7318
J. Phys. Chem. B 2007, 111, 7318-7320
Quasi-Isochoric Superheating of Nanoparticles Embedded in Rigid Matrixes C. C. Yang* and S. Li School of Materials Science and Engineering, The UniVersity of New South Wales, NSW 2052 Australia ReceiVed: March 12, 2007; In Final Form: April 27, 2007
A thermodynamic model for pressure-induced quasi-isochoric superheating of nanoparticles embedded in rigid matrixes was established quantitatively by introducing the size dependence of melting enthalpy. The accuracy of the developed model was verified with the reported experimental data of Sn and Pb nanoparticles encapsulated in fullerene-like graphitic shells (FGS) as well as Ge nanoparticles embedded in SiO2. The mechanism behind the smaller superheating for Al nanoparticles embedded in Al2O3 was also studied. It was found that the extent of the superheating is determined by the pressure, which is in turn related to the confinement effect and to the size of the nanoparticles. Through the knowledge obtained in this study, it can be concluded that the extreme superheating of nanoparticles can be achieved on the proviso that they are encased in a sufficiently rigid matrix, while the size of nanoparticles is small enough.
1. Introduction When the size of the low-dimensional materials approaches the nanometer scale, the electronic, magnetic, optic, catalytic, and thermodynamic properties of the materials diverge significantly from their bulk properties.1 In the thermodynamic properties of materials, thermal stability is critical for nanodevice applications as many mechanical, physical, and chemical properties of crystals are related to their corresponding melting temperatures. The pursuit of a general rule to describe the melting transition has been a subject of great interest for a century.2 As a variable in the determination of the melting thermodynamics, the size effect has been studied ever since it was observed by Takagi in 1954 that metallic nanoparticles melt below the melting temperature of their counterparts in the bulk material.3 It is believed that an understanding of the solid/liquid transition of low-dimensional crystals is not only beneficial to the theoretical explanation of phase transformation but also to applications in modern nanotechnology industries.1 In spite of the numerous experimental and theoretical investigations that have been carried out to study the melting behaviors of nanocrystals, several uncertainties still persist regarding this phenomenon. The melting denotes an equilibrium state of solid and liquid phases at a certain temperature and pressure. The essential thermodynamic parameters for melting are the melting temperature Tm and the melting entropy ∆Sm or the melting enthalpy ∆Hm.4,5 If D and ∞ represent the diameter of a nanocrystal and the corresponding bulk crystal, Tm, Tkh (Tkl), and Tfh (Tfl) are thermodynamic melting, kinetic melting, and freezing temperatures, respectively [the second subscript h (l) denotes high (low) heating or cooling rate]. The relationships of Tkh(∞) > Tkl(∞) ) Tmi(∞) and Tfh(∞) < Tfl(∞) < Tmi(∞) are established where the second subscript i denotes isobaric (isochoric) melting. The difference between Tkh(∞), Tfh(∞), and Tfl(∞) from Tmi(∞) is either kinetic superheating or supercooling depending on the sign.4,5 On the other hand, in isobaric (isochoric) melting process, as D decreases, Tmi(D) g Tmi(∞) or Tmi(D) e Tmi(∞), * To whom correspondence should be addressed. Fax: +61-2-93855956; e-mail:
[email protected].
which presents either thermodynamic superheating or supercooling, depending on the surface or interface conditions.4,5 The superheating of nanocrystals has been observed in many metallic systems when the embedded nanocrystals have coherent or semicoherent interfaces with the matrixes.5 Recently, however, it was found that nanoparticles embedded in rigid carbon or ceramic matrixes also exhibit evident superheating behavior despite the presence of incoherent interfaces between the nanoparticles and matrixes.6,7 In-situ transmission electron microscopy observations of the melting and solidification behavior of Sn and Pb nanoparticles trapped inside fullerene-like graphite shells (FGS) suggest that the melting points of Sn and Pb nanoparticles are substantially higher than their corresponding Tmi(∞) while the refreezing temperature of Pb nanoparticles is equal to the Tmi(∞).6 A similar case was also observed for Ge nanoparticles embedded in SiO2.7 Contrary to these findings, however, Al nanoparticles encapsulated in Al2O3 shells were observed to demonstrate smaller superheating when examined using in-situ X-ray diffraction.8 The melting behaviors described in these instances of confined nanoparticles diverge from current theoretical predictions, and as a result, a new model is required to describe the melting points in this class of materials. In this work, the superheating of nanoparticles embedded in rigid matrixes is considered as a form of quasi-isochoric superheating. On the basis of this consideration, the quasiisochoric melting temperatures Tmi(D) of Sn and Pb nanoparticles embedded in FGS and Ge nanoparticles embedded in SiO2 are calculated using a model adapted from Clausius-Clapeyron theory by introducing ∆Hmi(D). The mechanism behind the smaller superheating observed in the case of the Al nanoparticles embedded in Al2O3 shells is also explained with the developed model. 2. Methodology Through the introduction of the size-dependent melting enthalpy ∆Hmi(D), the classic Clausius-Clapeyron equation governing all first-order phase transitions of pure substances is adapted to determine the melting temperature-pressure (TmP) curve of the nanoparticles theoretically. The adapted model is expressed as
10.1021/jp072010t CCC: $37.00 © 2007 American Chemical Society Published on Web 06/01/2007
Quasi-Isochoric Superheating of Nanoparticles
dP )
∆Hmi(D) dTm ∆VmTm
J. Phys. Chem. B, Vol. 111, No. 25, 2007 7319
(1)
where ∆Vm ) VL - VC shows g-atom transition volume during the melting with the subscripts L and C denoting the liquid and the crystal, respectively. The size effect on ∆Vm has been neglected in eq 1 since the change of the equilibrium atomic distance is usually in the range of 0.1-2.5% even when D < 20 nm.9 Equation 1 can describe the joint rate of change dP/dT along the phase equilibrium lines and can estimate the derived properties of ∆H and ∆V. To utilize eq 1 for determination of Tm-P phase diagrams or a Tm(P) function, the integral of eq 1 is required. Thus, the ∆Hmi(D) function must be determined first. For an isolated nanoparticle or a nanoparticle embedded in a matrix with an incoherent interface, the Tm(D) and ∆Sm(D) functions can be determined as the size-dependent amplitude of atomic thermal vibrations of the nanocrystals with Lindemann’s criterion for the melting and Mott’s equation, respectively.4 Since ∆Hm ) Tm∆Sm, the ∆Hmi(D) function goes to
∆Hmi(D) ∆Hmi(∞)
[
) 1-
] [
]
2Svib(∞) 1 1 exp 3R D/(6x) - 1 D/(6x) - 1 (2)
where ∆Svib(∞) is the vibrational part of the overall melting entropy ∆Smi(∞), R is the ideal gas constant, and x is the equilibrium atomic distance. It is known that Lindemann’s criterion is valid for both bulk material and nanocrystals. In the criterion, a crystal melts when the root of mean square amplitude σ of the atoms reaches a certain fraction of x.4,5 It only describes melting behavior in solid state to avoid the complication of multistates in equilibrium. This criterion has been verified experimentally and is used to predict melting behaviors. In this case, the interfacial effects induced by the different interface energies of solid-matrix and liquid-matrix are not considered in our model. Owing to the significant differences in the Tmi(∞) and the thermal expansion coefficient R between nanoparticles and matrix materials in the case of Sn, Pb, and FGS as well as Ge and SiO2, the melting process of these embedded nanoparticles can be considered as a quasi-isochoric thermodynamic melt. The corresponding pressure P induced by the confinement effect can be expressed as P ) B(VL - VC)/VC where the bulk modulus B is assumed to be size-independent as a first-order approximation. Integrating P from zero to P and T from Tmi(∞) to Tmi(D) in terms of eq 1, we have
∫0P (VL - VC)dP ) ∆Hmi(D) ∫T T (∞)(D) T1m dTm mi
mi
and it yields
Tmi(D) ) Tmi(∞)exp[P(VL - VC)/∆Hmi(D)]
(3)
3. Results and Discussion The model predictions of Tmi(D) with eqs 2 and 3 and the experimental results of Tmi′(D) for Sn and Pb nanoparticles encapsulated in FGS6 as well as Ge nanoparticles embedded in SiO27 are listed in Table 1. The related parameters used in the modeling could also be found in this table. The comparison between the results obtained from the modeling and experimental data shows that the predicted Tmi(D) functions are in good agreement with the experimental results of Tmi′(D) where
TABLE 1: The Model Predictions of Tmi(D) with Eqs 2 and 3 and Experimental Results of Tmi′(D) for Sn and Pb Nanoparticles Encapsulated in FGS as Well as Ge Nanoparticles Embedded in SiO2 10
Tmi(∞) (K) ∆Hmi(∞) (kJ g-atom-1)10 ∆Smi(∞) (J g-atom-1 K-1)a VC (cm3 g-atom-1)10 VL (cm3 g-atom-1)b ∆Vm (cm3 g-atom-1)c ∆Svib(∞) (J g-atom-1 K-1) D (nm) x (nm)10 B (GPa)14 P (GPa)c ∆Hmi(D) (kJ g-atom-1)c Tmi(D) (K)c Tmi′(D) (K)
Sn
Pb
Ge
505 7.0 13.86 16.29 17.08 0.79 9.2212 146 0.3182 57.0 2.76 5.25 765 7706
601 4.8 7.99 18.26 18.9012 0.64 6.7712 306 0.3501 43.9 1.54 4.26 757 7406
1211 31.8 26.26 13.63 13.04 -0.59 4.6013 57 0.2450 75.0 -3.25 15.92 1366 13507
a ∆Smi(∞) ) ∆Hmi(∞)/Tmi(∞). b For Sn and Ge, VL ) M/FL with M being the g-atom weight and F being the density. For Sn, M ) 118.71 g g-atom-1,10 F ) 6.95 g cm-3,11 and for Ge, M ) 72.64 g g-atom-1,10 F ) 5.57 g cm-3.11 c Please see the text.
the deviation is only 0.6, 2.2, and 1.2% for Sn, Pb, and Ge nanoparticles, respectively. This exhibits the accuracy of the developed model. The results listed in Table 1 also raises the issue as to why only a smaller superheating of the Al nanoparticles encapsulated in Al2O3 shells was observed despite the large differences between their Tmi(∞) and bulk thermal expansion coefficient. The cause may be attributed to there being only a very small pressure acting upon the nanoparticles. For the three test samples A, B, and C of Al nanoparticles encapsulated in Al2O3 shells, the particle sizes (D) were 39.4, 36.6, and 33.0 nm with shell thicknesses (t) of 10.4, 12.1, and 12.5 nm, respectively. It was found experimentally that the derived pressures from the lattice spacing changes are only 0.13, 0.25, and 0.19 GPa for these samples.8 In general, the pressure increases with the increase of both t/(t + D/2) and (RN - RM) where the subscripts N and M denote the encapsulated nanoparticle and the matrix, respectively.8,15 It could therefore be expected that the pressure could become very high if the shell is rigid enough. However, for the Al nanoparticles encapsulated in Al2O3 shells, the thicknesses of the Al2O3 shells are less than 13 nm, and the ratio of t/(t + D/2) is only 0.3∼0.4 for all samples in the experiments.8 On the basis of a recent result, R(D)/R(∞) ) exp{[2Svib(∞)/(3R)]/ [D/(6x) - 1]},9 which carries the implication that the thermal expansion coefficient increases with decreasing nanoparticle size. In terms of our developed model, we predict R(D) and R(t) functions of the samples A, B, and C, as shown in Figure 1. It is found that with D (or t) decreasing, the thermal expansion coefficients of both Al2O3 shells and Al nanoparticles increase. Since t of Al2O3 (10.4∼12.5 nm) is much smaller than D of Al (33.0∼39.4 nm), the increase of the thermal expansion coefficient in Al2O3 is greater than that in Al, resulting in an insufficient differential between RN and RM. Therefore, the superheating of Al nanoparticles encapsulated in thin Al2O3 shells cannot be considered as a quasi-isochoric melting, and the observed smaller superheating is apprehensible. Summarizing the above finding, it can be seen that quasiisochoric superheating can be achieved when the matrix is rigid enough to cause a very high pressure. As shown in eq 3, the small enough value of ∆Hmi(D) could result in an extreme superheating. From eq 2, we know that the ∆Hmi(D) decreases with the decreasing of D. Thus, the extreme superheating is possible to be achieved if the size of encapsulated nanocrystals
7320 J. Phys. Chem. B, Vol. 111, No. 25, 2007
Yang and Li thermal vibration of atoms on the interface. Therefore, the σ of the interfacial atom is smaller than that of the interior one and there is a resultant superheating of the embedded nanoparticles.5 Although our approach is different from the BOLS correlation theory, both theories describe the superheating phenomenon with lower energetic state of surface or interface atoms. 4. Conclusions
Figure 1. R(D) and R(t) functions of samples A, B, and C of Al nanoparticles encapsulated in Al2O3 shells. For Al, R(∞) ) 23.1 × 10-6 K-1,10 ∆Svib(∞) ) 6.15 J g-atom-1 K-1,12 and x ) 0.2864 nm.10 For Al2O3, R(∞) ) 8.4 × 10-6 K-1,8 ∆Svib(∞) ≈ ∆Sm(∞) ) 9.71 J g-atom-1 K-1,16 and x ) Vc1/3 ) 0.6340 nm with Vc ) 0.2548 nm3 denoting the volume of the cell.11
is small enough. Such understanding may provide new insights into the fundamental mechanisms of pressure effects on the melting behaviors of nanomaterials, thus offering a new route to improve the thermal stability of metal or semiconductor materials, which is essential for the nanodevice applications. In the case of Pb nanoparticles embedded in FGS,6 the melting temperature is Tmi(D) now and Tmi(D) - Tfi(D) ) 139 K even if Tfi(D) ≈ Tmi(∞). The supercooling is also significant although in this instance it differs from the conventional definition.17,18 Similarly, the supercooling values are about 395 and 486 K for Sn6 and Ge,7 respectively. The supercooling tendencies of Sn, Pb, and Ge are different. This is due to the possible complication of kinetic solidification processes that deserves better explanations from the point of view of dynamics. These experimental results show that the nanoparticles embedded in the rigid matrixes demonstrate thermodynamic but not kinetic superheating. Recently, a bond-order-length-strength (BOLS) correlation mechanism based on the atomic coordination number imperfection has been developed to model the superheating of nanosolids.19 In BOLS mechanism, the superheating is attributed to the interfacial bond strengthening between embedded nanocrystals and matrixes as well as the bond nature evolution with atomic coordination reduction for nanoclusters.20 In this study, superheating is investigated using a different approach. The superheating in our work is attributed to the rising pressure upon heating because of the large difference in thermal expansion coefficients between embedded nanoparticles and rigid matrixes. The pressure buildup could reach several GPa as shown in Table 1. However, in the case of the metallic systems with coherent or semicoherent interfaces, the pressure effects would be less than the interfacial effects because of the interfacial bonding.21 Irrespective of the interfacial structures, however, the different thermal expansion coefficients are the origin of pressure because of the confinement effects. The confinement effects constrain
In conclusion, a model has been developed to estimate the quasi-isochoric superheating of Sn and Pb nanoparticles embedded in FGS as well as Ge nanoparticles embedded in SiO2 by considering their size-dependent melting enthalpies on the basis of the Clausius-Clapeyron theory. The model predictions are consistent with experimental results. It has been demonstrated that the smaller superheating of Al nanoparticles encapsulated in Al2O3 was caused by the insufficient difference of the thermal expansion coefficients between the nanoparticles and the thin shells. It is found that the extreme superheating could be achieved when the matrix is rigid enough while the size of nanoparticles is small enough. This study reveals that the observed superheating is a thermodynamic equilibrium while the solidification is a kinetic one. Acknowledgment. This project is financially supported by Australia Research Council Discovery Program (Grant No. DP0666412). References and Notes (1) Gleiter, H. Acta Mater. 2000, 48, 1. (2) Cahn, R. W. Nature 2001, 413, 582. (3) Takagi, M. J. Phys. Soc. Jpn. 1954, 9, 959. (4) Jiang, Q.; Shi, H. X.; Zhao, M. J. Chem. Phys. 1999, 111, 2176. (5) Jiang, Q.; Zhang, Z.; Li, J. C. Chem. Phys. Lett. 2000, 322, 549. (6) Banhart, F.; Herna´ndez, E.; Terrones, M. Phys. ReV. Lett. 2003, 90, 185502. (7) Xu, Q.; Sharp, I. D.; Yuan, C. W.; Yi, D. O.; Liao, C. Y.; Glaeser, A. M.; Minor, A. M.; Beeman, J. W.; Ridgway, M. C.; Kluth, P.; Ager, J. W., III; Chrzan, D. C.; Haller, E. E. Phys. ReV. Lett. 2006, 97, 155701. (8) Mei, Q. S.; Wang, S. C.; Cong, H. T.; Jin, Z. H.; Lu, K. Acta Mater. 2005, 53, 1059. (9) Yang, C. C.; Xiao, M. X.; Li, W.; Jiang, Q. Solid State Commun. 2006, 139, 148. (10) http://www.webelements.com/Web Elements Periodic Table, accessed on Mar 1, 2007. (11) Weast, R. C. CRC Handbook, of Chemistry, and Physics, 69th ed.; CRC Press: Boca Raton, FL, 1988-1989; pp B219, B217, B193-5. (12) Jiang, Q.; Zhou, X. H.; Zhao, M. J. Chem. Phys. 2002, 117, 10269. (13) Regel′, A. R.; Glazov, V. M. Semiconductors 1995, 29, 405. (14) Soler, J. M.; Beltra´n, M. R.; Michaelian, K.; Garzo´n, I. L.; Ordejo´n, P.; Sa´nchez-Portal, D.; Artacho, E. Phys. ReV. B 2000, 61, 5771. (15) Spaepen, F.; Turnbull, D. Scr. Metall. 1979, 13, 149. (16) Kim, S. S.; Sanders, T. H., Jr. J. Am. Ceram. Soc. 2001, 84, 1881. (17) Luo, S.-N.; Swift, D. C. Phys. ReV. Lett. 2004, 92, 139601. (18) Banhart, F.; Herna´ndez, E.; Terrones, M. Phys. ReV. Lett. 2004, 92, 139602. (19) Sun, C. Q.; Shi, Y.; Li, C. M.; Li, S.; Au Yeung, T. C. Phys. ReV. B 2006, 73, 075408. (20) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (21) Chattopadhyay, K.; Goswami, R. Prog. Mater. Sci. 1997, 42, 287.
