Size-, Shape-, and Dimensionality-Dependent Melting Temperatures

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Size-, Shape-, and Dimensionality-Dependent Melting Temperatures of Nanocrystals H. M. Lu, P. Y. Li, Z. H. Cao, 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: January 12, 2009; ReVised Manuscript ReceiVed: March 10, 2009

On the basis of a model for size-dependent cohesive energy, the size, shape, and dimensionality effects on melting temperatures of nanocrystals are modeled in a unified form. The model predicts that the melting temperature Tm(D,d,λ) decreases with reducing size D and dimensionality d or increasing shape factor λ. For nanoparticles with the same D values, there is Tm(icosahedron) > Tm(sphere or cube) > Tm(octahedron) > Tm(tetrahedron). Moreover, the ratio of depression of Tm(D,d,λ) is about 1:2λwire:3λparticle for thin films, nanowires, and nanoparticles when D is large enough, for example, 6 nm. The model is found to be in accordance with available experimental, MD simulation, and other theoretical results for Au, Ag, Ni, Ar, Si, Pb, and In nanocrystals. Introduction Nanocrystals are under considerable investigation worldwide because of their wide scientific and technological interest. Because of the unique properties of nanocrystals, the fabrication of nanostructural materials and nanodevices with desirable properties in atomic scale has become an emerging interdisciplinary field involving solid-state physics, materials science, chemistry, and biology.1 In this case, it is important and necessary to understand and predict the thermodynamics of nanocrystals for fabricating the materials for practical applications. It is well-known that nanocrystals are intrinsically characterized by a large ratio of the number of surface-to-volume atoms, which modifies some of the basic material properties. The most notable example different from the corresponding bulk thermodynamic behavior is probably the depression of the melting temperature of nanocrystals. Since the pioneering work of Pawlow in 1909,2 the variation of the melting temperature has been studied by experiments,3-10 molecular dynamics,11-21 and thermodynamics.22-35 A lot of thermodynamic and other theoretical models of nanoparticles melting assume spherical shape and yield a linear relationship between the melting temperature and the reciprocal of the particle size:2,22,27,30,32,34

Tm(D) ) Tm0(1 - β/D)

(1)

where Tm(D) and Tm0 are, respectively, the melting temperatures of the nanocrystal and the bulk with D being the diameter. β is a material constant, and many phenomenological models have been proposed to determine it.22,27,30,32,34 However, the quantities involved in the expression for estimating β may not be easy to evaluate, and even different models give different β values for the same metals.22,27 On the other hand, the shape of nanocrystals also affects the melting temperature of nanocrystals because the ratio of the atomic number at the surface to that in the volume is related to the shape, noting that the physical origin for the depression of melting temperature is thought of as the enormous ratio of the number of surface-to-volume atoms.11 It has been experimentally demonstrated that the change of the shape resulted in a * To whom correspondence should be addressed. Tel.: +86-025-83685585. Fax: +86-025-8359-5535. E-mail: [email protected].

considerable amount of depression in melting temperature.5 Moreover, through introducing the concept of a shape factor, some thermodynamic models have been proposed to describe the dependence on the shape. For example, Qi et al. have investigated the melting temperature of nanocrystals in some confined systems such as films and wires and further extended it to the nanoparticles of disk and sphere.28,29 However, a detailed study is lacking on many other shapes, which is important to understand the phenomena of the melting and to correlate the melting temperature of different shapes. Recently, Jiang et al. have proposed the following expression to determine the size-dependent cohesive energy of spherical nanoparticles:36

) (

2Sb0 E(D) 1 1 ) 1exp E0 2D/h - 1 3R 2D/h - 1

(

)

(2)

where h denotes atomic diameter, R is the ideal gas constant, and Sb0 ) ∆Hb0/Tb0 is the bulk solid-vapor transition entropy with ∆Hb0 and Tb0 being the bulk enthalpy of vaporization and the bulk boiling temperature, respectively. This model has been extended to determine the size-dependent melting temperature of zero-dimensional nanocavities where reasonable agreement could be found.37 In this Article, the model developed originally for zerodimensional spherical nanoparticles was extended to describe the size-, shape-, and dimensionality-dependent melting temperatures of nanocrystals through considering the effects of dimensionality and the ratio of surface atoms to the total atoms. The validity of the model is verified by the data of experiments, molecular dynamics simulations, and other theoretical results. Model It is known that the melting temperature and the cohesive energy are two parameters to describe the bond strength.27-29,31,32 Based on 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:38

Tm0 ) RE0

(3)

with R ) nf /(3kBZ), where n is the exponent of the repulsive part of the interaction potential between constituent atoms, f is 2

10.1021/jp900314q CCC: $40.75  2009 American Chemical Society Published on Web 04/08/2009

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the ratio of the atomic displacement at Tm0 to the interatomic separation at equilibrium, kB is the Boltzmann constant, and Z is the valency of the atoms. As eq 3 indicates, the melting temperature of a solid is directly proportional to its cohesive energy. Moreover, a similar linear relationship between these two parameters has also been deduced according to the Debye model at high temperature,39 the liquid drop model,27 and the bond order-length-strength correlation mechanism,25,26 respectively. When the nanocrystals have the same structure of the corresponding bulk,11 R can be assumed to be size-independent as a first-order approximation. In this case, eq 3 may be extended to nanometer size with the same form:

Tm(D) ≈ RE0(D)

(4)

Obviously, the Tm(D) function can be determined through combining eqs 2-4. However, noting that eq 2 was developed for zero-dimensional spherical nanoparticles, eq 2 thus needs modification before it is applied to other nanocrystals with different dimensionality and shape. Combining eqs 2-4 and the considerations of the shape effect on the ratio of surface atoms to the total atoms and the dimensionality effect, a unified model to describe size-, dimensionality-, and shape-dependent melting temperature of nanocrystals can be developed as

(

)

Tm(D, d, λ) E(D, d, λ) 1 ≈ ≈ 1× Tm0 E0 12D/D0 - 1

(

exp -

)

2λSb0 1 (5) 3R 12D/D0 - 1

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 spherical nanoparticles with dimensionality d ) 0 because 4π(D0/2)2h ) 4π(D0/2)3/3; (2) D0 ) 4h for cylindrical nanowires with d ) 1 because 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

(6)

It is evident that eq 5 combined with eq 6 with d ) 0 for spherical nanoparticles is the same as eq 2. λ, describing the shape effect on the ratio of surface atoms to the total atoms δ, can be determined as the ratio of δ between nanocrystals with other shape and those with basal shape (e.g., spherical nanoparticles and cylindrical nanowires for zero- and one-dimensional nanocrystals, respectively). To calculate the δ and λ values, the numbers of surface atoms and the total atoms should be determined first. To calculate the total number of the atoms N of the nanocrystals, the effect of lattice packing density ηL, which denotes the ratio of the volume of crystal occupied by atoms to the total volume, should be taken into account. ηL for some standard crystal structures such as face-centered cubic, bodycentered cubic, simple cubic, and hexagonal closed-packed structures can be found in the literature.40 In this case, N can be determined as

N ) ηLVC/Va

(7)

where VC and Va denote the volumes of nanocrystals and atoms. Va is equal to πh3/6 if the atom is regarded as ideal sphere, and VC values for nanocrystals with different shapes can be found in Table 1.

TABLE 1: Several Necessary Parameters Used in the Equations AC spherical nanoparticles regular tetrahedral particles cubic particles regular octahedral particles regular icosahedral particles cylindrical nanowire cubic nanowires hexagonal nanowires thin films

2

VC

λ (eq 10)

πD 31/2D2

3

πD /6 21/2D3/12

1 2.45

6D2 121/2D2

D3 21/2D3/3

1 1.23

751/2D2

5 × (3 + 51/2)D3/12

0.66

πDla 4Dl 6Dl

πD2l/4 D2l 271/2D2l/2

1 1 0.58 1

a Note that only lateral surface area is considered in calculating AC of nanowires because the length l of nanowires is assumed to be much larger than D. In this case, the basal surface area is negligible.

Similarly, the effect of surface packing density ηS, describing the ratio of the surface area occupied by atoms to the total surface area, should also be considered when determining the number of surface atoms n of the nanocrystals. For instance, the surface areas of the two-dimensional unit cell for (111) and (100) planes of face-centered cubic structure are 31/2h2/2 and h2, noting that the surface area Aa of single atom is πh2/4, and then ηS values for (111) and (100) planes are about 0.907 and 0.785, respectively. ηS for others can be calculated in the same way. Thus, one can determine n as follows:

n ) ηSAC/Aa

(8)

where AC is the surface area of nanocrystals, which can also be found in Table 1. In terms of eqs 7 and 8, δ can be described as

δ ) n/N ) (ηS/ηL)(ACVa/AaVC)

(9)

Assuming that the nanocrystals with other shapes take the same surface and lattice packing modes as those with basal shapes (e.g., spherical or cylindrical for zero- and onedimensional nanocrystals, respectively), ηS and ηL can be assumed to be shape-independent as a first-order approximation. In this case, λ can be written as

λ ) δ2/δ1 ) (AC2/AC1)(VC1/VC2)

(10)

where the subscripts 1 and 2 denote the nanocrystals with basal shapes and other shapes. According to the definition of λ, λ is equal to 1 for spherical nanoparticles, cylindrical nanowires, and thin films. Note that Va and Aa terms disappear in eq 10 because atomic diameter remains constant for the same nanocrystals with different shapes. Moreover, although the ηL term also disappears in eq 10, it is important to determine the grain size in terms of eq 7. Combining eqs 5, 6, and 10, size-, dimensionality-, and shapedependent melting temperatures of nanocrystals can be determined. Results and Discussion Figure 1 shows the comparisons between model predictions in terms of eq 5 with necessary parameters listed in Table 1 and experimental data,4,9,10,15 molecular dynamics (MD) simulation,19 as well as other theoretical results33 for Tm(D) of (a) icosahedral, (b) spherical, and (c) tetrahedral Ag nanoparticles. Note that the temperatures at which that Ag nanoparticles completely desquamate from the substrate are taken as the corresponding experimental results for spherical nanoparticles.10

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Figure 1. Comparisons of Tm(D) functions for (a) icosahedral, (b) spherical, and (c) tetrahedral Ag nanoparticles described by eq 5, and available experimental data,4,9,10,15 MD simulation results,19 and other theoretical calculations,33 where Tm0 ) 1235 K, Sb0 ) 104.7 J/mol · K, and h ) 0.289 nm.41

It is evident that our model predictions are in accordance with the corresponding experimental, MD simulation, and other theoretical results. Moreover, Tm(D) function is found to decrease with decreasing D, and it seems that the depression tendency of Tm(D) function takes the following sequence: ∆Tm(tetrahedron) > ∆Tm(sphere) > ∆Tm(icosahedron), at the same diameter or length with ∆Tm ) Tm0 - Tm(D) denoting the difference. Figure 2 compares the Tm(D) functions of spherical and icosahedral Au nanoparticles between the model predictions based on eq 5 and available experimental data4,6 as well as MD simulation results,13,16,18,42 where good agreements can also be found. The similar comparisons and accordances between model predictions in terms of eq 5 and MD simulation results14,21 for Tm(D) of spherical and icosahedral Ni nanoparticles are also displayed in Figure 2. As shown in Figure 2, Tm(D) function also decreases with decreasing diameter, and the drop becomes dramatic at D < 5 nm. Because no special assumption is made in the deductions of the model, eq 5 should be not only suitable for metallic nanoparticles but also applicable to materials with other types of bonds. Figure 3 shows the Tm(D) functions of (a) tetrahedral Si nanoparticles, (b) spherical, and (c) icosahedral Ar nanoparticles in terms of eq 5. As comparisons, available experimental data8,43 and MD simulation results12,44 are also listed in Figure 3. Although the shapes of Si nanoparticles are not mentioned definitely in the corresponding experiments,8,43 the agreements shown in Figure 3a suggest that Si nanoparticles may take the shape of tetrahedron. Note that the “O” and “b” in Figure 3c denote the melting temperatures of icosahedral Ar nanoparticles obtained using heating rates of 100 and 2 K/ns, respectively.12 It has been reported that the heating rate has a great influence on the melting temperature, and the Tm(D) value at a low heating rate is always smaller than that at a high heating rate.45 Li et al. ascribed the melting to diffusion of the local clusters, which would lead to some new defects in nanocrystals, noting that these defects would cause the decrease of the average binding energy and thus accelerate the melting.45 Moreover, the influence of the elastic energy is not negligible. The elastic energy increases with increasing heating rate, which means that the system temperature needs to be increased to overcome the elastic energy. Because our model is based on thermodynamics, the comparison with the lowest heating rate utilized in simulation is appropriate. As shown in Figure 3c, the model predictions are in agreement with MD simulation results (b) with low

Lu et al. heating rate. The same correspondence is also shown in Figure 3b for spherical Ar nanoparticles. As mentioned above, Tm(D,λ) functions are different for nanoparticles with different shapes. It is understandable because the shape determines the number of surface atoms whose reduced coordination number will affect the bond energy and thus the melting temperature. As shown in Figure 4, taking Au nanoparticles as an example, the size and shape effects on Tm(D) are clearly displayed in terms of eq 5. It can be found that Tm(D,λ) increases with increasing D or decreasing λ. Because λtetrahedron > λoctahedron > λsphere or cube > λicosahedron as listed in Table 1, the results of Tm(D, icosahedron) > Tm(D, sphere or cube) > Tm(D, octahedron) > Tm(D, tetrahedron) shown in Figures 1-3 are reasonable and understandable. Moreover, the correspondences shown in Figures 1-3 also indicate that our method in determining λ is right. Although the definitions and corresponding values of λ are different, our results of λtetrahedron/λsphere ≈ 2.45 for nanoparticles and λhexagon/λcylinder ≈ 0.58 for nanowires are nearly the same as those determined by Guisbiers et al.,34 noting that L in eq 2 of ref 34 to determine λ represents the radius for spherical structures or the length for nonspherical structures. These agreements confirm the validity of eq 10 again. In terms of eq 5, Tm(D) values are equal for spherical and cubic nanoparticles with the same D values when cubic nanoparticles take the same surface and lattice packing modes as spherical ones. However, when the total atomic numbers of spherical and cubic nanoparticles are the same, Vsphere ) Vcube, Tm(D) of spherical nanoparticles is larger than that of cubic ones due to Dsphere/Dcube ) (6/π)1/3 > 1. A similar relationship has been experimentally observed in the Curie temperature.46 Figures 1-4 show the size and shape effects on the melting temperature of nanoparticles. However, similar comparisons are not given for nanowires because no experimental or MD simulation results are available for nanowires with cross sections other than cylinder. Figure 5 shows the size and dimensionality effects on the melting temperature of Pb nanocrystals. It can be found that Tm(D) of spherical Pb nanoparticles decreases with decreasing size, and the drop becomes dramatic when D is below 15 nm. However, the sharp decrease happens at D < 10 nm for cylindrical nanowires. As shown in Figure 5, the model predictions are in agreement with the corresponding experimental and MD simulation results.3,11,47 Moreover, the Tm(D) functions of spherical In nanoparticles, cylindrical In nanowires, and In thin films are plotted in Figure 6 where the corresponding experimental results are also listed for comparison.3,47-51 It can be found that the model predictions correspond well to the experimental result for In nanocrystals, and the depression tendency of the Tm(D) function takes the sequence of ∆Tm(spherical nanoparticles) > ∆Tm(cylindrical nanowires) > ∆Tm(thin films) at the same D value. Considering the approximation of exp(-x) ≈ 1 - x when x is small enough (x < 0.1 or D > 20h), eq 5 can be rewritten as

Tm(D, d, λ) (3 - d)hSb0λ ∆Tm(D, d, λ) ≈1≈ or Tm0 9RD Tm0 (3 - d)hSb0λ (11) 9RD It is obvious that Tm(D,d,λ) decreases with increasing λ or decreasing D and d. Note that d and λ values range from 0 to 3 while D can vary from several to hundred nanometers, which suggests that the size effect is the principal factor while the

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Figure 2. Comparisons of Tm(D) functions for spherical and icosahedral Au and Ni nanoparticles between the predictions in terms of eq 5 and available experimental4,6 as well as MD simulation results,13,14,16,18,21,42 where Tm0, Sb0, and h values for Au and Ni are 1337 and 1728 K, 118.6 and 106.8 J/mol · K, and 0.249 and 0.288 nm.41

Figure 3. Comparisons of Tm(D) functions for (a) tetrahedral Si nanoparticles, (b) spherical, and (c) icosahedral Ar nanoparticles described by eq 5 and available experimental data8,43 and MD simulation results,12,44 where Tm0, Sb0, and h values for Si and Ar are 1687 and 83.8 K, 113.1 and 74.5 J/mol · K, and 0.235 and 0.372 nm.41

Figure 4. Tm(D,λ) functions of Au nanoparticles described by eq 5.

shape and dimensionality effects are the secondary ones in terms of eqs 5 and 11.

Figure 5. Comparison of Tm(D) functions of (a) spherical Pb nanoparticles and (b) cylindrical Pb nanowires described by eq 5, available experimental data,3,47 and MD simulation results,11 where Tm0 ) 600.6 K,41 Sb0 ) ∆Hb0/Tb0 ) 88.0 J/mol · K,41 and h ) 0.350 nm.41

According to eq 11, there is ∆Tm(D, thin films):∆Tm(D, nanowires):∆Tm(D, nanoparticles) ≈ 1:2λwire:3λparticle, noting that this expression can only hold when D is large enough (e.g., D > 20h or 6 nm) and the nanocrystals are free-standing or only weakly bound to a substrate. Because λ is equal to 1 for spherical nanoparticles and cylindrical nanowires, the ratio can be determined as 1:2:3, which is in agreement with the results of the liquid-drop model and other thermodynamic considerations.22,27 For regular icosahedral nanoparticles with λ ≈ 0.66, the ∆Tm(D) values will be nearly equal to that of cylindrical nanowires. As shown in ref 29, surface tensions of the liquid and the crystal play a major role in calculating the size-dependent melting temperature. Note that surface tensions of the liquid and the crystal are determined by the surface broken number and the cohesive energy,52 where the surface broken number is related to the geometry of the particles. Thus, it can be claimed that the validity of our model is based on the combination of size-dependent cohesive energy with the geometry of the particles, and hence the effect of surface tension has been implied in our theoretical model.

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Figure 6. Comparison of Tm(D) functions of (a) spherical In nanoparticles, (b) cylindrical In nanowires, and (c) In thin film described by eq 5 and corresponding experimental data,3,47-51 where Tm0 ) 430.0 K,41 Sb0 ) ∆Hb0/Tb0 ) 98.1 J/mol · K,41 and h ) 0.325 nm.41

Considering the cohesive energy is essential to describe the variations of thermodynamic and magnetic parameters of nanocrystals,52-54 the establishment of our model is of vital importance in quantitatively studying some basic problems of materials. In summary, a unified model has been developed to describe the size, shape, and dimensionality dependences of melting temperatures Tm(D,d,λ) of nanocrystals. It is found that Tm(D,d,λ) reduces with increasing λ or decreasing D and d, for example, Tm(D, icosahedron) > Tm(D, sphere or cube) > Tm(D, octahedron) > Tm(D, tetrahedron) for nanoparticles and ∆Tm(D, thin films):∆Tm(D, nanowires):∆Tm(D, nanoparticles) ≈ 1:2λwire:3λparticle when D is large enough. The validity of the model is verified by the available experimental, MD simulation, and other theoretical results for Au, Ag, Ni, Ar, Si, Pb, and In nanocrystals. Noting that Ar, Si, and other elements are the crystals with molecular, covalent, and metallic bonds, the agreements shown in Figures 1-3, 5, and 6 indicate that our model may be applicable for all elemental crystals. Acknowledgment. The financial support from the State Key Program for Basic Research of China (2004CB619305), the National Natural Science Foundation of China (50571044 and 50831004), and the Postdoctoral Science Foundation of China (20070410326 and 200801370) is acknowledged. References and Notes (1) Gleiter, H. Acta Mater. 2000, 48, 1. (2) Pawlow, P. Z. Phys. Chem. 1909, 65, 1. (3) Coombes, C. J. J. Phys. F: Metal Phys. 1972, 2, 441. (4) Castro, T.; Reifenberger, R.; Choi, E.; Andres, R. P. Phys. ReV. B 1990, 42, 8548. (5) Goswami, R.; Chattopadhyay, K. Philos. Mag. Lett. 1993, 68, 215. (6) Dick, K.; Dhabasekaran, T.; Zhang, Z. Y.; Meisel, D. J. Am. Chem. Soc. 2002, 124, 2312. (7) Shyjumon, I.; Gopinadhan, M.; Ivanova, O.; Quaas, M.; Wulff, H.; Helm, C. A.; Hippler, R. Eur. Phys. J. D 2006, 37, 409. (8) Hirasawa, M.; Orii, T.; Seto, T. Appl. Phys. Lett. 2006, 88, 093119. ´ .; Pe´rez-Tijerina, E.; Garcı´a, J. A.; et al. J. (9) Gracia-Pinilla, M. A Phys. Chem. C 2008, 112, 13492. (10) Tang, S. C.; Zhu, S. P.; Lu, H. M.; Meng, X. K. J. Solid State Chem. 2008, 181, 587.

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