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J. Phys. Chem. C 2008, 112, 16400–16404
Size-Dependent Phase Stability of Silver Nanocrystals C. C. Yang* and S. Li School of Materials Science and Engineering, The UniVersity of New South Wales, NSW 2052, Australia ReceiVed: July 14, 2008; ReVised Manuscript ReceiVed: August 11, 2008
Size dependences of three structural phase transitions, melting, evaporation, and solid-solid transition for silver (Ag) nanocrystals, were investigated with nanothermodynamics. The size-dependent melting temperature Tm(r) and evaporation temperature Tev(r) of Ag nanocrystals were determined based on a root-mean-squaredisplacement (rmsd) model. It is found that both Tm(r) and Tev(r) decrease with crystal size decreasing for the isolated Ag nanoparticles while Tm(r) increases for the Ag nanoparticles embedded in the Ni matrix. A nanothermodynamic model was also established to reveal the origin of size effect on the polymorphism (facecentered-cubic and 4H structures) behavior of Ag nanocrystals with the contributions of surface energy and surface stress to the total Gibbs free energy. It shows that the lower surface energy and surface stress coupled with a higher volume Gibbs free energy predominates the thermal stability of 4H-structured Ag in nanometer scale. The results may provide new insight into the fundamental understanding of high-performance nanostructural metals for their applications in nanodevices. 1. Introduction Nanocrystals have attracted intensive interest in the past decade as their unique electronic, magnetic, optic, catalytic, and thermodynamic properties with potential applications in nanodevices.1 Silver (Ag) nanocrystal, as one of the noble metals, has been widely used in several areas including catalysis, biological antimicrobial, information storage, surface-enhanced Raman spectroscopy, etc.2-5 Ag is also one of the most promising materials for interconnects in ultralarge-scale integration circuits due to its lower resistivity and higher resistance to electromigration.6 Recently, the size-dependent melting temperature Tm(r) and evaporation temperature Tev(r) as well as the polymorphism behavior of Ag nanocrystals have become interesting subjects due to their scientific and industrial importance where r is (1) the radius of nanoparticles and nanowires, (2) the half-thickness or half-width of nanobelts, and (3) the half-thickness of thin films. It is believed that (1) understanding the Tm(r) of low-dimensional materials benefits not only the theoretical exploitation of phase-transition mechanism, but also the applications of nanodevices in modern industries,7 (2) determining the Tev(r) of nanocrystals is useful for the gas-phase synthesis of nanocrystals through achieving the harsh requirements in high purity and thermal stability,8 and (3) the crystal structure is a critical parameter for the determination of materials properties of the nanocrystals.9 As a result, understanding the size-dependent phase stability and revealing the mechanism behind the new properties of Ag nanocrystals are important topics in the research community of metallic nanocrystals. The size control of materials in nanometer scale is one of the most challenging issues in nanoscience because the size plays an important role in determining the phase-transition temperature of metallic nanocrystals.7 Indeed, it was demonstrated experimentally that both Tm(r)10-12 and Tev(r)8,13 decrease with decreasing r of the isolated Ag nanoparticles. However, the nanometer-sized Ag particles embedded in the Ni matrix can be superheated above the corresponding bulk melting temper* Corresponding author. Fax: +61-2-93855956. E-mail: ccyang@ unsw.edu.au.
ature Tm(∞) due to their coherent or semicoherent interfaces between Ag and Ni, where ∞ denotes the bulk.14 In the past few years, several phenomenological models15-18 and computer simulations19,20 were implemented to calculate the values of Tm(r) and Tev(r) of metallic nanocrystals. These theoretical methods developed from various perspectives have established a framework for the phase-transition thermodynamics of nanocrystals, providing considerable references for investigating the mechanism of size effects on the properties of nanocrystals. The polymorphism behavior of metallic nanocrystals has been a hot topic in the research community. Recent researches show that a unique hexagonal 4H structural phase of Ag crystals (Ag4H) with an abcbabcba...-type of stacking sequence has been observed experimentally in the Ag nanoparticles, nanowires, and nanobelts.21-26 This structure is different from the common face-centered-cubic (FCC) structure of Ag crystals (Ag-FCC) with an abcabca...-type of stacking sequence in its bulk form. It is found that (1) Ag-4H and Ag-FCC could coexist at a particular condition and (2) the Ag-4H is the stable phase in Ag nanocrystals when their r remains smaller than a critical size rc at a certain temperature T.21-26 Approaching rc, the Ag4H and Ag-FCC reach an equilibrium, implying their difference in the Gibbs free energy ∆G(r,T) ) G4H(r,T) - GFCC(r,T) ) 0, where G(r,T) is size- and temperature-dependent Gibbs free energy, and the superscripts 4H and FCC denote the Ag-4H and Ag-FCC, respectively. Note that rc is temperature-dependent and the corresponding rc(T) function has been the subject of experimental and theoretical investigations recently.21,23-25 It is found that rc(T) progressively decreases with increasing T.21 Determining the rc(T) function and clarifying the phase transformation mechanism of Ag nanocrystals are essential for designing possible process routes to synthesize the metallic nanocrystals with desired properties. Although Tm(r) and Tev(r) have been investigated by phenomenological models15-18 and computer simulations,19,20 a general quantitative model needs to be developed to (1) reveal the intrinsic factor that dominates the variation of both Tm(r) and Tev(r) at the nanometer scale, (2) determine Tm(r) and Tev(r) of the isolated Ag nanoparticles as well as Tm(r) of the Ag
10.1021/jp806225p CCC: $40.75 2008 American Chemical Society Published on Web 10/01/2008
Size-Dependent Phase Stability of Ag Nanocrystals
J. Phys. Chem. C, Vol. 112, No. 42, 2008 16401
nanoparticles embedded in the Ni matrix, and (3) investigate the correlation between Tm(r) and Tev(r). A simple thermodynamic explanation has also been developed to calculate the rc of Ag nanowires at room temperature.23 To distinguish from the Ag-FCC, two energy levels are used to describe the Ag-4H nanostructure, such as lower surface energy and higher volume internal energy. However, it is generally accepted that both surface energy γ and surface stress f contribute to the stability of nanocrystalline phases.27 Moreover, (1) the rc(T) function has not been defined quantitatively over a wide temperature range, (2) the thermodynamic parameters of Ag-4H remain unavailable, and (3) the origin of the stability of Ag-4H is not clear. Therefore, a systematic investigation on the thermal stability, phase transformation mechanism, and thermodynamic parameters of the Ag-4H is eagerly awaited. In this work, the above-mentioned melting, evaporation, and solid-solid transition of Ag nanocrystals have been investigated through nanothermodynamics. On the basis of the root-meansquare-displacement (rmsd) model, both Tm(r) and Tev(r) functions can be determined quantitatively. The rc(T) function of Ag nanocrystals was also obtained with respect to the thermodynamic consideration of GFCC(r,T) ) G4H(r,T). Furthermore, several thermodynamic parameters of the Ag-4H are also determined with the developed model. 2. Methodology 2.1. Melting. Recently, a general rmsd model was established to elucidate size dependences of phase-transition functions.7 This model is based on the Lindemann′s melting criterion,28 Shi′s model for size-dependent melting temperature,29 and Mott′s expression for the vibrational entropy.30,31 With the rmsd model, the Tm(r)/ Tm(∞) function for the isolated nanocrystals can be written as
(
2Svib(∞) Tm(r) 1 ) exp Tm(∞) 3R (r/r0) - 1
)
(1)
where Svib(∞) is one of the terms of overall melting entropy Sm(∞) associated with vibration and R is the ideal gas constant. r0 in eq 1 is the critical radius of the low-dimensional material, which has all atoms located on its surface. In this case, r0 depends on the dimensionality d and the equilibrium atomic distance h through r0 ) (3 - d)h, where d ) 0 for nanoparticles, d ) 1 for nanowires and nanobelts, and d ) 2 for thin films. For the nanocrystals embedded in a matrix with coherent or semicoherent interface, assuming that the root of mean square amplitude of the interfacial atoms has an algebraic average value between that of the interior atoms and that of the matrix,7 the corresponding Tm(r)/Tm(∞) function can be expressed as
(
[Tm(∞)/TM(∞)](hM/h)2 - 1 Tm(r) 1 ) exp Tm(∞) 2 (r/r0) - 1
)
Hev(∞) and the evaporation temperature Tev(∞). Note that a factor of 12 has been taken in eq 3, which is different from eq 1. This is because the structural characteristic of a vapor is fully disordered and the phase state of a single atom/molecule cannot be identified.7 Distinct from our previous work,32 the dimensionality effect is considered in eq 3 with an universalized factor of 12r/r0 to replace 4r/h (the case of nanoparticles). This modification has been validated by recent research works.33,34 2.3. Polymorphism. In the case of Ag nanocrystals, the ∆G(r,T) function is given as ∆G(r,T) ) ∆Gv(∞,T) + ∆Gs(r) + ∆Ge(r) under ambient pressure where (1) ∆Gv(∞,T) is the temperature-dependent bulk volume free energy change, (2) ∆Gs(r) is the size-dependent surface energy change, and (3) ∆Ge(r) is the size-dependent elastic energy change induced by the internal pressure.27 As a first-order approximation, the melting entropy Sm(∞) of Ag-4H and Ag-FCC could be considered equivalent due to their small specific heat difference.27 Thus, ∆Gv(∞,T) ≈ ∆Hm(∞,T), where ∆Hm(∞,T) is the temperature-dependent melting enthalpy difference between Ag-4H and Ag-FCC. From the Helmholtz function, Hm(∞,T) ) Gm(∞,T) - T dGm(∞,T)/dT, where Gm(∞,T) denotes the melting Gibbs free energy. For metallic crystals, Gm(∞,T) ) 7Hm[∞,Tm(∞)]T[Tm(∞) - T]/{Tm(∞)[Tm(∞) + 6T]} in the temperature range of Tk < T e Tm(∞),35,36 in which Hm[∞,Tm(∞)] ) Tm(∞)Sm(∞) is the bulk melting enthalpy at Tm(∞), and Tk is the ideal glass transition temperature or the isentropic temperature with dGm(∞, T)/dT ) 0. Thus, Tk ) (71/2 - 1)Tm(∞)/6 and the Hm(∞,T) function can be expressed as Hm(∞,T) ) 49Hm[∞,Tm(∞)]T2/[Tm(∞) + 6T]2 at Tk < T e Tm(∞). When T e Tk, Hm(∞,T) ) Hm(∞,Tk) ) (28 - 73/2)Hm[∞,Tm(∞)]/ 18. Through the above considerations, the ∆Gv(∞,T) can be approximatelyobtainedby∆Gv[∞,Tk 1.7 As shown in the figure, our model predictions are in good agreement with the experimental data of Tm(r)/Tm(∞) for Ag/Ni. Although the experimental data of Tm(r)/Tm(∞) for the isolated Ag nanoparticles are not matched perfectly, they have the same trend within the measurement errors. It is believed that such a difference was caused by (1) different measurement techniques,10-12 (2) contamination such as oxygenation etc.,10 and (3) a wide spectrum of deviation in particle size.11 It is noted that the computer simulation results are also deviated from the experimental data. This should be associated with the accuracy of the surface energy determined by the simulation.20 However, our calculation results are between the experimental data and the computer simulation results, further evidencing the accuracy of the rmsd model. Figure 2 plots the calculation from eq 3 and the experimental results of Tev(r)/Tev(∞) for the isolated Ag nanoparticles. The related parameters used in the modeling are Sev(∞) ) 104.72 J mol-1 K-1 with Hev(∞) ) 255 kJ mol-1 and Tev(∞) ) 2435 K.39 It is discernible that the calculated Tev(r)/Tev(∞) values decrease with increasing 1/r or decreasing r, which is in good
Yang and Li
Figure 3. Comparisons among eqs 1, 3, and 5 for the isolated Ag nanoparticles. The solid lines denote the model predictions from eqs 1 and 3. The dotted and dash lines denote the model predictions from eqs 5a and 5b, respectively.
agreement with the experimental data, thereby confirming the accuracy of the developed model. For r/r0 > 10 or 1/r < 0.12 nm-1 for Ag nanoparticles (r0 ) 3h), exp{[-2Svib(∞)/(3R)]/[(r/r0) - 1]} ≈ 1 - 2hSvib(∞)/(Rr) and exp{[-2Sev(∞)/(3R)]/[(12r/r0) - 1]} ≈ 1 - hSev(∞)/(6Rr) as a first-order approximation. As a result, eqs 1 and 3 for the Ag nanoparticles can be rewritten as
Tm(r)/Tm(∞) ≈ 1 - 2hSvib(∞)/(Rr)
(5a)
Tev(r)/Tev(∞) ≈ hSev(∞)/(6Rr)
(5b)
Equation 5 obeys the thermodynamic law of low-dimensional materials, in which the alternation of size-dependent quantity is associated with the surface/volume ratio, or 1/r.7,15,29,32-34 This supports the notion that the size-dependent properties of nanocrystals are most likely affected by the severe bond dangling, which is induced by the crystal size reduction in nanoscale. The results calculated with eqs 1, 3, and 5 for the Tm(r)/Tm(∞) and Tev(r)/Tev(∞) of the isolated Ag nanoparticles are plotted in Figure 3 for comparisons. It is discernible that the results from these equations are overlapped when 1/r < 0.2 nm-1. In this case, eqs 1, 3, and 5 can be approximately combined into one model. Thus, both Tm(r)/Tm(∞) and Tev(r)/ Tev(∞) of the Ag nanoparticles have almost the same size dependence when r > 5 nm, providing the correlation between the melting and the evaporation of nanocrystals. As shown in the figure, the above three equations start to separate when 0.2 < 1/r < 0.3. When 1/r > 0.3, the outputs from eqs 1 and 3 show a remarkable difference, whereas the difference between eqs 5a and 5b is small. This is because Sev(∞) ) 13Svib(∞) for the Ag nanocrystals. This result is similar with Sev(∞) ) 12Svib(∞) when eq 5a ) eq 5b. 3.2. rc(T) of Ag Nanoparticles, Nanowires, and Nanobelts. Figure 4 plots the results calculated from eq 4 as well as the experimental and other theoretical results of the r-T phase diagram of the Ag nanoparticles, nanowires, and nanobelts. The related parameters used in the modeling are listed in Table 1. Moreover, several thermodynamic parameters of Ag-4H calculated with our model are also listed in Table 1. It was found that f > γ for both Ag-FCC and Ag-4H, which is consistent FCC with the general understanding. Furthermore, T4H m (∞) < Tm (∞), 4H FCC 4 FCC Hm (∞) < Hm (∞), and E H(∞) < E (∞), which reflect the stabilization of Ag-FCC in bulk form. For the Ag nanobelts, the length is a few orders of magnitude over the width and thickness,25 whereas the magnitudes of width and length of thin films are similar. In this case, the Ag nanobelts could be considered as one-dimensional materials. However, further
Size-Dependent Phase Stability of Ag Nanocrystals
J. Phys. Chem. C, Vol. 112, No. 42, 2008 16403 TABLE 1: Related Parameters Used in Modeling of rc with Eq 4 for Ag Nanocrystals Vm (cm · mol ) h (nm)a E(∞) (kJ mol-1) Hm[∞, Tm(∞)] (kJ mol-1) Tm(∞) (K) Sm(∞) (J mol-1 K-1)b Svib(∞) (J mol-1 K-1) B (GPa) f (J m-2)c 3
Figure 4. r-T phase diagram of (a) Ag nanoparticles (d ) 0) and (b) Ag nanowires and nanobelts (d ) 1). The solid lines denote the model predictions from eq 4. The dotted line shows the location of Tk. The symbols ∆21,24,25 and 321,24 are experimental results of Ag-4H and AgFCC as stable phases respectively at different temperatures. The symbols [23 and b23 in part b are experimental and theoretical results, respectively, of critical size for the stability transition between Ag-4H and Ag-FCC.
investigations on the orientation effect on the polymorphism behavior46 in the Ag nanobelts will be implemented. As shown in Figure 4, the rc(T) function decreases with increasing T when T > Tk for both d ) 0 and 1, which leads to higher thermal stability of Ag-4H than Ag-FCC for a smallsized Ag nanocrystal at lower temperature and vice verse. Note that the experimental data (marked as the symbols ∆21,24,25 and 321,24 in Figure 4) are the phase stability points of Ag nanocrystals rather than the equilibrium state or the phase boundary [rc(T) in Figure 4] as calculated by eq 4. Thus, our model predictions are in good agreement with experimental findings over a wide temperature range. The calculated rc(T) values are also in agreement with the experimental and other theoretical results for the Ag nanowires (marked as the symbols [23 and b23 respectively in Figure 4b), which validates the accuracy of the model. Figure 5 plots the ∆Gs(r), ∆Ge(r), ∆Gv(∞,T), and ∆G(r,T) functions of Ag nanocrystals at room temperature and thus the corresponding rc values can be determined. It is found that ∆Gs(r) < ∆Ge(r) (absolute values) for a certain r in the Ag nanoparticles, nanowires, and nanobelts. However, ∆Ge(r) is comparable with ∆Gs(r) especially for the Ag nanowires and nanobelts, which have lower ∆Gs(r) compared with that of Ag nanoparticles caused by the dimensionality effect (eq 4). As a result, we can infer that both γ and f are the main driving forces for the stability of the Ag-4H below rc while the contribution of f has been neglected in other thermodynamic explanations.23 Recent reports on the phase transformation of gold nanowires9 and pentacene films27 have also demonstrated the importance of f on the polymorphism of nanocrystals. The mechanism of the solid-solid phase transition between Ag-4H and Ag-FCC can be understood by the nanothermodynamics of nucleation. As shown in Table 1, both γ and f values of the Ag-4H are smaller than that of Ag-FCC, which facilitates the nucleation and growth of the Ag-4H in a small-sized Ag nanocrystal. This is because the critical crystal size and activation energy for the nucleation decrease with the decreasing of γ and f.27 With further increase in r, the number of surface atoms decreases and the role played by the surface in the nucleation and growth of Ag4H will not be accentuated due to small surface/volume ratio. In such a case, the nucleation of the Ag-4H will be depressed and the Ag-4H transforms into Ag-FCC. Moreover, the ∆Gv(∞,T) contributes to the temperature dependence of rc(T),
-1 a
γ (J m-2) ∆Hm[∞, Tm(∞)] (kJ mol-1)e
Ag-FCC
Ag-4H
10.259 0.2889 285.0039 11.3039 1234.9339 9.15 7.98 102.040 2.598 2.121 1.2542 0.21
10.856 0.2886 279.69b 11.09b 1212.02b 9.15 7.98b 105.7c 2.545 2.078 1.03d
a Vm ) Naν, where the Avogadro constant Na ) 6.02 × 1023 mol-1 and ν is the mean atom volume within the corresponding crystallographic structure. ν4H ) (33/2/2)a2c/12 with lattice constants a ) 0.2886 nm,23 and c ) 1.0000 nm.21,23,26 νFCC ) a3/4 with a ) 0.4085 nm.39 h ) (21/2/2)aFCC and h ) a4H for Ag-FCC and Ag-4H, respectively. b Each atom in the Ag-4H has 6 nearest-neighbor atoms with a distance of h1 ) 0.2886 nm and 6 nextnearest-neighbor atoms with a distance of h2 ) 0.3004 nm.23 On the basis of the BOLS correlation mechanism,15 E4H(∞)/EFCC(∞) ) 4H 4H (hFCC/h1 + hFCC/h2)/2. Hm [∞,Tm (∞)] can be obtained by 4H 4H FCC FCC Hm [∞,Tm (∞)]/Hm [∞ Tm (∞)] ) E4H(∞)/EFCC(∞) as a first order FCC FCC FCC FCC approximation. Furthermore, Sm (∞) ) Hm [∞,Tm (∞)]/Tm (∞), 4H FCC 4H FCC 4H Sm (∞) ) Sm (∞), Svib (∞) ) Svib (∞), and Tm (∞) ) 4H 4H 4H Hm [∞,Tm (∞)]/Sm (∞). c f ) h[3BSvib(∞)Hm(∞,Tm)/(4VmR)]1/2 for nanoparticles, nanowires, and nanobelts, whereas f ) h[BSvib(∞)Hm(∞,Tm)/(2VmR)]1/2 for thin films with B being the bulk FCC FCC 4H 4H 27 modulus.41 Combined with the relation fFCCVm /B ) f4HVm /B , fFCC, f4H, and B4H can be calculated. f values of 2.598 and 2.545 J m-2 are for nanoparticles, nanowires, and nanobelts while 2.121 and 2.078 J m-2 are for thin films. d Since the difference between h1 and h2 values is small, the Ag-4H can be considered as a combination of two quasi-hexagonal close packing structures during the determination of γ4H. On the basis of the revised broken-bond model considering the surface bond relaxations,43 γ(0001) ) [1 (ZS/ZB)1/2]E4H/(NaAs) ) 0.863 J m-2 with ZS ) 9,44 ZB ) 12, As ) (31/2/2)a2, where ZS is the coordinate number of surface atoms, ZB is bulk coordinate number, and As denotes the area of the two-dimensional unit cell. γ(101j0) ) 1.073 J m-2 can also be determined with this model where ZS ) 16/3,45 ZB ) 12, and As ) ac/2. γ4H is an algebraic average of all crystallographic facets determined by γ4H ) [2γ(0001)A(0001) + 6γ(101j0)A(101j0)]/ [2A(0001) + 6A(101j0)] where A(0001) ) (33/2/2)a2 and A(101j0) ) ac show the corresponding areas. e Please see the text.
which is also critical for the polymorphism of Ag nanocrystals. As a result, the lower γ and f coupled with a higher Gv(∞,T) predominates the thermal stability of 4H-structured Ag in nanometer scale. From Table 1, it can be seen that the f value of thin films (d ) 2) is smaller than that of nanoparticles, nanowires, and nanobelts. With eq 4, rc ) 0.39 nm for the Ag films at room temperature, which indicates that the Ag-4H can be found in ultrathin Ag films. As a result, the value sequence of rc(T) from large to small is nanoparticles, nanowires (nanobelts), and thin films at a particular temperature. Further experimental work will be employed to validate this point. 4. Conclusions In summary, the root-mean-square-displacement (rmsd) model was used to determine the size-dependent melting temperature Tm(r) and evaporation temperature Tev(r) of Ag nanocrystals. The temperature-dependent critical size rc(T) of solid-solid phase transition between 4H and face-centered-cubic structured Ag nanocrystals is also investigated with nanothermodynamics. It is found that (1) both Tm(r) and Tev(r) decrease with decreasing
16404 J. Phys. Chem. C, Vol. 112, No. 42, 2008
Figure 5. ∆G as a function of r at room temperature (T ) 298 K) for Ag nanoparticles (d ) 0) and Ag nanowires and nanobelts (d ) 1). The solid lines denote the calculation results of ∆Ge(r) and ∆Gv(∞,298 K) for both d ) 0 and 1, as well as ∆G(r,298 K) and ∆Gs(r) for d ) 0. The dash lines denote the calculation results of ∆G(r,298 K) and ∆Gs(r) for d ) 1. The dotted line shows the location of ∆G ) 0, which crosses with ∆G(r,298 K) at rc.
r for the isolated Ag nanoparticles, (2) Tm(r) increases with decreasing r for the Ag nanoparticles embedded in the Ni matrix, (3) Tm(r)/Tm(∞) and Tev(r)/Tev(∞) of the isolated Ag nanoparticles have almost the same size dependence behavior when r > 5 nm, (4) both surface energy and surface stress are the main driving forces for the solid-solid phase transformation of Ag nanocrystals, and (5) the value sequence of rc(T) from large to small is nanoparticles, nanowires (nanobelts), and thin films at a particular temperature. Acknowledgment. This project is financially supported by Australia Research Council Discovery Programs (Grant Nos. DP0880548 and DP0666412). References and Notes (1) Gleiter, H. Acta Mater. 2000, 48, 1. (2) El-Sayed, M. A. Acc. Chem. Res. 2001, 34, 257. (3) Sun, Y. G.; Xia, Y. N. Science 2002, 298, 2176. (4) Margueritat, J.; Gonzalo, J.; Afonso, C. N.; Mlayah, A.; Murray, D. B.; Saviot, L. Nano Lett. 2006, 6, 2037. (5) Kumar, A.; Vemula, P. K.; Ajayan, P. M.; John, G. Nat. Mater. 2008, 7, 236. (6) Hauder, M.; Gsto¨ttner, J.; Hansch, W.; Schmitt-Landsiedel, D. Appl. Phys. Lett. 2001, 78, 838. (7) Jiang, Q.; Yang, C. C. Curr. Nanosci. 2008, 4, 179. (8) Nanda, K. K. Appl. Phys. Lett. 2005, 87, 021909. (9) Diao, J. K.; Gall, K.; Dunn, M. L. Nat. Mater. 2003, 2, 656.
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