Size-Dependent Thermodynamic Properties of Metallic Nanowires

9444

J. Phys. Chem. B 2008, 112, 9444–9448

Size-Dependent Thermodynamic Properties of Metallic Nanowires H. M. Lu, F. Q. Han, and X. K. Meng* National Laboratory of Solid State Microstructures, Department of Materials Science and Engineering, Nanjing UniVersity, Nanjing 210093, P. R. China ReceiVed: April 3, 2008; ReVised Manuscript ReceiVed: May 20, 2008

Analytical models for size-dependent melting temperature Tm(D), melting enthalpy ∆Hm(D), and surface energy γsv(D) of metallic nanowires have been proposed in terms of the unified nanothermodynamical model where D denotes the diameter of nanowire. As D decreases, Tm(D), ∆Hm(D), and γsv(D) functions are found to decrease almost with the same size-dependent trend. Due to the inclusion of the effect of dimensionality, the developed model can be applied to other low-dimensional systems. It is found that the ratio of depression of these thermodynamic parameters for spherical nanoparticle, nanowire, and thin film is 3:2:1 when D is large enough (>20h with h being the atomic diameter). The validity of the model is verified by the data of experiments, molecular dynamics simulations, and other theoretical models. Introduction Recently, one-dimensional nanostructures such as wires, rods, belts, and tubes have become the focus of intensive research due to their unique applications in mesoscopic physics and fabrication of nanoscale devices. As one of the most important one-dimensional nanostructures, nanowires provide a good system to investigate the dependence of electrical and thermal transport or mechanical properties on dimensionality and size reduction (or quantum confinement).1–3 They are also expected to play an important role as both interconnects and functional units in fabricating electronic, optoelectronic, electrochemical, and electromechanical devices with nanoscale dimensions.3,4 To implement these nanoscale building blocks for nanoscale electronics,5,6 their thermodynamic properties have to be carefully determined because of their extremely fine sizes and high surface areas. The melting temperature of nanowires Tm, which is one of the basic properties of materials, has exhibited dramatically different character from their bulk counterparts both experimentally and theoretically.4,7–20 According to the thermodynamic model, the melting temperature of nanowires can be described as8

[ () ]

Tm(D) Fs 2 γsv )1Tm(∞) DFs∆Hm(∞) Fl

1⁄2

γlv

(1a)

where ∞ denotes the infinite size, D is the diameter of nanowires, ∆Hm is the melting enthalpy, γsv and γlv are the interfacial energies of solid-vapor and liquid-vapor interfaces (or called surface energy and surface tension), and Fs as well as Fl are the densities of solid and liquid, respectively. However, similar to the size dependences of ∆Hm, γsv, and γlv found in the nanoparticles,21–25 size dependences of these functions should also be considered for nanowires and thus included in eq 1a when D reduces to several nanometers.4,16 Moreover, the phonon-instability model and liquid-drop model deduced the following expression to determine the Tm(D) function of the nanowire7,12 * To whom correspondence should be addressed. Phone: +86-025-83685585. Fax: +86-025-8359-5535. E-mail: [email protected].

Tm(D) 2δ )1Tm(∞) 3D

(1b)

with δ being a material parameter. However, these two models gave different values of δ even for the same metals.7,12 Using differential scanning calorimetry the melting behavior of Zn nanowires was found to be curvilinear with the reciprocal of the diameter of nanowires, which was explained to result from the size-dependent melting enthalpy.16 However, Goswami and Nanda thought that this interpretation was incorrect.18 Moreover, they employed a one-dimensional model and zerodimensional model to explain the depressions of melting temperatures for Zn nanowires with diameters less than 65 nm and above 65 nm, respectively.18 It is noted that the zerodimensional model is usually used to describe the material with the size of three dimensions in the nanoscale, such as quantum dots; it is thus not suitable for nanowires with a large diameter (about 65 nm) and long length (about 100 µm).19 As a result, it is necessary to develop a quantitative thermodynamical model to determine the size dependence of the melting temperature and deepen our understanding of its nature. On the other hand, surface energy γsv, defined as the energy needed to cut a given crystal into two halves or energy consumed upon surface formation, is also one of the most important parameters of nanowires. The surface energy plays a key and central role in surface and nanosolid sciences because it links the microscopic bonding configuration at an interfacial region with its macroscopic properties, such as strength, wettability, diffusivity, thermal stability, etc.4,8,26 However, no experimental data on the surface energy of nanowires are available to our knowledge. Note that even the bulk surface energies of metals are also difficult to experimentally determine.27 Recently, a model28 established for the size-dependent cohesive energy E(D) has been extended to determine the sizedependent surface energies of nanoparticles23 and nanocavities29 where reasonable agreements could be found. In this contribution, this model, originally proposed for zero-dimensional nanoparticles, was extended to describe the size-dependent surface energy, melting temperature, and melting enthalpy of metallic nanowires through considering the effect of dimensionality. The accuracy of the developed model was verified

10.1021/jp802888t CCC: $40.75  2008 American Chemical Society Published on Web 07/15/2008

Properties of Metallic Nanowires

J. Phys. Chem. B, Vol. 112, No. 31, 2008 9445 R′ - R′1/2)]/(2 + 2β)}E(∞), where the prime denotes the nextnearest neighbors of the surface atoms and β is the total bond strength ratio between the next-nearest and the nearest neighbors.27 Although these three expressions are different, it can be found that the surface energy is directly related to the cohesive energy with the following form27,33–35

γsv(∞) ≈ nE(∞)

(2c)

where the parameter n is a constant as a function of the coordination number as described above. When the nanowires have the same structure of the corresponding bulk,4,8,14–16 R can be assumed to be size-independent as a first-order approximation. In this case, eqs 2a, 2b, and 2c may be extended to nanometer size with the same form Figure 1. E(∞) as a function of (a) Tm(∞) and (b) ∆Hm(∞) for metals.

E(D) ≈ kTm(D)

(3a)

E(D) ≈ m∆Hm(D)

(3b)

γsv(D) ≈ nE(D)

(3c)

by the experimental, molecular dynamics simulation, and other theoretical results.

and

Methodology

It is obvious that Tm(D), ∆Hm(D), and γsv(D) of nanowires can be calculated as long as E(D) is determined. Jiang et al. proposed the following expression to describe the size-dependent cohesive energy of nanoparticles28

It is known that the melting temperature and cohesive energy are two parameters to describe the bond strength.17 On the basis of Lindemann’s criterion of melting, the melting temperature is linear to the force constant of the lattice vibration where the latter can be expressed by the cohesive energy.30 As a result, it is remarked that the melting temperature of a solid is directly proportional to its cohesive energy.30 Guinea et al. also deduced the linear relation between these two parameters according to the Debye model at high temperature.31 As shown in Figure 1a, the bulk cohesive energy is plotted against the bulk melting temperature of metals where the values of both quantities are taken from ref 32. In this case, the relation between Tm(∞) and E(∞) can be linearly regressed as

E(∞) ≈ kTm(∞)

(2a)

where the linearly regressed slope k is equal to 0.24 kJ/mol-K with the correlation coefficient of the fit being 0.96. Similarly, a relation between the melting enthalpy and the cohesive energy should also exist since the melting enthalpy is also directly proportional to the bond strength. As shown in Figure 1b, the bulk cohesive energy is plotted as a function of the melting enthalpy of metals where the linear relation between them can be regressed as

E(∞) ≈ m∆Hm(∞)

(2b)

where the linearly regressed slope m ) 25 and the correlation coefficient of the fit is about 0.97. As for the bulk surface energy, also defined as the energetic difference between surface atoms and interior ones, Tyson and Haiss et al. related it to the multiplication of the broken bond number with the cohesive energy per bond, namely, γsv(∞) ≈ (1 - R)E(∞), with R being the ratio of the coordination number of surface atoms to that of the interior ones.33,34 A secondmoment tight-binding approximation conducted by Desjonque`res and Spanjaard suggested that the γsv(∞) value was proportional to the square root of the number of the nearest-neighboring broken bonds due to the lowering of the occupied states, i.e., γsv(∞) ≈ (1 - R1/2)E(∞).35 To obtain a more general expression, Jiang et al. proposed that an average of the approximations of Tyson and Desjonque`res and an extension to counting the contribution of the second nearest neighbors could be more comprehensive, namely, γsv(∞) ≈ {[(2 - R - R1/2) + β(2 -

) (

2Sb(∞) 1 E(D) 1 exp ) 1E(∞) 2D ⁄ h - 1 3R 2D ⁄ h - 1

(

)

(4a)

where h denotes the atomic diameter, R is the ideal gas constant, and Sb(∞) ) ∆Hb(∞)/Tb(∞) is the bulk solid-vapor transition entropy with ∆Hb(∞) and Tb(∞) being the bulk enthalpy of vaporization and bulk boiling temperature, respectively. It is noted that eq 4a was developed for zero-dimensional nanoparticles, while nanowires are one-dimensional; eq 4a thus needs modification before it is applied to nanowires. Recently, Yang et al. proposed a unified model to describe the sizedependent cohesive energy of nanocrystals through considering the effect of dimensionality36

) (

(

2Sb(∞) 1 E(D) 1 exp ) 1E(∞) 12D ⁄ D0 - 1 3R 12D ⁄ D0 - 1

)

(4b) where D0 is a critical size at which all atoms of crystal are located on its surface, which can be determined as (1) D0 ) 6h for nanoparticles with dimensionality d ) 0 since 4π(D0/2)2h ) 4π(D0/2)3/3, (2) D0 ) 4h for nanowires with d ) 1 since 2π(D0/2)h ) π(D0/2)2, and (3) D0 ) 2h for thin films with d ) 2. In short, the correlation between D0 and h is given by

D0 ) 2(3 - d)h

(5)

Evidently, eq 4b combined with eq 5 with d ) 0 for nanoparticles leads to eq 4a. Combining eqs 2a, 2b, 2c, 3a, 3b, 3c, and 4b, a univeral relation can be obtained as

Tm(D) ∆Hm(D) γsv(D) ≈ ≈ ≈ Tm(∞) ∆Hm(∞) γsv(∞)

(

1-

) (

2Sb(∞) 1 1 exp 12D ⁄ D0 - 1 3R 12D ⁄ D0 - 1

)

(6)

On the basis of eqs 1b and 6, the material parameter δ is determined as

9446 J. Phys. Chem. B, Vol. 112, No. 31, 2008

δ(D) ≈

[ (

)

3D 1 1- 1× 2 12D ⁄ D0 - 1

(

exp -

2Sb(∞) 1 3R 12D ⁄ D0 - 1

Lu et al.

)]

(7)

Results and Discussion Figure 2a shows the calculated Tm(D) function (solid line) of Zn nanowires in terms of eq 6 with necessary parameters listed in Table 1. It is evident that Tm(D) decreases with decreasing diameter while the drop is slow because the diameter is large (for example, the drop is only 11 K when D decreases from 250 to 37 nm). As a comparison, available experimental results (closed symbols) and predictions of eq 1b with different δ values (dashed lines) are also listed noted and the difference between these two δ values is small.7,12,16 As shown in Figure 2a, the model prediction corresponds well to eq 1b while slightly deviating from the experimental data. Figure 2b compares the Tm(D) function of Ni nanowires among the model prediction based on eq 6, available molecular dynamics (MD) simulation results,14,15 and the calculations of eq 1b,7,12 where good accordances can also be found. Obviously, Tm(D) decreases dramatically, and the drop is faster than that presented in Figure 2a. Note that the triangular symbols correspond to the heating rate of 1010 K/s rather than 2 × 1016 K/s.15 It has been reported that the heating rate has a great influence on the melting temperature of nanowires and that the Tm(D) value at a low heating rate is always smaller than that at a high heating rate.15 Li et al. ascribed the melting to diffusion of the local clusters, which would lead to some new defects in nanowires; it was noted that these defects would cause the decrease of the average binding energy and thus accelerate the melting.15 Moreover, the influence of the elastic energy is not negligible, which is important for a liquid nucleating within a solid.38 For example, migration of the liquid droplets to the grain boundaries is clearly a stress-driven event.39 The elastic energy increases with increasing heating rate, which means that the system temperature needs to be increased to overcome the elastic energy. However, the elastic energy is not involved in nanoparticles melting except when they are surrounded by the matrix with different elastic moduli. This is an important difference between nanoparticle melting and nanowire melting. A comparison of the Tm(D) function for Pb nanowires among the predictions of eqs 1b and 6 and MD simulation results8 is shown in Figure 2c. It can be found that the model prediction is in agreement with MD simulation results and that of eq 1b with δ ) 0.98 nm,7 while the calculation of eq 1b with δ ) 1.80 nm12 is evidently smaller than the results of the former three results. Similarly, the prediction of eq 1b with δ ) 1.80 nm for Pb nanoparticles is also smaller than the corresponding experimental data in the intermediate size range, although there is agreement for particles with lower and higher sizes.12 The authors explained that it was because the shape of the Pb nanoparticles resembled a disk and only one-half of the surface was free due to the existence of a substrate.12 Moreover, they claimed that Tm(D) in the intermediate size range could be accounted for by taking the value of δ to be one-half the value of 1.80 nm.12 However, in this case, the calculated result is obviously larger than the experimental data at D < 7 nm, as shown in Figure 3b of ref . To date, we cannot find the reason for the discrepancies among the liquid-drop model, the phononinstability model, eq 6, MD simulations, and experiments for Pb nanoparticles and nanowires. In contrast, the prediction of

Figure 2. Comparison of Tm(D) function described by various models and the corresponding experimental as well as MD simulation results for (a) Zn, (b) Ni, (c) Pb, and (d) Pd nanowires with D0 ) 4h according to eq 5.4,7,8,11,12,14–16

the liquid-drop model is consistent with the experimental results and theoretical calculation of Jiang et al. for Pb thin films.9 A similar comparison is also made for Pd nanowires as shown in Figure 2d where the prediction of eq 6 corresponds well to

Properties of Metallic Nanowires

J. Phys. Chem. B, Vol. 112, No. 31, 2008 9447

Figure 3. δ(D)/h as a function of D in terms of eq 7 for Pb, Pd, Ni, and Zn nanowires.

Figure 4. Comparison of ∆Hm(D) of Pb and Pd nanowires described by eq 6 and the corresponding MD simulation results.4,8

TABLE 1: Several Necessary Parameters Used in the Equations Where the Value of γsv(∞) Is Taken from Ref 4 while Others Are Cited from Ref 37

Zn Ni Pb Pd

h (nm)

∆Hm(∞) (kJ/mol)

Tm(∞) (K)

∆Hb (∞)

Tb (∞)(K)

Sb(∞) (J/mol-K)

γsv(∞) (J/m2)

0.267 0.249 0.350 0.275

7.3 17.2 4.8 16.7

693 1728 601 1828

115 370 178 357

1180 3187 2023 3237

97 116 88 110

1.81

those of eq 1b and MD simulation results.4 However, the reported experimental result11 of Tm(D ≈ 4.6 nm) ) 573 K is significantly lower than the MD simulation results4 of 1200 K at a smaller size of 2.6 nm and the predictions of both eqs 1b and 6. The cause of this dramatic depression remains unknown and also merits further investigation. The agreement shown in Figure 2a-d indicates that eq 6 can satisfactorily describe the size-dependent melting temperature of metallic nanowires. Since it is claimed that eq 1b can also be used to model the Tm(D) function, eq 7 derived from eqs 1b and 6 should also be able to determine the value of δ. δ(D)/h functions determined by eq 7 for Pb, Pd, Ni, and Zn nanowires are shown in Figure 3, where the material parameter is always positive and drops with decreasing size, which is contrary to the size dependence of the Tolman length for nanodroplets.24 Note that the δ(D)/h value remains almost constant at D > 5 nm. Considering the mathematical approximation of exp(-x) ≈ 1 - x when x is small enough (e.g., x < 0.1)

δ(D) Sb(∞) e h 3R

(8)

It is obvious that the size dependence of δ(D) strongly depends on the value of Sb(∞). Since the average value of Sb(∞)/R is about 12 for metallic elements as listed in Table 1, the mean of δ(∞) is thus about 4h, which is just the D0 value for nanowires as described by eq 5. In terms of eq 8, δ(∞) values of Pb, Pd, Ni, and Zn nanowires are, respectively, determined as 1.23, 1.21, 1.16, and 1.04 nm (or 3.5h, 4.4h, 4.6h, and 3.9h), which are in accordance with the corresponding values given by the phononinstability model and the liquid-drop model.7,12 The difference bewteen 4h and the calculated δ(∞) values originates from the deviation of Sb(∞) from 12R. Moreover, the ∆Hm(D) functions of Pb and Pd nanowires are plotted in Figure 4 according to eq 6 where available MD simulation results are also listed for comparison.4,8 It can be found that the model predictions correspond well to MD simulation results for Pb nanowire, while they are larger than that for Pd nanowires.

Figure 5. γsv(D) function of Pd nanowires in terms of eq 6, where the symbol is a MD simulation result.4

Figure 5 shows the comparison of the γsv(D) function of Pd nanowires between the prediction of eq 6 and MD simulation results,4 where the deviation is only 6%. Usually the surface energy is considered to consist of a chemical part γchem and a structural part γstru, namely, γsv ) γchem + γstru, where the former is induced by the surface atom dangling bond energy while the latter results from the surface stress derivation.20,25 It is noted that γstru is one or two orders smaller than γchem at several nanometer sizes;25 γstru is thus negligible as a first-order approximation. Moreover, the agreement shown in Figure 5 also indicates that this neglect does not result in a big error. As mentioned above, not only the surface energy of metallic nanowires but also that of bulk metals are difficult to experimentally determine;27 there are thus no other experimental data about the size-dependent surface energy of nanowires.4,8,10,16 As a result, no comparison for other metallic nanowires is shown in Figure 5. The agreement shown in Figures 2, 4, and 5 not only indicates that eq 6 can satisfactorily describe the Tm(D), ∆Hm(D), and γsv(D) functions of metallic nanowires but also suggests the physical origin of these size dependences: The coordination number and strength of single bond change with decreasing size, which results in that the atomic cohesive energy or the heat required to loosen the surface atoms being different from the bulk value. Such a difference is responsible for the depression of these thermodynamical properties.40 Considering the approximation of exp(-x) ≈ 1 - x again when x is small enough (e.g., x < 0.1 or D > 20h), eq 6 can be rewritten as

9448 J. Phys. Chem. B, Vol. 112, No. 31, 2008

Tm(D) ∆Hm(D) γsv(D) Sb(∞)D0 ≈ ≈ ≈1Tm(∞) ∆Hm(∞) γsv(∞) 18RD

Lu et al.

(9)

Equation 9 is in accordance with the general consideration that the change of the any size-dependent thermodynamic quantity is proportional to 1/D.41 However, as D decreases to a size comparable with the atomic diameter, namely, about several nanometers, the difference between eq 6 and eq 9 becomes evident. This is because the energetic state of interior atoms of the nanocrystals in a small size is higher than that of the corresponding bulk crystals.28 This change can even lead to the appearance of different structures in comparison with the corresponding bulk ones.42,43 Moreover, eq 6 should also be applied to other lowdimensional systems noted that eq 4a is the case of eq 4b with D0 ) 6h. Combining eqs 5 and 9 for a given size of the spherical particle, diameter of the wire, and thickness of the film, one can obtain

X(∞) - X(D) X(∞) - X(D) X(∞) - X(D) film : wire : sphere ≈ X(∞) X(∞) X(∞) 1 : 2 : 3 (10) where X denotes the thermodynamical function of γsv, Tm, or ∆Hm. Equation 10 suggests that the ratio of depression of the melting temperature for free-standing thin film, nanowire, and spherical nanoparticle (or, at least, when they are only weakly bound to a substrate) is 1:2:3, which is in agreement with the predictions of other thermodynamical considerations and the liquid-drop model.12,44 Conclusion The size dependences of Tm(D), ∆Hm(D), and γsv(D) functions are modeled for metallic nanowires. The values of these thermodynamical parameters decrease with decresing size, and the drop becomes dramatic at D < 5 nm. The model predictions are in accordance with the corresponding results of experiments, molecular dynamics simulations, and other theoretical models. These agreements enable us to determine the size dependence of the material parameter δ(D), which increases with increasing size and reaches the maximum value of about 4h. Because the effect of dimensionality is included, the model or eq 6 can also be applied to the nanoparticles and thin films. It is found that the ratio of depression of these three parameters for a spherical nanoparticle, nanowire, and thin film is found to be 3:2:1 when the size is large enough (e.g., D > 20h). Together with previous work on the size dependences of melting temperatures of nanosolids and nanocavities,12,29,40 surface energy of nanoparticles and nanocavities,23,29 surface tension of liquid droplets,24 nucleus-liquid interface energy of elements,45 and thermal conductivity and diffusivity of nanostructural semiconductors,36 the work here indicates the essentiality of cohesive energy, or eq 4b, in describing the effects of size on the thermodynamical properties of different lowdimensional systems. Acknowledgment. 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 Projects for PostdoctoralResearchFunds(grantno.0701029B)areacknowledged.

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