12914
J. Phys. Chem. C 2007, 111, 12914-12917
Surface Energy and Melting Temperature of Elemental Nanocavities H. M. Lu, D. N. Ding, Z. H. Cao, S. C. Tang, and X. K. Meng* Department of Materials Science and Engineering, National Laboratory of Solid State Microstructures, Nanjing UniVersity, Nanjing 210093, People’s Republic of China ReceiVed: May 9, 2007; In Final Form: July 1, 2007
A model for size-dependent surface energy of elemental nanocavities is proposed in terms of size-dependent cohesive energy. It is found that the surface energy of elemental nanocavities increases with decreasing of the cavity size to several nanometers. The newly built model is validated to be in agreement with that based on some other theories. Moreover, an analytic model is developed to describe the size-dependent melting temperature of elemental nanocavities for the first time. The analytic results show that the melting temperature increases with decreasing of the size of nanocavities. The theoretical predictions confirm well to the available data of argon obtained by molecular dynamics simulations.
Introduction In recent years, nanocavities have attracted much more interest because of important implications for microscopic physics, chemistry, biology, and medicines.1-3 In contrast to a nanoparticle, which is thought to be a cluster of atoms in a vacuum, a nanocavity in condensed matter can usually be considered as a cluster of vacancies.3 These two metastable condensed matter structures have an antisymmetry relation.3 For example, the melting temperature Tm of nanoparticles increases with the increasing particle size4-6 while that of nanocavities decreases with increasing of the cavity radius r and finally reaches a plateau at a large cavity size.7-9 Recently, Bai and Li have studied the mechanism of the melting of nanocavities via molecular dynamics simulation and found that the melting in each of the stages is governed by the interplay among different thermodynamic mechanisms.10 However, although there are relatively extensive molecular dynamics (MD) simulations of the size-dependent melting temperature Tm(r) of nanocavities, there is no analytic model to describe the Tm(r) function of nanocavities to the best of our knowledge. On the other hand, surface energy γsv is one of the most important quantities of nanocavities because the ratio of surface/ volume will increase with decreasing of the cavity size at a certain cavities numbers, and then surface energy will greatly affect the properties of materials. For examples, nanocavities have been reported to shrink under thermal activation and the shrinkage slows down when the pore size is reduced to 2 nm, where the cavity surface energy is considered as the driving force not only for the external shrinkage but also for the internal shrinkage.11 Moreover, cavity melting is thought to be governed by the variations and interplay of surface and interface energies, etc.10 However, no experimental data on surface energy of nanocavities are available to our knowledge, noted that even the bulk surface energy γsv0 of metals is also difficult to experimentally determine.12 Thus, surface energy of nanocavities remains little understood despite the fundamental thermodynamic importance of this quantity.13 * To whom correspondence should be addressed. Tel.: +86-025-83685585. Fax: +86-025-8359-5535. E-mail:
[email protected].
Through combination with the Tolman equation on surface tension14 and consideration of the negative curvature for the cavity, Ouyang et al. have deduced the follow expression to describe the size-dependent chemical surface energy γsv(r) of nanocavities:15
γsv(r)/γsv0 ) 1 + 2δ/r
(1)
Here δ is Tolman’s length, which is assumed to be equal to atomic diameter h in the deduction of eq 1.14,15 As a first-order approximation, although there is no direct experimental evidence to support eq 1, eq 1 should be also applicable to predicting the γsv(r) function of nanocavities since the structural difference between solid and liquid is very small in comparison with that between solid and gas or between liquid and gas.15 However, it is unknown whether r in eq 1 can be extended from micrometer size to nanometer size as noted that the Tolman equation is questionable for very small materials.16 Moreover, Tolman’s length has been extensively confirmed to be dependent on the size.17-19 Recently, a model for the size-dependent cohesive energy of nanocrystals E(r) has been established,20 and then size-dependent surface energy of nanoparticles have also been modeled on the basis of this model where reasonable agreements for Be, Mg, Na, Al, and Au nanoparticles are found.21 In this contribution, this model is extended to describe the γsv(r) and Tm(r) functions of elemental nanocavities. The developed models correspond to other theoretical results and available MD simulation data. Model Thirty years ago, Tyson proposed that the simplest approach to get a rough estimate of γsv0 values is to determine the broken bond number of surface atom and multiply this number with the cohesion energy per bond,22 namely,
γsv0 ) (1 - m)Eb
(2a)
where m denotes the ratio of coordination numbers of surface atom to that of the corresponding bulk one and Eb is bulk cohesion energy.
10.1021/jp073540s CCC: $37.00 © 2007 American Chemical Society Published on Web 08/14/2007
Surface Energy of Elemental Nanocavities
J. Phys. Chem. C, Vol. 111, No. 35, 2007 12915
Moreover, under the assumption that the total energy of a system can be expressed as a sum of energetic contributions of each atom that are proportional to the square root of the related coordination numbers, the classic broken-bond rule developed an expression to describe the surface energy of the 4d transition metals,23
γsv0 ) (1 - m1/2)Eb
(2b)
However, the former method does not consider the variation of bond strength with the coordination number or relaxation, while the latter is not complete because only attractive forces are taken into account. Recently, through modification of the classic broken-bond rule and consideration of the contributions of next-nearest coordination numbers, a simple formula has been established to estimate surface energies of elemental metals,12
Figure 1. γsv0 as a function of Tm0 for transition metals.
γsv0 ) [(2 - m - m1/2) + β(2 - m′ - m′1/2)]Eb/(2 + 2β) (2c) where the prime (′) denotes the next-nearest coordination number and β shows the total bond strength ratio between the nextnearest neighbor and the nearest neighbor. The predictions of eq 2 for 52 elemental crystals are in agreement with experimental results and the first-principles calculations.12,24 Although these three expressions and corresponding results are different, it can be found that the surface energy is directly proportional to cohesive energy, namely,
γsv0 ) kEb
(3a)
with k being a constant as a function of the coordination number. If the nanocrystals with cavities have the same structure of the corresponding bulk, k should be size-independent as a first order approximation. Thus, the linear relationship between the surface energy and the cohesive energy can be assumed to hold for nanocavities as well and the surface energy of nanocavities can be expressed as
γsv(r) ) kE(r)
(3b)
where E(r) for nanocrystals with positive curvature has been determined by20
[
E(r)/Eb ) 1 -
] (
2Sb 1 1 exp 4r/h - 1 3R 4r/h - 1
)
[
] (
2Sb 1 1 exp 4r/h + 1 3R 4r/h + 1
)
are cited from ref 25. In this case, the linear relation between γsv0 and Tm0 can be linearly regressed as
γsv0 ≈ nTm0
(6a)
where the linearly regressed slope n is equal to 1.06 × 10-3 J/(m2 K) with the standard deviation of 5.96 × 10-3 J/(m2 K) as noted that the correlation coefficient of the fit is 0.955. Also under the assumption that the elemental nanocavities have the same structure of the corresponding bulk, eq 6a may be extended to nanometer size with the same form,
γsv(r) ≈ nTm(r)
(6b)
(4) Combining eqs 5 and 6, there is
where h denotes atomic diameter, Sb ) Eb/Tb is the bulk solidvapor transition entropy with Tb being the bulk boiling temperature, and R is the ideal gas constant. Moreover, the minimal value of r is h/4, where the structure of solid and vapor is indistinguishable.20 If one considers the fact that the curvature of the cavity is negative and combines eqs 3 and 4, there is
γsv(r)/γsv0 ) 1 +
Figure 2. Comparisons of the r dependence of γsv(r)/γsv0 described by different models predictions in terms of eq 1 (the dotted lines) and eq 5 (the solid lines) for (a) Si and (b) Cu.
(5)
Since the surface energy denotes the bond energy difference between surface atoms and interior ones while the melting temperature is directly proportional to the bond strength, an empirical relation between surface energy and melting temperature should also exist. As shown in Figure 1, the surface energy is plotted against the melting temperature of transition metals where the experimental values of bulk surface energies are taken from ref 24 while those of the bulk melting temperature Tm0
[
Tm(r)/Tm0 ) 1 +
] (
2Sb 1 1 exp 4r/h + 1 3R 4r/h + 1
)
(7)
Results and Discussion In terms of eq 5, γsv(r)/γsv0 functions of Si and Cu nanocavities (the solid lines) are shown in Figure 2. As a comparison, the predictions of eq 1 with δ ) h are also shown. It is evident that our model predictions are in good agreement with those of eq 1. This correspondence in return confirms the validity of the assumption employed in the deduction of eq 3b. Moreover, although eq 3 is deduced in terms of the relationship between the bulk surface energy and the broken bond number of surface atoms for transition metals, the agreement for Si as shown in Figure 2a indicates that this relation should also be applicable to other types of materials. Usually, the surface energy of nanocavities is considered to consists of chemical part γchem and structural part γstru, namely γsv ) γchem + γstru. The former results from the dangling bond
12916 J. Phys. Chem. C, Vol. 111, No. 35, 2007
Lu et al.
TABLE 1: Several Necessary Parameters Used in the Equations atom
h (nm)
Eb (kJ/mol)
Tb (K)
Sb (J/(mol K))
Cu Si Ar
0.25625 0.15715 0.17625
300.325 384.225 6.425
283625 354025 87.325
105.9 108.5 73.3
energy at the inner surface of the nanocavity while the latter originates from elastic strain energy in the inner skin of one atomic layer thick of the nanocavity. However, γstru is 1 or 2 orders smaller than γchem at several nanometer sizes as shown in Figure 2a of ref 15. Thus, γstru is negligible as a first-order approximation. Furthermore, the agreement shown in Figure 2 also indicates that this neglect does not result in a big error. As shown in Figure 2, γsv(r) of nanocavities increases with the decreasing size, which is in contrast with the trend of γsv(r) of nanoparticles.20,26,27 This difference can be explained as the following: Due to the negative curvature in inner skin of nanocavity, the atomic bonding energy and the elastic strain energy are contributed to the total surface free energy. As shown in Figure 12 of ref 3, the atomic bonds at the surface layer of nanocavities and nanoparticles are schematically illustrated. Clearly, the density of atomic bonding energy increases with the decreasing size at negative curvature surface, while the positive curvature surface has the opposite trend. At the same time, the density of elastic strain energy also increases with the decreasing size. These results indicate that the excess surface free energy in inner skin of nanocavity is different from or contrary to that in the nanocrystal. Moreover, the radius of 2 nm seems to be the threshold as shown in Figure 2, at which the surface energy becomes stable. Since the intrinsic modulus in the cavity skin becomes larger than that of the bulk and local hardening may take place with the increase of the surface energy,15 the shrinking is faster when the cavity’s radius is larger than the threshold value of 2 nm and it will be slow down for the larger surface energy with smaller size, which is consistent with the observation of Zhu.11 Considering the mathematical relation of exp(x) ≈ 1 + x when x is small enough (e.g., x < 0.1), eq 5 can be rewritten as
γsv(r)/γsv0 ≈ 1 + Sbh/(6Rr)
(8)
Equation 8 is in agreement with the general consideration that the change of the any size-dependent thermodynamic quantity is proportional to 1/r.28 Moreover, Sb ≈ 12R for Cu and Si as shown in Table 1 leads to Sbh/(6Rr) ≈ 2h/r, which makes that eq 8 is the same as eq 1. This is the reason that the model predictions in terms of eq 5 are in good agreement with those of eq 1 at r/h > 10 for Cu and Si. Although δ is assumed to be constant as required by the deduction of eq 1 and the original Tolman equation, statistical thermodynamics have indicated that δ depends strongly on r.18 Since the results are based on rather complex numerical calculations, it would be helpless to express δ(r) analytically. Fortunately, eq 5 can be used to satisfy this requirement. Substituting eq 5 into eq 1 yields
δ(r) )
[(
) (
) ]
2Sb 1 1 r 1+ exp -1 2 4r/h + 1 3R 4r/h + 1
(9)
δ(r)/h functions determined by eq 9 for Cu and Ni are shown as a function of r in Figure 3. It is observed that Tolman’s length is positive for nanocavities and also decreases when the size is increased, being consistent with the trend of thermodynamics and other approach for liquid droplets.17-19 However, the value
Figure 3. δ(r)/h as a function of r in terms of eq 9 for Cu and Si.
Figure 4. Comparison of the r dependence of Tm(r)/Tm0 between the model predictions in terms of eq 7 (the solid line) and the results of MD simulation ([, b, and 4) for Ar.10
of δ(r)/h remains almost constant at r > 2 nm. Considering exp(x) ≈ 1 + x and the minimal value of r being h/4, the size of δ in eq 9 can be written as
[ ( ) ]
hSb/(12R) e δ e h 3 exp
Sb - 2 /16 3R
(10)
It is obvious that the size dependence of δ(r) strongly depends on the value of Sb. Figure 4 indicates the comparison of Tm(r)/Tm0 between the model predictions in terms of eq 7 (the solid line) and available results of MD simulation ([, 4, and b) for Ar where [, b, and 4 denote that the systems consist of 256 000, 108 000, and 32 000 atoms, respectively.10 Although it seems that the difference between the model predictions and the MD simulation results is large for the system with 32 000 atoms, the actual value is smaller than 10% at r > 1 nm. Even at the smallest radius of 0.58 nm, the difference is only 20%. Thus, it can be concluded that eq 7 can predict the size dependence of the melting temperature of nanocavities with good accuracy. The agreement shown in Figure 4 also in return confirms the validity of the assumption employed in the deduction of eq 6b. It is known that the curvature-induced compressive stress will build up on the surface of the inner wall of a nanocavity when the size of a nanocavity reduces to a nanometer range.3,11 This compressive stress will lead to the increase of intrinsic modulus in the cavity skin and a speeding up of the vibration of the surface dangling bonds,11,15 where the latter will thus increase the Debye temperature ΘD of a region surrounding the nanocavity. According to Lindemann’s criterion, ΘD ) c[Tm/ (MV2/3)]1/2,29 where c is a constant and M and V denote the molar weight and molar volume and can be assumed to be sizeindependent. Thus, the melting temperature will increase with the increase of Debye temperature since Tm(r) has the same
Surface Energy of Elemental Nanocavities size dependence of ΘD2(r) as a first-order approximation. On the other hand, the increase of the intrinsic modulus also suggests that Tm(r) should increase with decreasing of the size because two physical quantities characterize the bonding strength. Since the size dependence of the intrinsic modulus is reported to be directly proportional to that of the surface energy,15 the same size dependence for γsv(r) and Tm(r) is thus reasonable. It has been reported that cavity melting in four stages is governed by a unique mechanism that originated from the variations of and interplay between the following properties: surface energy; surface tension; solid-liquid interface energy; curvature of the void surfaces; elastic energy.10 Since all of surface energy, surface tension, solid-liquid interface energy, and elastic energy are related to the bond energy of surface atoms while the cohesive energy determines the size of bond strength, it is understandable and inevitable that eqs 6 and 7 developed according to the model as the size-dependent cohesive energy can describe the size dependence of the melting temperature of nanocavities. Conclusions A model established thermodynamically for size-dependent surface energy of elemental nanocavities shows that the surface energy increases with decreasing of the cavity size. Reasonable agreements between the model predictions and some other theoretical results in terms of eq 1 developed from Tolman’s equation are found for Cu and Si, which suggests that Tolman’s equation is applicable for elemental nanocavities. Moreover, an analytic model for the size-dependent melting temperature of elemental nanocavities is established for the first time, which indicates that the melting temperature also increases with decreasing of the size of cavities. The model predictions correspond to MD simulations results for Ar. The above agreements shown in Figures 2 and 3 for Cu with metallic bonds, Si with covalent bonds, and Ar with van der Waals forces suggest that our models should be applicable for nanocavities with different chemical bond natures. Different from the previous work on size dependence of surface energy of nanoparticles,21 surface tension of liquid droplets,19 and nucleus-liquid interface energy of elements,30 this work emphasizes the essential of cohesive energy to describe the effects of size on the thermodynamic quantities.
J. Phys. Chem. C, Vol. 111, No. 35, 2007 12917 Acknowledgment. The financial support from the State Key Program for Basic Research of China (Grant No. 2004CB619305), the Natural Science Foundation of Jiangsu Province (Grant No. BK2006716), the Postdoctoral Science Foundation of China (Grant No. 20070410326), and the Jiangsu Planned Project Funds for Postdoctoral Researchers is acknowledged. References and Notes (1) Gu, L.; Cheley, S.; Bayley, H. Science 2001, 291, 636. (2) Coyle, S.; Netti, M. C.; Baumberg, J. J.; Ghanem, M. A.; Birkin, P. R.; Bartlett, P. N.; Whittaker, D. M. Phys. ReV. Lett. 2001, 87, 176801. (3) Zhu, X. F.; Wang, Z. G. Int. J. Nanotechnol. 2006, 3, 492 and references therein. (4) Bertsch, G. Science 1997, 277, 1619. (5) Schmidt, M.; Kusche, R.; Issendorff, V. B.; Heberland, H. Nature 1998, 393, 238. (6) Jiang, Q.; Zhang, Z.; Li, J. C. Acta Mater. 2000, 48, 4791. (7) Solca, J.; Dyson, A. J.; Steinebrunner, G.; Kirchner, B. J. Chem. Phys. 1998, 108, 4107. (8) Velardez, G. F.; Alavi, S.; Thompson, D. L. J. Chem. Phys. 2004, 120, 9151. (9) Agrawal, P. M.; Raff, L. M.; Komanduri, R. Phys. ReV. B 2005, 72, 125206. (10) Bai, X. M.; Li, M. Nano Lett. 2006, 6, 2284. (11) Zhu, X. F. J. Phys.: Condens. Matter 2003, 15, L253. (12) Jiang, Q.; Lu, H. M.; Zhao, M. J. Phys.: Condens. Matter 2004, 16, 521 and references therein. (13) Moody, M. P.; Attard, P. Phys. ReV. Lett. 2003, 91, 056104. (14) Tolman, R. C. J. Chem. Phys. 1949, 17, 333. (15) Ouyang, G.; Tan, X.; Cai, M. Q.; Yang, G. W. Appl. Phys. Lett. 2006, 89, 183104. (16) Koga, K.; Zeng, X. C.; Shchekin, A. K. J. Chem. Phys. 1998, 109, 4063. (17) Bartell, L. S. J. Phys. Chem. B 2001, 105, 11615. (18) McGraw, R.; Laaksonen, A. J. Chem. Phys. 1997, 106, 5284. (19) Lu, H. M.; Jiang, Q. Langmuir 2005, 21, 779. (20) Jiang, Q.; Li, J. C.; Chi, B. Q. Chem. Phys. Lett. 2002, 366, 551. (21) Lu, H. M.; Jiang, Q. J. Phys. Chem. B 2004, 108, 5617. (22) Tyson, W. R.; Miller, W. A. Surf. Sci. 1977, 62, 267. (23) Galanakis, I.; Papanikolaou, N.; Dederichs, P. H. Surf. Sci. 2002, 511, 1. (24) Vitos, L.; Ruban, A. V.; Skriver, H. L.; Kollr, J. Surf. Sci. 1998, 411, 186 and references therein. (25) Periodic Table of the Elements; Sargent-Welch Scientific Co.: Skokie, IL, 1980; p 1. (26) Ouyang, G.; Tan, X.; Yang, G. W. Phys. ReV. B 2006, 74, 195408. (27) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (28) Jiang, Q.; Shi, H. X.; Zhao, M. J. Chem. Phys. 1999, 111, 2176. (29) Dash, J. G. ReV. Mod. Phys. 1999, 71, 1737. (30) Lu, H. M.; Wen, Z.; Jiang, Q. Colloids Surf., A 2006, 278, 160.
