Numerical Investigation of the Cubic-to-Tetragonal Phase Transition in

Feb 4, 2010 - Cross-sectional geometry dependence of spontaneous phase transformation of copper nanowires. Xiao-Yu Sun , Wen-Ping Wu , Xue-Lin Dong ...
0 downloads 0 Views 814KB Size
Download PDF
3364

J. Phys. Chem. C 2010, 114, 3364–3370

Numerical Investigation of the Cubic-to-Tetragonal Phase Transition in Ag Nanorods Francesco Delogu* Dipartimento di Ingegneria Chimica e Materiali, UniVersita` degli Studi di Cagliari, Piazza d’Armi, I-09123 Cagliari, Italy ReceiVed: September 22, 2009; ReVised Manuscript ReceiVed: December 10, 2009

Molecular dynamics simulations have been employed to study the phase behavior of Ag nanorods that were 10 nm long with square cross-sections and side lengths in the range between 1 and 5 nm. The nanorods were embedded in a model liquid organic phase, employed as a pressure transmitter. The systems were then studied under different hydrostatic pressure conditions. It is shown that the smallest nanorods undergo a spontaneous transition from the face-centered cubic phase to a body-centered tetragonal one as a consequence of the compressive stress induced by the intrinsic surface stress. It is also shown that the phase behavior is affected by hydrostatic pressure, the increase of which determines an increase of the minimum cross-sectional area required for nanorods to undergo the transition. I. Introduction Nanometer-sized systems have comparable fractions of surface and bulk atoms.1-3 Their properties are thus significantly affected by surface effects, which originate from the incomplete coordination shell of surface atoms.1-5 Surface effects become increasingly important as the fraction of surface atoms increases, which occurs as the system size decreases.1-5 Such relationship provides a satisfactory rationale for a number of thermodynamic and kinetic features including the stability of specific phases and the mobility of specific atoms in nanometer-sized structures.4,5 Among the others, surface effects can favor the occurrence of structural transformations unusual for bulks.6-8 A clear example is given by the cubic-to-tetragonal phase transition of Pd systems under high-pressure conditions.6 In this case, the compression of nanocubes in a diamond anvil cell destabilizes the face-centered cubic (fcc) phase and induces the rearrangement of atoms into a face-centered tetragonal one inaccessible to bulk Pd under the same pressure conditions.6 Similarly, simulations predict for isolated Au nanowires (NWs) a bodycentered tetragonal (bct) structure that is completely unstable in bulk Au.7,8 Calculations also indicate that the bct phase is stabilized in NWs by the surface stress, which allows a local minimum in potential energy.7,8 Conversely, the bulk bct phase exhibits a fatal elastic instability due to a vanishing shear stress.7,8 The evidence that nanometer-sized systems can exhibit unusual crystalline structures consequent to particular pressure or stress conditions emphasize the need of investigating their response to specific distortions in view of their possible technological applications. Along this line, the present work focuses on the behavior of Ag nanorods (NRs) submitted to hydrostatic compression while embedded in a liquid organic phase. Aimed at a qualitative investigation of the system behavior, this study represents an ideal prosecution of previous work.6-9 It will be shown that a specific size range exists for NRs to undergo a transition from the initial fcc to a bct phase as a consequence of the intrinsic surface stress. It will be also shown that the size range favorable to the transition is affected * To whom correspondence should be addressed. E-mail: delogu@ dicm.unica.it.

by the application of hydrostatic pressure. The details of calculations are given below. II. Computational Outline The forces operating between Ag atoms were reproduced by employing a semiempirical tight-binding (TB) potential based on the second-moment approximation to the density of electronic states.10 The cohesive energy of the system containing N Ag atoms, N

E)

{

N

∑ ∑ Ae i)1

j)1

-p

[

]}

( ) - ∑ξ e ( ) r -1 r0

N

j)1

2 -2q

r -1 r0

1/2

(1)

is expressed as a function of the distance r between atoms i and j, of the distance r0 of nearest neighbors (NNs) at 0 K and of a set of characteristic parameters A, ξ, p, and q. Their value was taken from previous work.10 The TB potential exhibits a remarkable capability of reproducing structural, thermodynamic, and mechanical properties of both bulk and nanometer-sized Ag systems.5,10,11 It is however worth noting that the present work has only qualitative purposes and that the results obtained should be considered as good approximations at best. Different Ag NRs with the main axis pointing along the 〈100〉 crystallographic direction and square cross-section were created starting from a bulk containing 256 000 atoms arranged in 40 × 40 × 40 cF4 fcc elementary cells. The bulk system was relaxed at 300 K in the NPT ensemble with number N of Ag atoms, pressure P, and temperature T constant.12,13 The Parrinello-Rahman scheme was used to allow possible phase transitions.14 Periodic boundary conditions (PBCs) were applied along the three Cartesian directions.15 The equations of motion were solved by employing a fifth-order predictor-corrector algorithm15 and a time step of 2 fs. The fluctuations of the system volume as well as of the potential and kinetic energy were used to follow the relaxation process, which attained completion after approximately 0.1 ns. A parallelepipedic region of the desired size was then selected at the center of the bulk phase. Afterward, it was isolated from the surrounding by linearly decreasing to zero the A and ξ

10.1021/jp9091199  2010 American Chemical Society Published on Web 02/04/2010

Cubic-to-Tetragonal Phase Transition in Ag Nanorods TABLE 1: Side Length s, the Cross-Sectional Area ANR and the Number NNR of Atoms of the Different NRs Investigateda s (nm)

ANR (nm2)

NNR

0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0 4.4 4.8 5.2

0.64 1.44 2.56 4.00 5.76 7.84 10.24 12.96 16.00 19.36 23.04 27.04

384 864 1537 2400 3455 4704 6143 7776 9602 11615 13824 16225

a Data refer to the initial fcc configuration. All the NRs are about 10 nm long. The first four NRs spontaneously undergo the fcc-bct phase transition.

potential parameters for the interactions between atoms inside and outside the selected region. The decrease was completed in 50 ps. Twelve different Ag NRs were created with side length s of 1 to 5 nm and total length l of about 10 nm. Essentially, the sides of two consecutive NRs differ from an elementary crystallographic cell. The NR main axis was chosen coincident with the 〈100〉 crystallographic direction, so that the free surfaces have (100), (010), and (001) crystallographic structure. A few of the NR characteristics are summarized in Table 1. Once created, the NRs were embedded in a liquid CCl4 phase, which was prepared starting from a simple cubic lattice of CCl4 molecules. For simplicity, CCl4 molecules were regarded as spheres interacting with each other via a 12-6 Lennard-Jones (LJ) potential.15 The values of the LJ radius σCCl4 and energy well depth εCCl4 were taken from literature.16 A system of 49 883 molecules arranged in a volume of about 20 × 20 × 20 nm3 enclosed by PBCs was simulated in the NPT ensemble. Starting from 200 K, the CCl4 temperature was rapidly raised to 300 K. Melting occurred at about 258 K, in accordance with previous work,16 and the liquid was relaxed for 50 ps. A parallelepipedic cavity of the desired size was then created at the center of the CCl4 bulk to contain the corresponding Ag NR. Following previous work,17 cross interactions between Ag and CCl4 were described by a 12-6 LJ potential by representing Ag as Xe, which has a similar atomic mass. In fact, energy exchanges between Ag atoms and CCl4 molecules are mostly governed by mass and not significantly affected by the finer details of the potential curve.16 Previous studies have shown that the approximation introduced by this method is acceptable.16 The LJ parameters σAg-CCl4 and εAg-CCl4 for the Ag-CCl4 interactions were estimated according to the Lorentz-Berthelot rules.18 The obtained system was then studied under NPT conditions for at least 0.1 ns. It must be noticed that the presence of a surrounding liquid phase does not affect the behavior of NRs, in the sense that Ag NRs of suitable size undergo a rapid transition from the fcc to the bct phase independent of the presence of absence of the liquid phase. In fact, the phase transformation starts as soon as the NRs are selected out from the parent matrix. Under such circumstances, the NRs and the liquid phase do not form an equilibrated system. Instead, in other cases, the NR and the liquid phase remain in contact with no ongoing phase transition for a time long enough to permit the system equilibration. Finally, in various cases, no transformation takes place. However, embedding the NRs of different size in the organic phase is a necessary step to submit them to hydrostatic pressure. The liquid indeed represents the pressure transmitter.

J. Phys. Chem. C, Vol. 114, No. 8, 2010 3365 Once the systems formed by Ag NRs and CCl4 liquid phase attained equilibration, a hydrostatic pressure was applied with values in the range between 0 and 50 GPa. The dynamics of each given NR was studied at twenty different pressure values. Each value was reached by using a compression rate of about 0.02 GPa ps-1 and the system successively relaxed for time intervals ranging from 100 to 200 ps depending on the pressure range. At each pressure value, the system was monitored for at least 0.1 ns. An analogous procedure was carried out to perform simulations at 250 and 350 K, temperatures at which the CCl4 phase keeps its liquid character. The NR structure was studied by comparing the mutual distances of groups of neighboring atoms. By simple crystallographic arguments,19 this allows one to point out the departure from the equilibrium fcc crystalline arrangement and to determine the attained crystalline structure, also evaluating the elementary cell parameters of the new phase. The stress at the surface of the NRs was estimated by evaluating the suitable first derivatives of the interatomic potential as indicated in previous work.20 The method is relatively simple and direct, but considerably sensitive to the shape of the interatomic potential curve. For this reason, calculations carried out with even slightly different potential parameters can lead to different results. However, the f estimates obtained are reliable, as shown by the comparison with f values obtained by two different methodologies based on the virial stress and the surface energy respectively.21,22 These methods are more cumbersome than the first one. Their detailed description as well as the detailed comparison of the results obtained with the different methods will be given in a future work. It suffices here to point out that the f values obtained with the different methods exhibit differences on the order of ( 5% only. For this reason, only the average f values obtained by the first cited method will be reported and discussed in the following. The estimation of the average surface stress f permits the indirect evaluation of the induced compressive stress σNR along the NR axis. This is approximately expressed by the ratio 4 f l/ ANR,7 where l and ANR represent the total length and the crosssectional area of the NR, respectively. The quantity σNR will be used to quantify the effects of the surface stress f on the general behavior of the NRs. Indeed, it permits a direct comparison between the behavior of NRs and a bulk phase, a necessary step to understand the importance of the surface stress on the nanometer scale. For sake of comparison between systems of different size, the large bulk phase from which the different NRs were selected out was also submitted to both compressive and tensile uniaxial stresses along the 〈100〉 crystallographic direction. This permitted some light to be shed on the role played by size on the stability of the fcc lattice under different deformation conditions. Regarding this latter point, it should be noted that the relative thermodynamic stability of two different phases can be in principle ascertained only through the evaluation of either the Helmholtz or the Gibbs free energy of the two phases.23,24 However, precisely the evaluation of free energies represents one of the most challenging problems in numerical simulations.25,26 The reason is that numerically obtained trajectories in the phase space generally sample only the most probable microstates, i.e., the atomic configurations with the most favorable free energy.25,26 A thorough sampling of the accessible states in the phase space is indeed prevented by the small size of simulated systems and the very short times accessible to numerical integration,25,26 which are intrinsic limitations of numerical simulations. There-

3366

J. Phys. Chem. C, Vol. 114, No. 8, 2010

fore, no reliable estimation of free energy differences is possible by the direct simulation of a pair of correlated systems. Despite the impossibility of carrying out a direct estimation, a few methods have been developed to perform the indirect estimation of free energy differences between two phases.25,26 These are based on a variety of approaches ranging from thermodynamic integration and particle insertion or deletion methods to acceptance ratio and umbrella sampling ones.25,26 All of the cited methods have specific advantages and disadvantages, with intrinsic limitations and difficulties to tackle in order to obtain trustworthy results.25,26 This makes the evaluation of free energy differences quite hard for bulk systems and even more complicated for small systems, where the unphysical sampling of local regions of the phase space is facilitated.25,26 Due to all of these aspects, the estimation of free energy differences for NRs based on the available methods was ruled out. As a consequence, the same decision was also taken for the bulk Ag phase. Therefore, the relative thermodynamic stability of any given pair of phases was dealt with only on an approximate basis by resorting to their potential energy difference. Of course, this method cannot supply anything but a rough estimate of the thermodynamic stability. Nevertheless, it is easy to apply and can be reliably employed to compare different cases. Actually, it is widely utilized in the field of numerical simulations to address thermodynamic stability issues and it has been already applied to the case of nanometer-sized systems including nanowires.5,7,8 All of the results discussed below were obtained from a minimum of two simulations. In fact, calculations were repeated at least two times in order to ascertain the reliability of results and their dependence on the initial configuration. III. Behavior of Bulk Ag under Different Stress Conditions The compression along the 〈100〉 direction determines a corresponding expansion of the system along the perpendicular 〈010〉 and 〈001〉 directions, whereas the opposite tensile deformation results in a contraction. In both cases, the system experiences a triaxial stress state defined by the imposed stress along the 〈100〉 direction and the equal response stresses along the 〈010〉 and 〈001〉 directions. From a crystallographic point of view, the uniaxial stress along the 〈100〉 direction coincides with the Bain path of deformation that allows the transformation of the crystalline lattice from the fcc to the bct arrangement. This is the reason for which only the compression along the 〈100〉 direction, and not others, is in principle able to induce the fcc-bct transition. As explained in the previous section, the possible occurrence of the above-mentioned structural transformation was investigated by monitoring the potential energy difference ∆U ) U - Ufcc between the average potential energies U and Ufcc of the strained and of the unstrained fcc lattice, respectively. It represents a rough measure of the relative stability of the strained and unstrained systems and in various cases permitted the identification of the structural characteristics of the new relatively stable phases formed as a consequence of the system deformation.5,7,8,10 In fact, the existence of relatively stable phases is readily pointed out by local minima in the curve describing the variation of ∆U as a function of strain.5,7,8,10 The obtained ∆U values are shown in Figure 1 as a function of the lattice spacing along the 〈100〉 direction, which will be hereafter indicated by the symbol c. The symbol a will be instead used to indicate the bct lattice spacing along the two

Delogu

Figure 1. The relative potential energy ∆U as a function of the lattice spacing c along the 〈100〉 direction. Data refer to the bulk Ag system.

crystallographic directions perpendicular to the 〈100〉 one. Of course, in the absence of deformation, both a and c are equal to the elementary fcc cell parameter a0, equal to 0.408 nm for a pure Ag fcc lattice.10 It can be seen that ∆U exhibits a minimum when c is roughly equal to 0.286 nm. The corresponding a value amounts to about 0.492 nm. This pair of a and c values identifies a bulk phase with bct structure characterized by elementary cell parameters abct and cbct equal to about 0.347 and 0.286 nm. It is here worth noting that the minimum uniaxial stress σ along the 〈100〉 crystallographic direction necessary to allow the formation of the bct phase, i.e., the minimum σ value that makes the ∆U decrease accessible, amounts to about 11 GPa. It follows that lower stress values are not able to induce the fcc-bct transition. It should be also noted that the formation of the bct phase is only due to the application of mechanical forces. In fact, as the compressive stress along the 〈100〉 direction is removed, the system fully relaxes undergoing a reverse transformation into the initial fcc phase. In contrast with the effects of the uniaxial stress, the hydrostatic compression of the bulk Ag lattice does not induce any transition. No local minimum is observed in the ∆U curve. A uniform contraction of the bulk is only obtained, with a consequent general increase of the average potential energy U. Submitting the system to triaxial stress states can instead promote the phase transformation. Although a variety of stress states can be generated by imposing independent uniaxial stresses along the three Cartesian directions, the present work will focus only on the ones that mimic the situation of a NR submitted to hydrostatic compression. For such NR, the hydrostatic pressure sums up to the uniaxial compressive stress originated by the intrinsic surface stress. A similar stress state can be obtained for the bulk by superimposing additional stress components along the three 〈100〉, 〈010〉, and 〈001〉 crystallographic directions to the uniaxially compressed system. However, the results obtained for the bulk Ag system indicate that the application of additional stress components to the ones operating along the 〈100〉 crystallographic direction results invariably in a further stabilization of the fcc phase. In other words, any additional stress or hydrostatic pressure applied to the bulk generally favors a deepening of the minimum in the ∆U curve associated with the fcc structure. IV. Behavior of Undoped Ag NRs at Zero Pressure The thermodynamic behavior of Ag is greatly affected by the system size. Whereas in the case of bulk Ag the transition of the fcc structure to the bct one can be obtained only through the application of a uniaxial stress along the 〈100〉 crystallographic direction, the NRs with sides shorter than about 2 nm undergo a spontaneous fcc-bct phase transition at zero pressure and 300 K. In the other cases, the NRs keep their initial fcc structure. The observed behavior can be rationalized by

Cubic-to-Tetragonal Phase Transition in Ag Nanorods

Figure 2. The relative potential energy ∆U as a function of the lattice spacing c along the 〈100〉 direction. Data refer to the NRs with sides s 0.8 (0), 1.6 (O), 2.4 (4), 3.2 (3), and 5.2 (]) nm long. The vertical arrows indicate the relative displacement of local minima as the NR size increases.

J. Phys. Chem. C, Vol. 114, No. 8, 2010 3367

Figure 4. The rate ν of the fcc-bct phase transition as a function of the number NNR of atoms in the NR. Data refer to the transforming NRs, i.e., to the four smallest NRs investigated. A best-fitted line is also shown.

Figure 3. The number NNR,bct of Ag atoms with bct coordination as a function of time t. The best-fitted line is also shown. Data refer to the case of the NR with sides about 1.2 nm long.

Figure 5. The planar order parameter S(k) of the different crystallographic planes as a function of the number m of planes. Data refer to the case of the NR with sides about 1.2 nm long at the three different times indicated.

looking at the potential energy differences ∆U for the different NRs. A few representative ∆U curves are shown in Figure 2 as a function of the elementary cell parameter c, which is a measure of the lattice spacing of the bct structure along the 〈100〉 crystallographic direction. In this case, the potential energy differences ∆U were evaluated by imposing to NRs the desired c value along the 〈100〉 crystallographic direction and leaving them free to relax along the 〈010〉 and 〈001〉 ones. It can be seen that the ∆U curve shape depends on the system size. The smallest NRs exhibit a single global minimum around a c value of about 0.286 nm, which is then coincident with the lattice spacing cbct of the bct structure formed in the case of the bulk Ag system. The ∆U value of the above-mentioned minimum increases as the NR size increases. At the same time, a second minimum develops around a c value of about 0.41 nm and becomes increasingly deeper as the NR size increases. This second c value corresponds to the equilibrium lattice spacing a0 of the fcc Ag lattice. Therefore, the ∆U curve points out the existence of a stable fcc structure for NRs with sides s longer than about 2 nm. As a whole, the modifications of the ∆U curves with the NR size indicate that the initial fcc crystalline arrangement can be preserved only by the systems with characteristic size larger than 2 nm. In fact, the ∆U curves exhibit in this case two local minima, the deeper of which is the one associated with the fcc structure. On the contrary, as the side length s becomes shorter than roughly 2 nm the minimum associated with the fcc arrangement disappears and only the one corresponding to the bct phase is observed. The above-mentioned data demonstrate that the capability of NRs of undergoing the fcc-bct transformation is governed by the system size. However, the system size not only determines for a given NR the possibility to undergo the fcc-bct transition or not, but also affects the transformation rate V. The rate V at which the structure of a given NR changes from the fcc lattice to the bct one was measured by evaluating the number NNR,bct of Ag atoms with a bct arrangement of nearest neighbors as a function of time t. It can be seen from Figure 3 that NNR,bct undergoes a roughly linear change with t, which allows extrapolating an average rate value V from the slope of

the plot. The V data obtained in the different cases are shown in Figure 4 as a function of the number NNR of atoms in the NR. It appears that V depends linearly on NNR and takes values in the range roughly between 200 and 600 nm ns-1. However, being that the NR length is constant for all of the NRs investigated, such linear dependence suggests that the phase transformation rate V is substantially related to the NR crosssectional area ANR, in agreement with previous observations regarding the Au NWs.7 In turn, this suggests that the mechanism of the fcc-bct transition is governed by the rearrangement of Ag atoms in the individual 〈100〉 crystallographic planes along the NR main axis. The aforementioned inference is supported by the evidence that the fcc-bct transformation involves successive adjacent crystallographic planes. The process can be suitably monitored by defining a sort of planar order parameter S(k), where k is a vector accounting for the reciprocal distances of atoms in each given plane. S(k) was set equal to 1 and 0, respectively, for the fcc and bct arrangements. The S(k) values for each individual crystallographic plane m at three different times are shown in Figure 5 for the NR with sides s about 1.2 nm long. Similar plots are obtained for the other transforming NRs. Such plots clearly point out that the phase transformation nucleates at the free (100) surfaces along the NR main axis and then propagates inward. A similar evidence can be worked out from the analysis of NR images obtained at different times, a few of which are shown in Figure 6 for the same NR. An estimate of the rate at which the phase transformation propagates along the NR main axis can be obtained from both S(k) data and NR images. The rate values obtained for the NR with sides s of about 1.2-nm in length amount respectively to about 425 and 436 nm ns-1, which are substantially coincidental with the ones of the rate V evaluated by calculating the number NNR,bct of Ag atoms with bct coordination at different instants. The latter quantity was then used in successive analyses. Simulating the Ag NRs at different temperatures permitted the gathering of information about the apparent activation energy Ea of the fcc-bct phase transformation. As usual, the Ea values were estimated by reporting the average propagation rate ν of

3368

J. Phys. Chem. C, Vol. 114, No. 8, 2010

Figure 6. Three atomic configurations of the NR with sides about 1.2 nm long at the different times indicated. The bct and fcc phase domains can be easily identified by the different width.

Figure 7. The logarithm of the average phase transformation rate ν, ln ν, as a function of the reciprocal of the temperature T, T-1. Data refer to the case of the NR with side length s equal to about 1.2 nm. The best-fitted line is also shown.

the phase transformation as a function of the reciprocal of the temperature T, T-1. The data concerning the case of the NR with side length s equal to about 1.2 nm are shown in Figure 7. Similar evidence are obtained for all the other transforming NRs. It can be seen that the V values arrange according to a linear trend. Correspondingly, the estimated activation energy Ea amounts to about 8.4 kJ mol-1. The Ea value is roughly the same for all of the different transforming NRs, which means that the mechanism of transformation is not affected by the cross-sectional area ANR. In addition, the Ea values obtained are significantly smaller than the ones typically associated with structural transformations, which are roughly on the order of 100 to 500 kJ mol-1. These evidence can be suitably rationalized by considering that Ea represents the apparent activation energy of a phase transformation mediated by collective atomic rearrangements, which is typically lower than the activation energy for transformations governed by the migration of individual atoms via lattice defects.27 Following previous work,7,22 the occurrence of a spontaneous fcc-bct phase transformation in NRs with cross-sectional area ANR smaller than about 4 nm2 has been related to the surface stress f. For the (100) free surfaces of the Ag NRs, the average surface stress f amounts roughly to -1.3 J m-2. This value is on the same order of magnitude of the values obtained for Au NWs.7,8 The negative sign indicates that the NRs contract along their main axis of an amount dependent on the intensity of the induced compressive stress σNR. This latter quantity changes with the NR size as shown in Figure 8, where the σNR values are reported as a function of the reciprocal of cross-sectional -1 area ANR of NRs, ANR . Since the largest NR that undergoes a spontaneous fcc-bct phase transition is the one with crosssectional area ANR equal to about 4 nm2, the minimum induced compressive stress σNR able to promote the fcc-bct transition should amount to about 13 GPa. Support for such a hypothesis is given by the fact that the above-mentioned σNR value is approximately equal to the minimum uniaxial stress σ of about 11 GPa necessary to stabilize the bct phase with respect to the fcc one in a bulk Ag system. According to this evidence, it seems that the NRs tend to exhibit a behavior similar to the bulk phase, at least in the

Delogu

Figure 8. The induced compressive stress σNR as a function of the -1 reciprocal of cross-sectional area ANR, ANR . Data refer to the NRs undergoing a spontaneous fcc-bct transformation.

Figure 9. The relative potential energy ∆U as a function of the lattice spacing c along the 〈100〉 direction. Data refer to the NRs with sides s about 2.4 nm long at 0 (0), 0.05 (O), 0.10 (4), 0.15 (3), and 0.20 (]) GPa. The vertical arrow indicates the relative displacement of the local minimum associated with the fcc phase induced by the pressure increase.

sense that the fcc-bct transformation occurs at approximately the same critical uniaxial stress value. Nevertheless, it should be noted that the surface stress f required to induce the fcc-bct transformation is intrinsic to the NR systems with sides s shorter than about 2 nm. It follows that these can undergo a spontaneous transformation, whereas the bulk Ag lattice can undergo the fcc-bct transition only as a result of external forces. V. Behavior of Undoped Ag NRs at Different Pressures Pressure is able to significantly affect the phase transformation behavior of NRs. More specifically, the maximum crosssectional area ANR,max that allows a spontaneous fcc-bct transition increases as the pressure increases. The effect of pressure originates from its capability of modifying the relative thermodynamic stability of the fcc and bct phases. This can be readily seen from the shape of the potential energy difference ∆U curves for any given NR at different pressure P values. The ∆U values for the NR with cross-sectional area ANR equal to about 5.8 nm2 are shown in Figure 9 as a function of P. Each curve was constructed by equilibrating the NR at the selected pressure P value and then changing systematically the lattice spacing c along the 〈100〉 crystallographic direction. The NR was left free to relax along the 〈010〉 and 〈001〉 ones. The obtained ∆U curves indicate that any pressure increase results in a progressive destabilization of the fcc lattice with respect to the bct one. In fact, the depth of the minimum associated with the fcc phase becomes increasingly small and finally disappears. On the contrary, the minimum associated with the bct phase keeps its characteristic features substantially unaltered. As a consequence, at a pressure P value dependent on the NR size, the bct phase becomes the most stable one. In the case of the data in Figure 9, referring to the NR with crosssectional area ANR equal to about 5.8 nm2, this occurs when the pressure P attains a value around 0.15 GPa. Correspondingly, at such pressure, the NR undergoes a spontaneous fcc-bct phase transition. A similar behavior is observed in the other cases.

Cubic-to-Tetragonal Phase Transition in Ag Nanorods

Figure 10. The maximum cross-sectional area ANR,max below which the NRs undergo the fcc-bct transformation as a function of the pressure P at which the phase transformation actually took place. The bestfitted line is also shown. Data refer to the NRs with side length s roughly in the range between 2 and 4 nm.

J. Phys. Chem. C, Vol. 114, No. 8, 2010 3369

Figure 12. The logarithm ln V of the fcc-bct phase transformation rate V as a function of the pressure P. Data refer to the case of the NRs with sides about 1.2 (0) and 1.6 (O) nm long at 300 K. Best-fitted lines are also shown.

Figure 11. The average surface stress f as a function of the hydrostatic pressure P. The best-fitted line is also shown.

Figure 13. The number NNR,bct of Ag atoms with bct arrangement as a function of time t. Vertical dotted lines indicate the application and the removal of hydrostatic pressure. Data refer to the case of the NR with a cross-sectional area ANR equal to about 5.76 nm2.

For example, the NR with cross-sectional area ANR equal to about 10.2 nm2 undergoes the transformation at pressures higher than about 0.5 GPa. The effect of pressure is summarized in Figure 10, where the different values of maximum cross-sectional area ANR,max for the different NRs are shown as a function of the minimum pressure P value allowing a spontaneous fcc-bct phase transformation. The numerical findings indicate that, at any given pressure P, all of the NRs with cross-sectional areas ANR smaller than ANR,max undergo the fcc-bct transition. The relationship between ANR,max and P is approximately linear. However, the relatively small number of points in the plot does not permit to rule out different behaviors at higher pressure. The capability of pressure P of affecting the phase transformation behavior of NRs can be connected with factors substantially related to the elementary cell volume and to the surface stress. In the former case, it must be noted that the overall volume of the NRs decreases of about the 25% when the fccbct transition occurs. Therefore, the increase of pressure P should somewhat favor the formation of the bct crystalline phase. However, the effect of hydrostatic pressure must be mostly ascribed to its influence on the surface stress f intrinsic to NRs. This can be seen from Figure 11, where the average surface stress f is shown as a function of pressure P. It appears that the average surface stress f increases according to a roughly linear trend as P increases. The f increase is common to all the different NRs investigated at different pressures and the f values shown in the plot were calculated by averaging over all the f estimates obtained from the different NRs. Regarding the effects of pressure P on the surface stress f, it should be also noted that the different f values substantially correspond to the same value of the induced compressive stress σNR. In fact, the obtained σNR estimates roughly range between 12.5 and 14.1 GPa. Therefore, all of them are quite close to the value of uniaxial stress σ that allows the occurrence of the fccbct transition in bulk Ag. The hydrostatic pressure P also affects the rate V at which the fcc-bct transformation takes place. Indeed, for any given

transforming NR, V decreases as P increases. Data shown in Figure 12 indicate that the logarithm of the transformation rate V, ln V, exhibits an approximately linear dependence on pressure P. In addition, the plots referring to the different transforming NRs have substantially the same slope. It follows that the effect of pressure is roughly the same for all the transforming NRs. The linear dependence of ln V on pressure P can be exploited to obtain information on the activation volume νa governing the rearrangement of atoms from the fcc to the bct crystalline lattice. In fact, νa can be related to V and P by the expression νa ≈ -kB T (∂ln V/∂P)T27, where kB is the Boltzmann constant. Then, an estimate of the activation volume νa can be worked out from the slopes of the plots in Figure 12. The average νa value obtained by considering all the transforming NRs roughly amounts to 5.5 × 10-3 nm3 at-1. Thus, it is significantly smaller than the average volume Ω characteristic of Ag atoms in the fcc lattice, which amounts to about 1.7 × 10-2 nm3 at-1. Such difference supports the hypothesis that the phase transformation is mediated by a cooperative mechanism in which the atoms attain the bct arrangement starting from the fcc one via a sequence of collective short-range displacements. Indeed, such kind of cooperative mechanisms generally exhibit activation volumes νa on the order of 0.3-0.4 Ω, whereas transformations mediated by vacancies exhibit νa values of about 0.7-1.0 Ω.27 Furthermore, it can be inferred that the activation energy for the fcc-bct phase transformation also depends on the pressure according to the expression Ea,P ≈ Ea - γ P νa, where γ is a proportionality constant. However, this point has not been further investigated herein. It must be finally noted that the fcc-bct phase transitions taking place under given hydrostatic pressure conditions exhibit complete reversibility. In fact, the bct crystalline arrangement is replaced by the initial fcc one as soon as the hydrostatic pressure is removed. This behavior can be readily observed in Figure 13, where the number NNR,bct of Ag atoms with bct arrangement is shown as a function of time t. Data refer to the case of a NR with a cross-sectional area ANR equal to about 5.8 nm2. A suitable hydrostatic pressure field was repeatedly applied and removed in order to monitor the structural changes

3370

J. Phys. Chem. C, Vol. 114, No. 8, 2010

undergone by the NR lattice. It can be seen that the application or the removal of the hydrostatic pressure correspond respectively to the occurrence of the fcc-bct or the bct-fcc phase transformations. Also, it appears that the observed transformation rates are always comparable. In addition, no hysteresis is observed. In all of the investigated cases, the fcc lattice transforms completely into the bct one and vice versa. A similar behavior can be obtained for all the transforming NRs. The evidence discussed above definitely suggest that the hydrostatic pressure P represents in principle a useful tool to affect and eventually govern the structure of NRs with suitable cross-sectional area ANR. Indeed, the complete reversibility of the fcc-bct phase transition permits to carry out series of transformation cycles free of hysteretic features, thus envisaging for NRs a possible technological application as mechanical actuators on the nanometer scale. VI. Conclusions The results obtained from molecular dynamics simulations indicate that Ag NRs can exhibit phase behavior dependent on their size. This can be ascribed to the effects of the intrinsic surface stress originating from the small NR size. Indeed, the induced compressive stress operating on NRs can be high enough to produce a significant contraction of NRs along their main axis and a corresponding expansion along the perpendicular direction. Under such conditions, the NRs with crosssectional areas smaller than a given threshold undergo a spontaneous phase transition and take an unusual bct structure. The above-mentioned bct structure is analogous to the one taken by bulk Ag when submitted to a suitable uniaxial deformation. In addition, the uniaxial stress allowing the fccbct phase transition in bulk Ag is approximately equal to the minimum induced compressive stress, arising from the intrinsic surface stress, necessary to produce the same behavior in NRs. However, the bulk phase is able to keep the bct structure only as far as a mechanical stress is applied. The phase behavior of the Ag NRs can be also affected by pressure. In fact, the increase of hydrostatic pressure permits a corresponding increase of the minimum cross-sectional area allowing a spontaneous fcc-bct phase transition. Interestingly, such transition exhibits complete reversibility, since the removal of pressure determines the inverse bct-fcc transformation. Accordingly, the control of hydrostatic pressure could provide a method to govern the NR phase behavior, which could be

Delogu important for possible applications of NRs or similar systems as mechanical actuators on the nanometer scale. Acknowledgment. Financial support has been provided by the University of Cagliari. A. Ermini, ExtraInformatica s.r.l., is gratefully acknowledged for his technical support. References and Notes (1) Moriarty, P. Rep. Prog. Phys. 2001, 64, 297. (2) Jortner, J.; Rao, C. N. R. Pure Appl. Chem. 2002, 74, 1491. (3) Ozin, G. A.; Arsenault, A. C. Nanochemistry. A Chemical Approach to Nanomaterials; RSC Publishing: Cambridge, U. K., 2005. (4) Baletto, F.; Ferrando, R. ReV. Mod. Phys. 2005, 77, 371. (5) Ferrando, R.; Jellinek, J.; Johnston, R. L. Chem. ReV. 2008, 108, 845. (6) Guo, Q.; Zhao, Y.; Mao, W. L.; Wang, Z.; Xiong, Y.; Xia, Y. Nano Lett. 2008, 8, 972. (7) Diao, J.; Gall, K.; Dunn, M. L. Nat. Mater. 2003, 2, 656. (8) Gall, K.; Diao, J.; Dunn, M. L.; Haftel, M.; Bernstein, N.; Mehl, M. J. J. Eng. Mater. Tech. 2005, 127, 417. (9) Delogu, F. J. Nanosci. Nanotech. 2009, 9, 2944. (10) Cleri, F.; Rosato, V. Phys. ReV. B 1993, 48, 22. (11) Smith, D. R.; Fickett, F. R. J. Res. Natl. Inst. Stand. Technol. 1995, 100, 119. (12) Andersen, H. C. J. Chem. Phys. 1980, 72, 2384. (13) Nose`, S. J. Chem. Phys. 1984, 81, 511. (14) Parrinello, M.; Rahman, A. J. Appl. Phys. 1981, 51, 7182. (15) Allen, M. P.; Tildesley, D. Computer Simulations of Liquids; Clarendon Press: Oxford, 1987. (16) Radhakrishnan, R.; Gubbins, K. E.; Watanabe, A.; Kaneko, K. J. Chem. Phys. 1999, 111, 9058. (17) Westergren, J.; Gronbeck, H.; Rosen, A.; Nordholm, S. J. Chem. Phys. 1998, 109, 9848. (18) Munster, A. Statistical Thermodynamics; Springer: Berlin, 1974. (19) Klug, H. P.; Alexander, L. E., X-ray Diffraction Procedures; John Wiley & Sons: New York, 1974. (20) Ackland, G. J.; Finnis, M. W. Philos. Mag. A 1986, 54, 301. (21) Streitz, F. H.; Cammarata, R. C.; Sieradzki, K. Phys. ReV. B 1994, 49, 10699. (22) Diao, J.; Gall, K.; Dunn, M. L. J. Mech. Phys. Solids 2004, 52, 1935. (23) Fermi E. Thermodynamics; Dover Publications: New York, 1956. (24) Berry S. R., Rice S. A., Ross J. Matter in Equilibrium: Statistical Mechanics and Thermodynamics, 2nd ed.; Cambridge University Press: Cambridge, 2004. (25) Frenkel, D. “Free-Energy Computations and First-Order Phase Transitions” In Molecular Dynamics of Statistical Mechanical Systems; Ciccotti, G., Hoover, W. G., Eds.; Proceedings of the Enrico Fermi International School of Physics, Vol. 97, North-Holland, Amsterdam, 1986, pp. 151-188. (26) Vega, C,.; Sanz, E.; Abascal, J. L. F.; Noya, E. G. J. Phys.: Condens. Matter 2008, 20, 153101. (27) Mehrer, H. In Diffusion in Solid Metals and Alloys; Landolt Bo¨rnstein, New Series, Group III; Mehrer, H. , Ed.; Springer: Berlin, Germany, 1990; Vol. 26, p 1.

JP9091199