given incremental temperature using the Nose
Hoover thermostat21,22 for 50 ps with a time step of 0.001 ps. A temperature increment of 5 K is used to determine the Tcritical value in the present case. The temperature at which the nanowires fully transform from Æ100æ/{100} to Æ110æ/{111} configuration has been considered the critical temperature (Tcritical). The equations of motion are integrated using the velocity Verlet algorithm. All simulations are performed using an MD code called LAMMPS,23,24 and for the analysis of local crystal structure we use the common neighbor analysis (CNA) as implemented in LAMMPS.25 No periodic boundary conditions are used at any stage of the simulation, which is to capture accurately the relevant surface effects. The stresses are calculated using the virial theorem.26 The average virial stress over a volume Ω with total number of atoms N is calculated as 0 1 N N 1 @1 N ð3Þ Π¼ rij X f ij
mi u: i X u: i A Ω 2 i ¼ 1 j ¼ 1, j6¼ i i¼1

∑ ∑

∑

where mi is the mass of atom i. The displacement of atom i with respect to the reference position is designated as ui. u_ i = dui/dt represents the material time derivative of ui, “X” denotes the tensor product of two vectors, and rij = rj
ri. The force vector f can be calculated as f =
∂Utotal/∂rij, where Utotal is the total potential energy which includes both the pairwise and many body interaction terms. Further, in order to find out the state of the stability of nanowires, initial residual stress is also calculated. To calculate the initial residual stress, the Æ100æ/{100} FCC metallic nanowires are created by generating atomic positions as in the bulk corresponding to the FCC crystal structure with a known lattice constant, as mentioned above. Then the nanowire is started relaxing using molecular static simulation using the conjugant gradient method at a temperature of 0 K. The state of stress (along the length axis of nanowire, i.e., Πxx) which is 10395

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Figure 1. Initial residual stresses in Æ100æ/{100} FCC metallic nanowires with varying cross-sectional size.

calculated before the start of the relaxation process is called the initial residual stress of the nanowire.

3. RESULTS AND DISCUSSION The initial residual stress in the nanowires is computed using molecular static simulation using the conjugant gradient method at a temperature of 0 K, as shown in Figure 1. It is found that with an increase in the cross-sectional size of the nanowires the initial residual stresses decrease for all the FCC crystalline nanowires, as shown in Figure 1. Figure 1 is further divided into two different regions, i.e., (i) unstable region and (ii) metastable region. In the unstable region, an initial Æ100æ/{100} FCC crystalline nanowire transforms into a new HCP structure during the energy minimization process. The new HCP structure observed during the energy minimization is reported for the first time in this paper. It is observed that a large value of initial residual stress drives the FCC f HCP phase transformation during the energy minimization process. The second region, which is marked as metastable, shows no change in the initial Æ100æ/{100} FCC crystalline structure during the energy minimization process. This indicates that the initial residual stress, which reduces with an increase in the cross-sectional size of the nanowire, is not enough for the FCC f HCP phase transformation. It can be seen from Figure 1 that, for a given cross-sectional size, the residual stress varies for different materials. For example, at a cross-sectional size of ∼15 Å, the Pd nanowire gives ∼17 GPa of initial residual stress whereas Al nanowire gives ∼3 GPa. This result indicates that Al nanowires are the most stable and Pt nanowires are the least stable structure for a given cross-sectional size. In order to find out the effect of varying size and initial residual stress of different crystalline FCC nanowires, a detailed study is performed next. 3.1. Size Dependent Transformation Temperature. It is observed that a cross-sectional dimension of even 8.18  8.18 Å2 shows a metastable Æ100æ/{100} Ag nanowire, as can be seen from Figure 1. With thermal heating the metastable Ag nanowire, interestingly, is transformed into a new FCC configuration of Æ110æ orientation with {111} side surfaces, instead of getting transformed into the HCP structure. Such an FCC Æ100æ/{100} to Æ110æ/{111} transformation is observed in all the metallic nanowires above the critical cross-sectional dimensions, i.e., dcritical. Further, details of a size dependent Æ100æ/{100} to Æ110æ/{111}transformation can be seen in Figure 2. The nanowires which show metastability during the energy minimization, (i.e., do not show any phase transformation/reorientation during

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Figure 2. Transformation temperature of the FCC crystalline nanowires which transform from Æ100æ/{100} to Æ110æ/{111} orientations.

Figure 3. Atomistic stress distributions of Au nanowires for a given initial cross-sectional size of 12.24  12.24 Å2.

Figure 4. Process of FCC f HCP phase transformation during the energy minimization for Au nanowire of a given initial cross-sectional size of 12.24  12.24 Å2. Potential energy values are in eV.

the molecular static simulation at a temperature of 0 K) are further heated to find out the critical temperature (Tcritical). The main idea of heating the nanowires is to find out the transformation temperature at which those nanowires become stable. The various cross-sectional sizes of the crystalline FCC nanowires which show metastability can be found in Figure 1. Result presented in Figure 2 show that an increase in temperature is required with increasing cross-sectional size of the nanowire for a Æ100æ/{100} to Æ110æ/{111} transformation. This Æ110æ/{111} configuration represents a low energy state of FCC metallic nanowires, hence, more stability, and has been observed frequently in experiments and atomistic simulations for Au, Cu, and Ag nanowires.9
11 Results reported in the present 10396

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Figure 5. (a) Unrelaxed FCC and (b) relaxed HCP configurations of the Au nanowire. Cross sections at the center regions of the unrelaxed and relaxed wires, observed from z directions. Only two adjacent lattice planes of atoms are shown, and atoms in different lattice planes are shown in different colors. (c) High-magnification TEM micrograph of HCP ordered superlattice formed by Au particles prepared by the SMAD method (image is taken from ref 15 with permission). (d) Schematic representation of the HCP ordering viewed along [0001] (image is taken from ref 15 with permission).

Figure 6. (a) Initial FCC Al nanowire, which gets transformed into a newly observed HCP structure during the energy minimization process. The snapshots (b) and (c) show the progress of phase transformation. (d) Completion of FCC f HCP phase transformation. Following is the color coding: blue, FCC lattice; green, HCP lattice; and red, unknown lattice. The histogram shown indicates the CNA patterns of different lattice structures: 1 indicates FCC lattice, 2 indicates HCP lattice, and 5 indicates unknown lattice.

paper show that only surface stresses cannot drive such transforms from an initial Æ100æ/{100} configuration to the Æ110æ/ {111} configuration spontaneously. Thermal heating of nanowires drives such spontaneous transformation through a lattice reorientation process, exhibiting a contraction in the length direction and an expansion in the lateral directions; details can be found in refs 9 and 10. 3.2. FCC f HCP Phase Transformation in FCC Crystalline Nanowires. Next, we illustrate the FCC f HCP phase transformation in nanowires, which takes place below a critical crosssectional size, i.e., dcritical. Values of dcritical for all the FCC metallic nanowires considered in the present work are shown in Table 1. Such a spontaneous FCC f HCP phase transformation occurs due to very high tensile surface stresses. The atomistic stress distributions of Au nanowires for a given 12.24  12.24 Å2 are

shown in Figure 3, which shows very high tensile stress distribution at the surfaces as compared to the core of the nanowire. Figure 4 shows such an FCC f HCP phase transformation in a Æ100æ/{100} Au nanowire of an initial cross-sectional size of 12.24  12.24 Å2 during the energy minimization process at T = 0 K. Figure 4a shows an initial Æ100æ/{100} FCC Au nanowire. The process of FCC f HCP phase transformation during the energy minimization can be seen in Figure 4b
d. Such behavior is observed in almost all the FCC metallic nanowires. It can be seen from Figure 4 that the highly stressed surface atoms, which show high values of potential energy, get relaxed during the energy minimization process. Detailed relaxed and unrelaxed structural details for Au nanowire are further shown in Figure 5. Only two adjacent lattice planes of atoms are shown, and atoms in different lattice planes are shown in different colors. Results are 10397

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The Journal of Physical Chemistry C also compared with the high-magnification TEM micrograph of HCP ordered superlattice formed by Au particles prepared by the SMAD method,15 as shown in Figure 5c,d. Further, we employ common neighbor analysis (CNA)25 to analyze the phase transformation quantitatively, as shown in Figure 6. This technique identifies the local structure in atomic systems by using information about the mutual relation between neighbors of atom pairs. Figure 6a shows an initial FCC Al nanowire of cross-sectional dimensions 7.97  7.97 Å2. The isometric view as well as the view of the x-plane shows that all the nanowires have the FCC lattice as shown in blue. It is also important here to mention that the surface atoms show an unknown lattice, which is due to the fact that the surface atoms do not have the same neighbors of atom pairs (due to the free surfaces) compared to the core part of the nanowire. Further, we have plotted the histogram for accurate quantifications on the phase transformations, as shown in the subset of Figure 6. It can be seen from Figure 6a that the histogram shows the CNA patterns of an initial FCC Al nanowire, which shows only the FCC lattice (having CNA patterns 1) and unknown lattice at surface sites of the nanowires (having CNA patterns 5). For further details on the CNA patterns, see refs 24 and 25. Further during the energy minimization, the process of FCC f HCP phase transformation is shown in Figure 6b
d. The isometric view of Figure 6b,c clearly shows that the FCC f HCP phase transformation initiated from the free end of the nanowire and progressed toward the center of the nanowire. The corresponding histogram also shows that the number of atoms which includes the FCC lattice is reduced and at the same time the number of atoms which includes the HCP lattice (having CNA patterns 2) is increased due to the FCC f HCP phase transformation. The complete phase transformed HCP lattice is shown in Figure 6d. The histogram also shows that all the atoms with FCC lattice get transformed into HCP lattice during the energy minimization process.

4. CONCLUSIONS The effects of initial residual stress on phase transformation from FCC to HCP in group VIIIB
IB transition metal based FCC nanowires are explored in detail. The unstable nanowires with initial Æ100æ/{100} FCC crystalline nanowires show transformation into a newly formed HCP structure. The evidence of HCP structure is also confirmed with the experimental data. The quantitative FCC f HCP phase transformation is demonstrated using CNA patterns and using histograms. It is also shown that the metastable region which does not show any change in the initial Æ100æ/{100} FCC crystalline structure during the energy minimization process get transforms into Æ110æ/{111} FCC orientation during thermal heating. Such FCC f HCP phase transformation could be used in nanoscale shape memory applications of nanodevices.

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’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ 91-80-25087117. E-mail: [email protected]. Present Addresses #

Aeronautical Development Agency, Post Box No: 1718, Vimanapura Post, Bangalore, India 560017. 10398

dx.doi.org/10.1021/jp2006815 |J. Phys. Chem. C 2011, 115, 10394–10398