Shape, Structural, and Energetic Effects on the Cohesive Energy and

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Shape, Structural, and Energetic Effects on the Cohesive Energy and Melting Point of Nanocrystals Ali Safaei* Institute of Nano-ParthaVa, Mashhad, Iran ReceiVed: April 26, 2010; ReVised Manuscript ReceiVed: June 2, 2010

A nonlinear, lattice type-sensitive model, free of any adjustable parameter, has been developed to account for the shape and size dependency of the cohesive energy of free-standing nanocrystals (nanoparticles, -wires, and -films). In this model, the effects of the averaged structural and energetic properties of the surface and the volume of nanocrystals along with the first and second nearest-neighbor atomic interactions have been taken into consideration and gathered in a new parameter named as the surface-to-volume energy contribution ratio. This model has been compared to the experimental data of the cohesive energy of W and Mo nanoparticles, and the melting points of Au, Pb, Al, and Sn nanoparticles and Pb and In nanofilms. Moreover, the model has been corrected to account for the effect of substrate on the melting point of substrate-supported Sn nanodisks. It has been found that the present model has generally a good agreement with those experimental data measured by different techniques under different experimental conditions. 1. Introduction Nanosolids include a wide variety of shapes such as spherical dots, rods, thin plates, or voids of sizes in the nanometer range.1 The increasing desire to work with these materials requires a detailed understanding of the size dependency of their physical/ chemical properties. One of their most important properties is their cohesive energy, which is proportional to their melting point2 and is a significant coupling parameter for other physical/ chemical properties such as the Debye temperature, the diffusion activation energy, the vacancy formation energy, etc.3 It has been well established through theoretical4–16 and experimental17–29 investigations that the cohesive energy and so the melting point of free-standing nanocrystals decrease with decreases in their sizes. Recently, a model has been developed for the size-dependent cohesive energy, considering the effect of the lattice structure with no consideration on the energetic effects.13,14 The most important parameter of this model13,14 is the surface-to-volume coordination number ratio (q), defined as the ratio of the coordination number of atoms at the crystal surface to that of the lattice. This parameter was considered as an adjustable parameter to fit the experimental data at larger sizes with a value of q ) 0.5 and at lower size ranges (approximately less than 10 nm) with q ) 0.25.13,14 On the basis of our very recent work,15 in this paper, we present a nonlinear, lattice type-sensitive model, free of any adjustable parameter, for the shape- and size-dependent cohesive energy and thereafter for the melting point of nanocrystals (crystalline nanoparticles, -wires, and -films), by considering the effects of the averaged structural and energetic properties of the surface and volume of the nanocrystal, being the surface j S) and surface and volume average average packing factor (P coordination numbers. These effects along with the effects of the first nearest-neighbor (1NN) and the second nearest-neighbor (2NN) atomic interactions are gathered in a new parameter named the surface-to-volume energy contribution ratio of the nanocrystal (Γ).15 Unlike the model of ref 14, which is only * Tel and Fax: +985118793911. E-mail: [email protected].

applicable for spherical nanoparticles, the present model is applicable for all shapes of nanocrystals, including spherical and nonspherical nanoparticles, nanowires, -films, and -disks. The effect of different shapes has been simply introduced in this model through considering a nanodisk as a fundamental nanocrystal from which the physical/chemical properties of other shapes of nanocrystals can be easily obtained. More importantly, the adjustable parameter of the model of refs 13 and 14 has been removed in the present model through using a new boundary condition for the minimum value of Γ in small sizes. Because of these two advantages over previous lattice typesensitive models,13,14 the present model can be considered as a generalization of the previous models,13,14 with improved accuracy in the prediction of the experimental data. We only consider the perfect nanocrystals without any kinds of defects such as interfaces, grain boundaries, voids, or impurities. Finally, we compare our present model with the experimental data of the cohesive energy of W and Mo nanoparticles and with the experimental data of the melting point of Au, Pb, Al, and Sn nanoparticles and Pb and In nanofilms as well as the results of liquid drop model (LDM),4 the homogeneous melting model (HMM),5–7 and the model developed by Jiang et al.10–12 Our obtained model has been corrected to account for the substrate effect upon the melting point of substrate-supported nanocrystals and has been applied to the case of substrate-supported Sn nanodisks. Our main aim of comparing the model with the results of different experiments is to decide which experimental system, technique, and data analysis is in agreement with our present model prediction and, more importantly, to obtain the limitations and abilities of the model. It has been found that the present model has good consistency with the experiments. Moreover, it has been discussed that the size range of the applicability of our presented model is limited down to nanoparticles consisting of a total number of about 103 atoms. 2. The Model Considering the effects of the 1NN and the 2NN atomic interactions, defining some structural-averaged quantities, and neglecting any difference between the bond energies of different

10.1021/jp1037365  2010 American Chemical Society Published on Web 07/23/2010

Cohesive Energy and Melting Point of Nanocrystals

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volume atoms of nanocrystals and also between the bond energies of their surface atoms located at surface faces, surface edges, and surface corners, we have found the below formula for the cohesive energy of nanocrystals (Ecn) in our recent work:15

( )

j S nS jS j SεS + Z¯S′ εS′ Ecn E E Z )1- 1,Γ ) ) Ecb j V nt jV j VεV + Z¯V′ εV′ E E Z

(1)

Equation 1 is a general relation for the size-dependent cohesive energy of all types of crystalline nanosolids (nanocrystals), where nt is the total number of atoms of the nanocrystal and nS is its total number of surface atoms. Here, Γ is a new parameter, named as the surface-to-volume energy contribution ratio of the nanocrystal, which has been defined for the first time in ref 15. j S are, respectively, the average contributions of each j V and E E volume atom and surface atom of the nanocrystal to its total j VεV + Z j V′ εV′ )/2 and E jS ) j V ) (Z cohesive energy, defined as E j S′ εS′ )/2. Also, Z j S and Z j V are, respectively, the surface j SεS + Z (Z and the volume average coordination numbers of the nanocrysj V′ are the j S′ and Z tal, based on the 1NN definition, and Z corresponding parameters for the 2NN atomic neighbors, respectively. For the detailed definitions of these averaged quantities, which are based on the consideration of the averaged quantities of different surface sites (faces, edges, and corners), we bring the reader’s attention to ref 15. Also, εS and εV are the bond energies of the 1NN interactions of atoms located at the surface and within the volume of the nanocrystal, respectively. Similarly, εS′ and εV′ are those parameters for the 2NN atomic interactions, respectively. It should be noted that in this paper, the subscript “b” indicates the corresponding parameters for the bulk material. j SεS/ The 1NN-approximation of Γ can be obtained as Γ ≈ Z jZVεV through neglecting the effect of the 2NNs, that is, εS′ /εS ≈ 0 and εV′ /εV ≈ 0. Considering εS ≈ εV, the second-order j V) is equal to the surface-to-volume j S/Z approximation of Γ (≈Z coordination number ratio used in ref 13; hence, eq 1 would be similar to that obtained in ref 13. As it has been well discussed in Appendix A, the surfaceto-volume ratio of all shapes of nanocrystals can be easily obtained from that of a nanodisk with the height H and the diameter D. Hence, the following general equation has been obtained in Appendix A for the cohesive energy of all nanocrystals:

Ecn X ) 1 - 2(1 - Γ) , X ) 1/H + 2/D Ecb X0 + X

(2)

This equation can be used for nanodisks with X ) 1/H + 2/D, for nanowires with X ) 2/D, for spherical nanoparticles with X ) 3/D, and for nanofilms with X ) 1/H. Here, X0 is a material constant, which can be obtained from eq 6A in Appendix A. j V, named as the j S/Z The simplest approximation of Γ is Z surface-to-volume coordination number ratio.13,14 It has been well discussed that the surface-to-volume coordination number ratio decreases with size, and treating it as an adjustable parameter to fit the experimental data of melting point of nanoparticles, it has been found that the best fitting value of this parameter for large nanoparticles is 1/2 and for particles smaller than 10 nm is 1/4.13 Considering these fitting values of this parameter as boundary conditions, a size-dependent function

has been derived for the surface-to-volume coordination number ratio of spherical nanoparticles in ref 14, and it has been applied to obtain the melting point of spherical nanoparticles. Although the derived model of ref 14 has an acceptable accuracy in predicting the melting point depression of free-standing spherical Au, Pb, and Sn nanoparticles, it has two important shortcomings. The first one is the presence of an adjustable parameter in that model to fit the experimental data, and the second is its limited applicability only for the case of spherical nanoparticles. The adjustable parameter in that model is the diameter of the nanoparticle for which the value of the surface-to-volume coordination number ratio has been assumed to be 1/4.14 Here, in Appendix B, we try to derive a more general function, free of any adjustable parameter, for the surface-to-volume energy contribution ratio and then apply it to eq 2 to find a general formula, free of any adjustable parameter, for Ecn of all shapes of nanocrystals. According to Appendix B, we have

(

)

Ecn Tmn X X ) ) 1 - 2(1 - Γb) · 1 + C · , Ecb Tmb X0 X0 9[1 - exp(-2Smb /3R)] C) -3 2(1 - Γb) j Sd X ) 1/H + 2/D, X0 ) 3PL /2P

(3a) Ecn Tmn X/X0 ) ) 1 - 2(1 - Γb) , Ecb Tmb X 1+C X0 2(1 - Γb) C) - 3 (3b) [1 - exp(-2Smb /3R)] Equations 3a and 3b are two analytically derived relations for the cohesive energy and melting point of nanodisks (X ) 1/H + 2/D), spherical nanoparticles (X ) 3/D), nanowires (X ) 2/D), and nanofilms (X ) 1/H). As it can be seen, unlike the model of ref 14, there is no adjustable parameter in both eqs 3a and 3b, and also, it has a general unique form for all shapes of nanocrystals. In the following, it will be seen that there is no significant difference between eq 3a and eq 3b for the discussed cases. 3. Results and Discussion In the present model, Γ is a parameter including the averaged structural and energetic properties of the nanocrystal surface. Therefore, it can be used for comparing the relative thermal stabilities of different surface crystalline faces of nanocrystals.15 We have tabulated the values of the 1NN and 2NN coordination numbers of the lattice and crystalline faces of common lattice structures in Table 1. To calculate Γ(hkl) for the face (hkl), assuming εS ≈ εV ≡ ε and εS′ ≈ εV′ ≡ ε′, we need the value of ε′/ε for each lattice structure. Jiang et al.33 have estimated ε′/ε, for example, for the body-centered cubic (BCC) lattice structure, as (ε′/ε)BCC ) 2/3. Hence, as an example, for (110), (100), and (111) surface faces of a bulk BCC crystal, we have Γb(110) ) 4/9 > Γb(100) ) 2/9 > Γb(111) ≈ 0. Therefore, the (110) face of BCC crystals has the highest thermal stability, and the (111) face has the lowest one. The usefulness of Γ in comparing the relative thermal stabilities of different shapes of nanoclusters and different crystalline faces of a nanocrystal has been fully discussed in ref 15.

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Safaei

TABLE 1: Face and Volume (Lattice) Coordination Numbers (Zf and ZV) for Different Surfaces and Lattice Structures According to the 1NN and 2NN Definitionsa surface coordination numbers 1NN lattice structure SC BCC FCC

(100) (110) (110) (100) (111) (111) (100) (110)

volume coordination numbers

2NN

1NN

2NN

nb

Zf

nb′

Zf′

ZV

1 2 2 4 4 3 4 5

4 2 4 0 0 6 4 2

4 5 2 1 3 3 1 2

4 4 2 4 0 0 4 2

6

a)d

8

a(31/2)/2 ) d

6

a ) 1.16d

12

a(21/2)/2 ) d

6

a ) d(21/2)

Lbond

ZV′ 12

Lbond ′ a(21/2)

a Also, for each crystalline face, the numbers of its 1NN and 2NN broken bonds are shown, which are, respectively, defined as nb ) (ZV - Zf)/2 and nb′ ) (ZV′ - Zf′)/2. Lbond and Lbond ′ are the bond lengths between the 1NN and 2NN atoms, respectively, and a and d are the lattice constant and the atomic diameter, respectively.

j S, PL, and Γb. To plot eqs 3a and 3b, we need the values of P j S, according to eq 4A, it is necessary to have the To calculate P area fractions of the surface-faces, -edges, and -corners as well as the average face and edge packing factors. Although the surface of a nanocrystal is composed of some different faces and edges with different numbers of each type of those faces and edges, it seems reasonable to assume that the nanocrystal surface is composed of the most-compacted crystalline faces and edges (closest-packed ones). This assumption overestimates the numbers of the closest-packed faces and edges actually located at the surface of the crystal; however, it can be j e, and Z j S of j f, P considered as an approximation to estimate P the nanocrystal surface. If we had the exact numbers of different crystalline faces and edges, located at the crystal surface, then, j e (eq j f (eq 2A), P we could calculate the accurate values of P j S. However, because of the lack of any decisive 1A), and Z quantified information about the surface structure of nanocrystals (i.e., the fraction of different surface faces and edges), we use Nf j f ≈ ∑j)1 PfCP · AfCP/Af ≈ PfCP, the following approximations: P Nf (j) CP where Af ) ∑j)1Af in eq 2A. Here, Pf and ACP f are the packing factor and the area amount of the closest-packed crystalline face (CP face), and because of the above assumption, we have Af ≈ Nf AfCP. Similarly, considering eq 1A, we can approximately ∑j)1 Ne CP CP CP CP j e ) ∑k)1 write P PCP e · Le /Le ≈ Pe , with Pe and Le being the packing factor and the length of the closest-packed surface Ne Ne CP L(k) edge (CP edge), and also, Le ) ∑k)1 e ≈ ∑k)1Le . Therefore, using eq 4A and assuming the closest-packed atomic arrangej S of the nanocrystal can be ment at the nanocrystal surface, P approximated as below: area area j S ≈ PCP P + PCP + 1 × xarea f xf e xe c

(4)

Because the numbers and the type of surface faces, edges, and corners are dependent upon the size and shape of the nanoarea area are also size- and shapecrystal, the values of xarea f , xe , and xc j S is also dependent. Hence, from eq 4A, it can be found that P size- and shape-dependent. Because of the fact that the higher the coordination number of surface atoms is, the higher the surface packing factor is, we may conclude that there may be a close relation between these two quantities; that is, the surface packing factor may also decrease with a decrease in the crystal size as the coordination number of the surface atoms does.31 However, because of difficulties in determining the accurate area values of xfarea, xarea e , and xc , here, we neglect the numbers of surface edge and corner atoms as compared to the number of ≈ 0, xarea ≈ 0, and xfarea ≈ face atoms and then assume that xarea e c

1. Considering these assumptions along with eq 4 and also applying them for the case of the surface average coordination j S) defined in ref 15, we can have the below number (Z approximations: CP area j S ≈ PCP j P ≈ 0, xarea ≈ 0, xarea ≈ 1) f , ZS ≈ Zf , (xc e f

(5)

where ZfCP is the coordination number of atoms at the CP face of the nanocrystal’s surface. In spite of the possible size dependencies of PfCP and ZfCP, we use their corresponding bulk values for plotting eqs 3a and 3b; for example, for an FCC CP ) 0.91. crystal, we use PL ) PLb ) 0.74, and PfCP ) Pfb j S ≈ ZfCP and the definition of Γ in eq 1 and Now considering Z assuming εS ) εV ) ε and εS′ ) εV′ ) ε′, the surface-to-volume energy contribution ratio of the bulk crystal can be approximated as:

Γb ≈

CP ZCP fb εb + Z'fb εb′ j Vbεb + Z¯Vb ′ εb′ Z

(6)

Neglecting the effect of the edge and corner atoms may cause some deviations for very small nanocrystals. For example, for icosahedral (ICO) clusters obtained by truncating an FCC lattice in the [111] direction, the numbers of the surface face, edge, and corner atoms have been obtained by Montejano et al.31 in terms of the cluster order (ν: the number of the crusts of the cluster). They have found that the ICO clusters have only triangular faces with the closestpacked atomic arrangement,31 which is close to our assumptions, because we have considered that the nanoparticles’ surface is composed of the closest-packed faces. Therefore, we can assume that FCC nanoparticles are equivalent to the ICO clusters described by Montejano et al.31 From their work,31 we can find that for FCC clusters with ν e ν1 ) 1.4, the corner atoms have the largest contribution to the total number of the surface atoms (nS), that is, nS ≈ nc. Also, for the ICO clusters with the size of ν1 e ν e ν2 ) 2.7, both of the edge and corner atoms are most important, that is, nS ≈ nc + ne; for ν2 e ν e ν3 ) 5.0, both the edge and the face atoms have significant contribution to nS, that is, nS ≈ ne + nf; and for ν3 e ν, only the face atoms have the highest contribution to nS, that is, nS ≈ nf. If we consider these ranges of size and use D ) d(1 + 2ν), for example, for Au nanoparticles with a FCC lattice, we have D1 ) 1.21 nm, D2 ) 2.06 nm, and D3 ) 3.52 nm, implying that for very small Au nanoparticles with diameter D e D1, the corner atoms can have significant effects and for D g D3,



Figure 1. Comparison between the present model prediction for the cohesive energy of (a) W and (b) Mo nanoparticles and the corresponding nano/bulk-normalized experimental data.17 Equations 3a and 3b have been plotted for spherical and regular octahedral nanoparticles using X ) 3/D j S ≈ Pf(110) ) 0.83, and Γb ) 0.5 (with only and X ) 1.5(61/2)/D, respectively. The lattice type of W and Mo is BCC; hence, we used PL ) 0.68, P 1NN interactions). Other parameters are from Table 2.

TABLE 2: All of the Parameters of the Bulk Crystal Required for Plotting Eqs 3a and 3b, Jiang et al.,10–12 and the HMM,5–7 Which Is Only for Spherical Particles as Tmn/Tmb ) 1 - (4/GsHmb)[σsv - σlv(Gs/Gl)2/3] · (1/D)a element

Tmb (K)

Hmb (kJ/mol)

r (nm)

Fsb (g/cm3)

Fsl (g/cm3)

σsv (mJ/m2)

σlv (mJ/m2)

Sn Au Pb In W Mo Al

505.08 1337.3 600.61 429.75 3680 2890.0 933.47

7.148 12.724 4.7863 3.284 52.3020 37.484 10.7890

0.1862 0.1594 0.1935 0.1843 0.1554 0.1549 0.1583

7.2819 18.45 11.345

6.9719 17.285 10.6645

68519 1409b 5 56023

55019 11355 45223

a

The bulk melting point (Tmb) and melting enthalpy (Hmb) are from ref 45, the atomic radius (r) is deduced from the molar volume taken from ref 4, and the references of the densities and the surface tensions of bulk solid and liquid phases have been written below their values in the table. To plot eqs 3a and 3b, the value of the bulk melting entropy is obtained as: Smb ) Hmb/Tmb. b Obtained as an adjustable parameter by Buffat and Borel.5

the effect of the edge and corner atoms can be completely neglected. Therefore, the above assumptions of neglecting the effect j S ≈ ZfCP are j S ≈ PfCP and Z of the edge and corner atoms to obtain P applicable for particles lager than approximately D g D2 (or ν g ν2) and may cause some deviations for very small nanoparticles having sizes smaller than D1 ) d(1 + 2ν1) (ν1 ) 1.4). The accuracy of these assumptions will be tested below when we compare the present model with experimental data. 3.1. Cohesive Energy of Nanoparticles. Figure 1a,b shows the prediction of our model as compared to the experimental data of the cohesive energy of W and Mo nanoparticles.17 The most compacted surface crystalline face of large nanocrystals of W and Mo with the BCC lattice is (110) with the packing j S ≈ Pf(110) ) 0.83 and factor of Pf(110) ) 0.83. Hence, we use P PL ) 0.68 for W and Mo nanoparticles. Considering only the j Vb ) 8 and for its closest1NNs, we have for the BCC lattice Z (100) packed face Zfb ) 4. Then, from eq 6, Γb ) 0.5. As it can be seen from the figure, the present model prediction for spherical nanoparticles is consistent with the data, and it falls very close to the data of W nanoparticles if we assume a regular octahedral shape [X ) 1.5(61/2)/D] for the particles. The total surface area

of a regular octahedral with the diameter D is AS ) D2121/2 and its volume is Vc ) 21/2D3/3; then, AS/Vc ) 3(61/2)/D. A nanocrystal having a regular octahedral shape can be considered as a nanodisk with the same surface/volume ratio, that is, (AS/Vc)disk ) 3(61/2)/D. Because for a disk (AS/Vc)disk ) 2X, then, we can obtain X ) 1.5(61/2)/D to plot eqs 3a and 3b for regular octahedral particles. It is here worth mentioning that because for hexagonal closed-packed (HCP), FCC, and BCC crystals, we have PfCP/PL ) (3/2)1/2, then for nanoparticles having these j S/PL ≈ (3/2)1/2. Now, using eq lattice structures, we will have P 6A, we can find for these three types of lattice structures: X0 ) j S ≈ (3/2)1/2, which is independent from the lattice type. 3PL/2P Therefore, the effect of any possible structural transition between BCC, FCC, and HCP lattices at small sizes, for example, the transition of small nanocrystals of W form BCC to FCC at small sizes,17 has been implicitly taken into account in our model. Now, we discuss the size limit of the applicability of our model. Assuming the approximation of nt ≈ PL(D/d)3, one can find that the total number of atoms of a tungsten spherical nanoparticle of the diameter D ) 6 nm is nt ≈ 4894 and that of a Mo spherical nanoparticle with D ) 4 nm is nt ≈ 1464. Hence,

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it can be inferred from Figure 1a,b that the present model is capable of predicting the cohesive energy of W nanoparticles larger than nt ≈ 4894 and of Mo particles having nt > 1464. This is the size range of the applicability of the model, inferred directly from the comparison made with the experimental data in Figure 1a,b. Moreover, because the present model deals with crystalline nanosolids, it is not applicable for nanoparticles smaller than the size of the smallest crystalline nanoparticles.34 For instance, the diameter of the smallest crystalline tungsten nanoparticle has been estimated to be 0.92 nm, which contains the total number of about nt ≈ 32 atoms.34 On the basis of our previous work,34 tungsten nanoparticles smaller than this size are not in the crystalline state, which is well in accord with the reported experimental value of nt ≈ 30 for the size of the amorphous tungsten particle.17 Hence, a possible reason for the deviation of the present model from the experimental value of Ecn for the tungsten particle of D ) 1 nm is its amorphous state reported in ref 17. On the other hand, the deviation of the present model for very small particles may be caused by neglecting the effects of the edge and corner atoms of the surfaces of the particles. As we discussed above, for FCC clusters with ν < ν3 ) 5.0, the neglected effect of the edge and corner atoms can cause some deviations in the present model. Hence, considering a FCC structure for very small tungsten nanoparticles17 and using D3 ) d(1 + 2ν3), we can find D3 ) 3.42 nm, which implies that eqs 3a and 3b are safely applicable for tungsten nanoparticles containing more than nt ≈ 985 atoms (D > D3). Similarly, we can find that the size limit of the safe applicability of our model for Mo nanoparticles is nt > 987 atom, which is in accord with the comparison made in Figure 1b. In light of both of the above view points, that is, (I) the size limit of the existence of the crystallinity of nanoparticles and (II) the size limit of the applicability of our assumptions of neglecting the effects of the edge and corner atoms at the surface of nanoparticles, one can deduce that eqs 3a and 3b are applicable for nanoparticles containing the total number of nt > 1000 atoms. Another possible limitation of the present model for very small sizes is the neglected size dependency of the atomic bond energy of nanoparticles, which is significant at very small sizes.30 Hence, generally, eqs 3a and 3b are not applicable on the deep nanometer scale. 3.2. Nanodisk Melting Point. Here, we compare the present model prediction with the experimentally obtained melting point data of Sn nanodisks.19 The most compacted surface crystalline faces of Sn nanocrystals with the body-centered tetragonal (BCT) lattice are (100) and (110) with the packing factors of jS Pf(100) ) 0.71 and Pf(110) ) 0.50, respectively. Here, we use P ≈ Pf(100) ) 0.71 and PL ) 0.53 for Sn nanodisks. Considering (100) ) 4, only the 1NNs, for its closest-packed face, we have Zfb j and also, ZVb ) 8. Then, from eq 6, we find Γb ) 0.5. These parameters are used to plot eqs 3a and 3b. Allen et al.19 performed the melting point experiments with the platelet-shaped deposits of tin. They calculated the surface area/volume ratios and thicknesses of tin platelets by equating the volume of the solid disklike particle (Vs) with the measured volume of the spherical liquid particle (Vl) formed on melting. Their consideration (Vs ) Vl and, hence, Fs ) Fl, Fs/l: the density of the solid/liquid phases) leads to HAllen ) Vl/(πD2/4), which overestimates the actual value of the platelet thickness (H) and consequently underestimates its surface area/volume ratio. Here, D is the diameter of the disklike particle and HAllen is its thickness calculated by Allen et al.19 Considering FsVs ) FlVl, we modify their calculations as: H ) (Fl/Fs)HAllen. The surface area/volume ratio of a disklike particle is equal to 2X. Hence,

Safaei

Figure 2. Variation of melting point of Sn nanodisks in terms of their X() 1/H + 2/D), from the present model, as compared to the corresponding experimental data.19 The bars indicate the ranges of the measured melting points of substrate-supported Sn nanodisks19 normalized by Tmb ) 505.08 K. Equations 3a and 3b have also been plotted for a substrate-free Sn nanodisk for comparison. The effect of the substrate has been accounted for through eq 9 with different values of ∆1 and ∆2(with Svib,b and Smb). Please see the text for details. The lattice j S ≈ Pf(100) ) 0.71 and type of Sn is BCT; hence, we used PL ) 0.53, P Γb ) 0.5 (with only 1NN interactions). Other parameters are from Table 2.

using H ) (Fl/Fs)HAllen, we have X ) XAllen + (Fs/Fl - 1)(1/ HAllen), where XAllen ) 2/D + 1/HAllen. With the use of the reported values of the thicknesses (HAllen) and the surface area/ volume ratios (2XAllen) of the tin platelets,19 we have calculated our modified values of X. In Figure 2, the measured ranges of the experimental melting points of the tin platelets have been plotted in terms of our calculated values of X. As it can be seen from the figure, the present model predictions for free-standing platelets, with no substrate effects obtained from eqs 3a and 3b, are higher than the experimental data by a maximum deviation of about 11-12 K (2.4%). The lower measured melting point of these tin nanodisks than that predicted by eqs 3a and 3b can be attributed to the platelet shape of the deposited particles. A platelet-shaped deposit has a large amount of the particle/substrate interface area, which leads to a significant effect of the substrate on the measured melting point of the particle. We explain this effect by considering the change of the free energy of the particle upon its melting as ∆Gsfl ) subs subs subs subs - Afree ∆Gv(T) + Afree l σlv + Al σl s σsv - As σs , where ∆Gv is the temperature-dependent volume free energy change in the and Asfree are the area amounts of solid-to-liquid transition, Afree l the free surfaces of the melted particle and the solid particle, respectively. Asubs and Assubs are the area amounts of the liquid/ l substrate and solid/substrate interfaces, respectively. σlv, σsv, , and σssubs are, respectively, the surface energies per unit σsubs l area of the liquid/vapor, solid/vapor, liquid/substrate, and solid/ substrate interfaces. At the melting point of the nanocrystal (Tmn), we have ∆Gsfl ) 0. Hence, using the approximations of Vl = Vs and ∆Gv/Vs ) (Hmb/Vmol)(Tmb - T)/Tmb, where Hmb is the molar melting enthalpy of the bulk crystal and Vmol is its molar volume, we can obtain

Tmb - Tmn Vmol free ) (A σ + Assubsσssubs - Alfreeσlv Tmb VsHmb s sv Alsubsσlsubs)

(7)

Equation 7, including the terms of free-standing and substratesupported nanocrystals, can be rewritten in the following form:


( ) Tmn Tmb

free

( ) ( ) Tmn Tmb

)

subs

)

(

Vmol Asfree Alfree )1σ σ Hmb Vs sv Vs lv

Tmn Tmb

-

free

(

Vmol Assubs subs Alsubs subs σs σ Hmb Vs Vs l


(8a)

)

(8b)

where (Tmn/Tmb)free and (Tmn/Tmb)subs are, respectively, the sizedependent nano/bulk-normalized melting points of the freestanding and the substrate-supported nanocrystals both having the same size. It should be noted that, for a spherical particle free of diameter D (Afree s /Vs ≈ Al /Vl ≡ 3/D), eq 8a is approximately 5 similar to the HMM with the assumption of Fs ≈ Fl. Moreover, it is a point of note that eq 8b decomposes the melting point of a substrate-supported nanocrystal in terms of the melting point of its equally-sized substrate-free nanocrystal and the effect of the substrate/particle interaction. On the basis of eq 8b, it can be inferred that the solid/substrate interface with the area of Assubs has a driving force of Assubsσssubs for the substrate-supported particle’s melting and thus reduces its melting point by the amount of (Vmol/Hmb)(Assubs/Vs)σssubs as compared to the unsupported particle. For spherical particles, are sufficiently small that we may neglect the Assubs and Asubs l effect of the substrate on the particle melting, but for disklike particles, Assubs is large; therefore, Assubsσssubs may have a large value and hence a significant effect on melting, which results in lowering of the melting point. It is noteworthy that the shape of a solid particle after its melting can be assumed to be approximately spherical; hence, Alsubs is generally negligible, meaning that the liquid/substrate interface has generally no significant effect. This is particularly true for the case of the liquid particles of tin, which, reportedly, were in a spherical shape after melting of the platelet-shaped solid tin particles.19 Thus, the only dominant factor in lowering the measured melting point of the platelet tin particles is the solid/substrate effect. For disklike tin particles, we can neglect the area amounts of their side surfaces and assume that Assubs ≈ AS/2, where AS is the total surface area of a disklike particle. Because for a disk we have AS/Vc ) 2X, then the ratio of the solid-substrate interface area to the volume of a substrate-supported disklike solid particle is approximately equal to Assubs/Vs ≈ X. Hence, the solid/substrate interface reduces the measured melting point of disklike particles approximately by the amount of ∆ ≈ (Vmolσssubs/Hmb)X. Thus, eq 8b can be rewritten for substratesupported disklike particles as:

( ) ( ) Tmn Tmb

)

subs

Tmn Tmb

- ∆, ∆ ≈ free

Vmolσssubs X Hmb

(9)

Sn Now, if we assume σsubs ≈ σSn s ss , with σss being the surface energy per unit area of the solid/solid interface of tin, then its value can be estimated from the phenomenological equation of σssubs ≈ 1.3 dHmb/Vmol (d is the atomic diameter).39,40 Now, substituting this value into ∆ ≈ (Vmolσssubs/Hmb)X, we can find ∆1 ) 1.3dX. Using this value of ∆1 and eq 3a for (Tmn/Tmb)free, eq 9 has been plotted, in Figure 2, in terms of X for tin nanodisks. Another Sn Sn can also be obtained through the formulas σss value of σss ≈ 2σslSn 40 and σslSn ) 2dSvib,bHmb/(3VmolR),40,41 where Svib,b is the vibrational component of the overall melting entropy of the bulk crystal at its melting point, and R is the ideal gas constant. In light of these formulas, we can obtain ∆2 ) (4dSvib,b/3R)X.

Considering Svib,b ) 9.22 J/mol K for Sn,42 eq 9 has also been plotted in Figure 2 using ∆2. Moreover, the result of replacing Svib,b by Smb in ∆2 has also been shown in Figure 2. It is now evident from the figure that the prediction of the present model, through taking into consideration the effect of the substrate, has been corrected and fairly accords with the two ranges of the experimental data. However, the reason of the generally constant trend of the experimental data is not obvious, although the rapid increase of the measured melting point from X ) 0.0194 to 0.0201 and then its rapid decrease from X ) 0.0201 to 0.0228, perhaps, shows that the accuracy of the experimental data may be questionable. 3.3. Nanoparticles Melting Point. Here, we compare the prediction of the present model for some elemental nanoparticles with different sets of experimental data measured by different techniques to obtain the abilities and limitations of the model. As discussed above, for spherical nanoparticles, Assubs and Asubs l are sufficiently small; thus, we can neglect the effect of the substrate caused by the amount of the area of the particle/ substrate interface and, accordingly, qualitatively discuss only about the effect of the particle/substrate interaction depending and σssubs. Because Assubs ≈ 0 and Asubs ≈ on the amount of σsubs l l 0, only for very large amounts of σssubs and σsubs , the values of l Assubsσssubs and Asubs σsubs may have considerable effects on the l l melting points of spherical particles. Hence, it is conceivable that for normal cases, we neglect the particle/substrate effects upon the melting points of spherical particles. 3.3.1. Au Nanoparticles. Using the approximations given in eq 5, for Au nanoparticles with a FCC lattice structure (PL ) (111) j S ≈ Pf(111) ) 0.91 and Z j Sb ≈ Zfb 0.74), we have P ) 6. j Considering ZVb ) 12 and only the 1NNs in eq 6, we have Γb ) 0.5. Using these values, eqs 3a and 3b have been plotted for Au spherical nanoparticles in Figure 3a,b as compared with the experimental data5,18,20 and MDs results35,36 as well as the HMM,5–7 LDM,4 and Jiang et al.’s model.10–12 All of the experimental data of ref 20 are completely shown in Figure 3b. Generally, eq 3a is similar and very close to eq 3b, especially for large nanoparticles. It is obvious that the present model has good agreement with the compared experimental data and the MDs results. Here, we briefly discuss each set of the experimental data. The experimental data of ref 5, measured by means of scanning electron diffraction technique, and of ref 20 are for gold nanoparticles on the amorphous carbon and graphite substrates, respectively. These substrates have low affinity with gold, and also, it has been reported that the graphite substrate has a negligible effect on the melting point of the nanometersized gold particles.37 Therefore, both sets of these experimental data can be considered as the substrate-insensitive measured data, which are in good agreement with the present model prediction. This agreement can be explained in the way that the present model (eq (3a,b)) treats the melting phenomenon of individual surface-faceted nanocrystals without considering any effect of interactions in the substrate/particle and the particle/ particle interfaces. The little deviation of our model from the experimental data for very small nanoparticles may be attributed to the large effect of the substrate on the melting point of very small gold particles.37 Moreover, this deviation for nanoparticle diameters in the range of D < D3 ) 3.52 nm may be due to neglecting the effect of edge and corner atoms, which may have significant contributions to the total number of the surface atoms for gold particles in this size range. The other set of the experimental data, in fairly good accordance with our model, is the melting point data of silica-encapsulated gold nanoparticles

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Safaei

Figure 3. Dependence of the melting point of (spherical) nanoparticles of Au on their size. The present model is compared to the HMM,5–7 LDM,4 j S ≈ Pf(111) and Jiang et al.10–12 as well as the experimental data5,18,20 and MDs results.35,36 The lattice type of Au is FCC; hence, we use PL ) 0.74, P ) 0.91, and Γb ) 0.5. Other parameters are from Table 2.

measured by Dick et al. using differential thermal analysis.18 Although the silica shell enhances the thermal stability of the gold nanoparticles, it has been reported that, because of having high porosity, the silica shell had only a small effect on the melting point of the gold nanoparticles.18 Furthermore, it has been stated by Dick et al. that for their nanoparticles, the thickness of the silica shell decreased upon decreasing the Au core size.18 Therefore, much more importantly, one can conclude that the effect of the silica shell on the melting point of the gold nanoparticles should decrease with decreasing the Au core size. This situation can also be easily understood from the better consistency of the present model prediction with the experimental data of ref 18 for small silica-encapsulated nanoparticles as compared to the larger ones. Accordingly, it is obvious that the conditions of the experiment, performed by Dick et al.,18 especially for the small, silica-encapsulated, gold nanoparticles, can be approximately considered as that of the indiViduated particles’ melting experiment, being nearly similar to the present model conditions. In addition to the experimental data, our model is also in good agreement with the MD results of refs 35 and 36. It is noticeable that the reported diameters of the gold nanoparticles by Shim et al.35 had been calculated using D ) n1/3 t d, while we have calculated their diameters through the relation nt ) PL(D/ d)3 + 2PfCP(D/d)2.13 This calculation of the diameters gives a better consistency for the results of Shim et al.35 Moreover, the present model gives a better prediction than that of the HMM, LDM, and Jiang et al. model. The HMM has a good agreement for large particles, but for particle diameters smaller than approximately 10 nm, it deviates from the experimental data. This may be caused by neglecting the size dependency of the surface tension of the gold nanoparticles, which may be important for very small nanoparticles.37 Furthermore, this deviation of the HMM implies that a linear relation cannot sufficiently fit the melting point data, especially for the small sizes.

3.3.2. Pb Nanoparticles. Figure 4a,b illustrates the melting point prediction of the present model (eqs 3a and 3b) for spherical nanoparticles of Pb as compared to the experimental data21–24 and MD results38 as well as the theoretical models. In view of the lattice structure of Pb (FCC), we have used PL ) j S ≈ 0.91, and Γb ≈ 0.5 to plot eqs 3a and 3b. As 0.74, P illustrated in Figure 4a, among all of the compared models, the present model prediction has the best consistency with the experimental data measured by Skripov et al.,21 besides having an acceptable agreement with the experimental data obtained by Coombes22 and Peters et al.23 and with the MD results.38 All of the other theoretical models underestimate the melting point of Pb nanoparticles in comparison with the data. Here, we briefly discuss about the experimental data of Peters et al.23 and Kofman et al.24 shown in Figure 4b. As shown in Figure 4a, the present model prediction is fairly consistent with the experimental results of Peters et al.23 for nanoparticles approximately larger than D ) 14 nm. They investigated the size-dependent melting point of Pb nanoparticles supported on the native oxide surface of a Si [532]-oriented substrate.23 Their samples had no contaminant materials and also at the particle/substrate interface, Pb did not reduce nor wet the silica.23 Because of these facts, we can assume that their investigation was approximately similar to the conditions of the melting of indiVidual particles with no external effects and therefore nearly close to the present model considerations. Also, Peters et al.23 reported that their particles were in nearly spherical shape, similar to that assumed in our present model (i.e., X ) 3/D). However, the reason of the deviation of their data for particles with D < 14-15 nm is not obvious. It should be noted that in ref 23, the data were plotted against the area-averaged crystallite size (〈L〉), but here, we used the reported relation of 〈D〉 ) 1.5〈L〉 23 to plot the data in terms of the average diameter (〈D〉). Furthermore, as shown in Figure 4b, our present model has a fairly acceptable consistency with the experimental data



Figure 4. Tmn(D) variation of (spherical) Pb nanoparticles in terms of their diameter. The present model is compared to the HMM,5–7 LDM,4 and Jiang et al.10–12 as well as the experimental data21–24 and MDs results.38 The experimental data of ref 22 are shown for particles deposited on the carbon and silicon monoxide substrates prepared at room temperature. In ref 23, the data were plotted against the area-averaged crystallite size (〈L〉), but here, we used 〈D〉 ) 1.5〈L〉 23 to plot the data in terms of the average diameter. The lattice structure of Pb is FCC; so, we use PL ) 0.74, j S ≈ Pf(111) ) 0.91, and Γb ) 0.5 (considering only the 1NN interactions). Other parameters are from Table 2. P

obtained by Kofman et al.24 They investigated the melting behavior of Pb inclusions confined in amorphous SiO by means of dark field electron microscopy (DFEM). As they reported, the melting was initiated from the nanocrystal surface and progressively continued into the core of the crystal.24 By increasing the temperature, the finally remained spherical solid core of diameter D, surrounded by its liquid layer, melted abruptly at a temperature that was considered as the melting point of that final spherical solid core [Tmn(D)].24 It should be accentuated here that in the present model no surrounding environment has been considered for particles, contrary to Pb inclusions studied by Kofman et al.,24 which were surrounded by a SiO matrix when they were in low temperatures and by a Pb liquid layer when they remained as the final solid cores near their melting point.24 Considering these conditions and the good agreement of the present model prediction with the experimental data, we can conclude that the surrounding liquid layer had a small effect on the melting point of the final Pb solid cores. This conclusion can be supported by the report of Kofman et al.24 on the energy of the Pb/SiO interface. Importantly, they reported that the Pb/SiO surface energy must be of the same order of magnitude as that of the Pb/vacuum.24 Hence, these experimental conditions were nearly close to the conditions of an individual particle melting with no significant external effects and, therefore, are in a fairly agreement with our model prediction. 3.3.3. Al Nanoparticles. Here, to show the flexibility and capability of the present model in predicting the melting point of nonspherical nanoparticles, we consider the case of Al nanoparticles investigated by Lai et al.43 and by Sun and Simon.44 As it has been shown in our recent work,15 all of the LDM,4 HMM,5–7 and Jiang et al.’s model10–12 and the model of ref 14 underestimate both sets of the melting point experimental data of Al nanoparticles measured by Lai et al.43 and by Sun

and Simon.44 It has been fully discussed in ref 15 that this discrepancy between the theories4–7,10–12,14 and the experiments43,44 is probably due to the difference between the assumed spherical shape of Al nanoparticles in the theories and their real shapes in the experiments. This inconsistency is also seen for the case of the present model when we plot eqs 3a and 3b for free-standing Al spherical particles (X ) 3/D). For Al with a j S ≈ 0.91, and Γb ≈ FCC lattice structure, we used PL ) 0.74, P 0.5 to plot eqs 3a and 3b. A possible explanation for this disagreement between the theories and the experimental data has been given in ref 15. On the basis of the reports of refs 43 and 44 on the experiments’ conditions, it has been explained that this discrepancy is not due to any effects of oxidation and/ or uncertainty in the measured experimental data, but it is likely due to the possibly inaccurate assumption, used in the theories, about the shape of the Al nanoparticles.15 Recently, through considering the geometrical characteristic of nanoclusters described by Montejano et al.,31 we have conjectured that the shape of the Al nanoparticles in the experiments was probably ICO clusters rather than spherical particles.15 Now, we apply the present model (eqs 3a and 3b) to the case of ICO nanoparticles of Al to test again, different from the method of ref 15, the recently obtained result on the possible shape of the Al nanoparticles.15 The surface area of a regular ICO particle of diameter D is equal to AS ) 751/2D2 and its volume is Vc ) 5 × (3 + 51/2)D3/12, and then, its surface/volume ratio will be AS/Vc ) 12 × 751/2/(5 × (3 + 51/2)D). A nanocrystal having a regular ICO shape can be considered as a nanodisk with the same surface/volume ratio, that is, (AS/Vc)disk ) 12 × 751/2/(5 × (3 + 51/2)D). Because for a disk (AS/Vc)disk ) 2X, then, we can obtain X ) 6 × 751/2/(5 × (3 + 51/2)D) to plot eqs 3a and 3b for regular ICO particles. As it can be seen form Figure 5, first, the prediction of the present model is more consistent with the data of ref 43 than the data of ref 44, which

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Figure 5. Tmn(D) variation of Al nanoparticles in terms of their diameters. The present model (eqs 3a and 3b) is plotted for free-standing spherical (X ) 3/D) and ICO (X ) 6 × 751/2/(5 × (3 + 51/2)D)) nanoparticles in comparison with the nano/bulk-normalized experimental data.43,44 The lattice type of Al is FCC; hence, we use PL ) j S ≈ Pf(111) ) 0.91, and Γb ) 0.5. Other parameters are from 0.74, P Table 2.

reflects the fact that the data of ref 43 were measured, for the first time, for Al clusters with a free surface. Second, the model prediction is improved through assuming an ICO shape, instead of a spherical shape, for the Al nanoparticles of refs 43 and 44. This result, obtained through the present model, and also our recent conclusion on the possible shape of the investigated Al nanoparticles,15 obtained through using a different model and the geometrical characteristic of nanoclusters,15 indicate that these Al particles had, more probably, ICO shapes in the corresponding experiments. However, the exact shape of the Al nanoparticles has not been specified in refs 43 and 44. This discussion reflects a very important conclusion of note, that is, the shape of nanocrystals, significantly affecting their cohesive energy, should be clearly indicated in the corresponding experiments of measuring their physicochemical properties, to allow an accurate interpretation of the experimental results. Moreover, it is here worth mentioning that the simple flexibility of the present model in being applicable for nonspherical particles supports my previous statement that a nanodisk can be considered as a fundamental shape at the nanoworld from which the physicochemical properties of other shapes of nanocrystals can be easily obtained. 3.3.4. Sn Nanoparticle. As mentioned previously, Sn with a BCT lattice has two of the most-compacted crystalline faces, (100) and (110), with the packing factors of Pf(100) ) 0.71 and jS ≈ Pf(110) ) 0.50, respectively. For Sn nanodisks, we used P Pf(100) ) 0.71, but for Sn nanoparticles, because of the fact that jS their surface is more faceted than that of nanodisks, we use P (100) (110) ≈ (Pf + Pf )/2 ) 0.61 to account for the effect of the presence of both of (100) and (110) faces. This consideration gives better consistency with the melting point experimental data of Sn nanoparticles, in comparison with the case of using j S ≈ Pf(100) ) 0.71. This is because of approximately equal P packing factors of these two crystalline faces and implies that the surface of Sn nanoparticles may be composed of both (100) and (110) faces. Also, to plot eqs 3a and 3b, we used Γb ≈ 0.5. Shown in Figure 6a,b are the predictions of eqs 3a and 3b and the other models for spherical Sn nanoparticles as well as a few sets of the experimental data.19,25,26 Here, we discuss briefly the experimental data and the specific conditions. The best agreement is obtained between the present model and the experimental data measured by Lai et al.25 (see Figure

Safaei 6a). They investigated the size-dependent melting point of Sn nanoparticles deposited on a SiN substrate by a calorimetry technique.25 Given the fact that this investigation involved the particles in a spherical shape with a free surface and no significant particle/substrate interaction and without particle oxidation,25 one can conclude that it is approximately similar to melting of indiVidual particles with no external effect, as considered in the present model. From Figure 6a, for particles with diameters smaller than approximately 20-25 nm, the present model has a fairly good agreement with the experimental data obtained by Wronski,26 but for larger particles, it falls below the experimental results. Wronski26 studied the melting point of spherical tin particles deposited on carbon and silicon monoxide thin films, by means of transmission electron diffraction and microscopy. From Figure 6a, the similarity between the experimental data for both two substrates (carbon and silicon monoxide) and their fairly good agreement with the present model, respectively, indicate that these substrates had equal and no significant effects on the tin particles melting point. Although there was no significant substrate effect, the reason of the deviation of the present model from the data for particles of D > 20-25 nm is not obvious. Among the important factors affecting the accuracy of experimental results is the selected physical parameter whose behavior is used to study the solid-liquid transition. Wronski determined the melting point of tin particles by means of diffraction patterns, as the temperature at which the rings of tin solid particles in the diffraction patterns became obscured by those halo rings of liquid tin.26 To our knowledge, this criterion, which is based on the vanishing of edges and facets of small crystals upon their melting, barely determines a more precise value for the melting point because of the difficulty of intercomparison of diffraction patterns. Hence, some deviations of the data may have been caused by this selected criterion for the melting determination. This idea has also been supported by Kofman et al. that for nanocrystals, edges or facets cannot easily be seen; therefore, this criterion is difficult to use for the melting point measurements.24 Allen et al.19 measured the melting point of tin nanoparticles deposited on thin carbon film by means of dark field transmission electron microscopy as the temperature at which the dark field image greatly dimmed. Figure 6b shows their measured data for the melting point of newly deposited tin particles (full circles: with no oxide layer) and of the same particles melted a second time (open circles: with an oxide layer). As shown, the present model prediction is higher than the measured melting point of the newly deposited particles but in good agreement with the melting point data of the particles taken under more contaminated conditions (open circles in Figure 6b). For the second set of the Sn particles, a surface oxide layer formation was observed.19 It is important to emphasize that the present model is for particles with no impurity and no external surrounding environment. Thus, the measured melting point of the particles with an oxidized surface layer must be higher than that predicted by the present model. In view of this, the observed consistency of the present model prediction with the melting point data of oxidized tin particles and its disagreement with the data of newly deposited particles indicate that the experimental data of ref 19 may be lower than the expected melting point values for tin particles. This idea can also be accentuated when we compare the data of the newly deposited tin particles obtained by Allen et al.19 with the melting point data obtained by Lai et al.25 and by Wronski26 shown in Figure 6a.



Figure 6. Tmn(D) variation of Sn (spherical) nanoparticles in terms of their diameter. The present work is compared to the HMM,5–7 LDM,4 and Jiang et al.10–12 as well as the experimental data.19,25,26 The data of ref 19, represented by full circles in Figure 6b, are for newly deposited tin particles with no oxide layer, and those by open circles are for the same particles melted a second time (with an oxide layer). The lattice structure j S ≈ 0.61, and Γb ) 0.5 (considering only the 1NN interactions). Other parameters are from Table 2. of Sn is BCT, so we usePL ) 0.53, P

3.4. Nanofilms. 3.4.1. Pb Nanofilms. As can be seen form Figure 7a,b, among the compared models, the present model has the best consistency with the experimental data27 and the MD results28 of the melting point of Pb thin films. Tsuboi et al.27 measured the melting points of Pb thin films, consisting of grains, with thicknesses ranging from 10 to 200 nm, confined between different kinds of wall materials (Ge, MgF2, Cu, Cr, Al, and Mn), by means of differential scanning calorimetry (DSC). They have found that melting began at the grain boundaries of Pb films.27 On the basis of their analysis, the effect of the grain boundaries in lowering the melting point of Pb films can be explained through the term ∆Tmg ) Tmb - Tmf ) 2mTmbσsl/(HmbdgFs), where Tmf is the melting point of thin films, m is the atomic weight of Pb, σsl is its solid/liquid interfacial energy, Hmb is its bulk latent heat of fusion (per mole), dg is the diameter of the Pb grains in Pb films, and Fs is the mass density of solid Pb.27 Considering dg independent from the thickness of the films, Tsuboi et al. estimated that, for Pb films confined between Ge walls (Pb/Ge), the contact angle of Pb liquid on the wall in the presence of Pb solid and the average grain diameter are 65° and 120 nm, respectively.27 These values for the system of Pb/Al were estimated to be 45° and 560 nm, respectively.27 Considering the value of σsl ) 0.046 J/m2 for Pb23 and its atomic weight m ) 207.245 and using the values of the other parameters from Table 2, the contribution of grain boundaries of Pb grains of the average size of 120 nm in lowering the melting point of Pb films of the system of Pb/Ge can be calculated as ∆Tmg ) 0.0029Tmb ) 1.7640 K. Also, using dg ) 560 nm, for Pb films constrained between Al walls (Pb/ Al), we have found ∆Tmg ) 6.294 × 10-4Tmb ≈ 0.0006Tmb or ∆Tmg ) 0.3780 K. Both of these results indicate that the melting point depressions of Pb films due to the presence of grain boundaries in films in both systems of Pb/Ge and Pb/Al are negligible. Moreover, in light of the fact that the contact angle

of Pb liquid on walls for the Pb/Al films is (45°) less than that for Pb/Ge films (65°), we can conclude that the effect of the confinement of Pb films by the Al wall on their melting point is less than that of the Ge wall, because of the higher wetting of Pb liquid for the case of Pb/Ge as compared to the case of Pb/Al. Given the fact that the present model is applicable for unsupported nanofilms with no external effects, it can be seen that this conclusion is well consistent with the comparison made in Figure 7b in which the present model is closer to the experimental data of Pb/Al films rather than that of Pb/Ge films. Furthermore, the smaller effect of the wall material for the case of Pb/Al than that for Pb/Ge can also be well understood when we look through Figure 7b in which it can be easily seen that, with increasing the film’s thickness, the experimental data of Pb films of the Pb/Al system have more tendency toward the bulk melting point than that of the Pb/Ge system. This point has also been mentioned in ref 27. The experimental data related to the other wall materials are closer to those of the system of Pb/Al. An interesting point of note is the observation of (111) crystal textures parallel to the film surface in Pb films before their melting.27 Interestingly enough, this observation is well consistent with our assumption that the surface of nanocrystals, particularly nanofilms, is composed of the most compacted crystalline faces, for example, for the case of Pb films with a FCC lattice structure, the film surface should be composed of the closest packed face (111). It should also be noted that the experimental data plotted in Figure 7b are taken from ref 27 by dividing the therein reported data of melting point depressions (∆Tm ) Tmb - Tmf) by the value of Tmb ) 600.61 K. 3.4.2. In Nanofilms. Shown in Figure 8 are the experimental melting point data of thin films of indium deposited on a passive substrate of Ge (100),29 as compared to the present model (eqs 3a and 3b with X ) 1/H) and the other models. The thicknesses of indium films reported in ref 29 were in terms of monolayers

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Figure 7. Tmf(H) variation of lead nanofilms. The present model (eqs 3a and 3b with X ) 1/H) is compared to the LDM4 and Jiang et al.10–12 and (a) to the MD results of ref 28 and (b) to the experimental data27 of thin Pb films constrained by different kinds of materials (Ge, MgF2, Al, Cu, Cr, and Mn). The experimental data plotted in panel b are taken from ref 27 by dividing the therein reported data of melting point depressions (∆Tm ) Tmb - Tmf) by the value of Tmb ) 600.61 K. Other required parameters are as those of Figure 4.

Safaei SND · Va/PL, where Va is the atomic volume (the volume of an atom) of indium, PL ) 0.69 is its lattice packing factor, nt is the total number of indium atoms deposited on the Ge surface, and AS is the total surface area of the film. By these calculations, the measured melting point data of indium films, illustrated in Figure 8, are in fairly acceptable agreement with the present model. We have also shown the data of ref 29 cited by Jiang et al.11 for comparison, which is in better agreement with our model than our cited data whose thicknesses have been calculated by H ) SND · Va/PL. We should mention that eqs 3a and 3b have entirely overlapped each other. Here, we should briefly compare our present model with that presented previously in ref 14. If we plot the graphs of the model of ref 14 for spherical nanoparticles of Au, Pb, and Sn in comparison with those of the present model, it can be easily seen that generally both of these models have the same accuracy in predicting the melting point variation of these nanoparticles, but there are some advantages that can give preference to the present model. The first and the most important advantage of the present model over the previous model of ref 14 is that the present model has no adjustable parameter to fit the experimental data, while in the model of ref 14 (eq 14 of ref 14), the j V is 1/4, has j S/Z nanoparticle diameter, for which the value of Z been chosen as an adjustable and fitting parameter. This assumption has resulted from the discussion given in ref 13 in j V ) 1/2 gives j S/Z which it has been found that the value of Z good consistency with the melting point experimental data of j V ) 1/4 is j S/Z spherical nanoparticles larger than 10 nm, while Z suitably fitting for particles smaller than 10 nm. Also, if we plot the graph of the model of ref 14 for Sn spherical nanoparticles, we can see that the consistency of the present model (eqs 3a and 3b) with the experimental data for Sn nanoparticles smaller than 10 nm is somewhat better than the model of ref 14. Another advantage of the present model over the model of ref 14 is that the applicability of the previous model14 is restricted only to spherical nanoparticles, while the present model can be easily used for nonspherical nanoparticles, nanowires, and nanofilms. Because for nanowires and nanofilms j V may be different from 1/4, j S/Z the suitable fitting value of Z one can conclude that the model of ref 14 will be different for the case of nanowires and nanofilms, whereas the present model has a unique form for all nanocrystals. Hence, the present model is a generalization of the previous lattice type-sensitive models.13,14 4. Conclusion

Figure 8. Tmf(H) variation of indium nanofilms. The present model compared to the LDM,4 Jiang et al.,10–12 and the experimental data29 (denoted by squares). The data marked by circles are the experimental data of ref 29 cited by Jiang et al.11 The lattice structure of In is BCT j S ≈ 0.82, and Γb ≈ 0.5 (considering only the 1NN). with PL ) 0.69, P Other parameters are from Table 2.

(ML), and 1 ML was defined therein as the site number density (SND) of the unreconstructed Ge (100) surface, that is, SND ) 6.24 × 1014 atom/cm2.29 Considering PLASH ) ntVa and SND ) nt/AS, we calculated the thicknesses of indium films as H )

We developed a new nonlinear, generalized lattice typesensitive model, free of any adjustable parameter, for the shapeand size-dependent cohesive energy and melting point of freestanding nanocrystals taking into account the averaged energetic and structural effects of the surface and the volume of the crystal, especially the effect of the average surface packing factor and average coordination number. The adjustable parameter of the model of refs 13 and 14 has been removed here through introducing a new parameter named as the surface-to-volume energy contribution ratio (Γ) and using a new boundary condition based on its minimum value. We compared our model with several types of melting point experimental data measured by different techniques under different experimental conditions, for example, substrate-supported nanoparticles, encapsulated nanoparticles, those confined as inclusions in other materials, and nanofilms constrained between walls of different materials. It has been found that the present model has generally a good agreement with those experimental data, especially with the data obtained by calorimetry techniques. Moreover, the model has



been corrected to account for the effect of the substrate on melting point of substrate-supported nanocrystals and has been applied to the case of substrate-supported tin nanodisks. It is also worth mentioning that the present model can be straightforwardly used for different shapes of nanocrystals (nanodisk, spherical and nonspherical nanoparticles, nanowires, and nanofilms). It is evident from the given discussions that, to be able to accurately interpret any experimental results, we need experimenters to clearly specify the shape of nanocrystals at experimental conditions. The present model treats the melting phenomenon of individual nanocrystals without any external and surrounding effects, such as the effects of the substrate/ particle and particle/particle interactions, the contaminants and impurities effects, and also with no effect of the crystal defects, such as atomic vacancies, interfaces, grain boundaries, or voids. Acknowledgment. I am greatly thankful to the Zeus Engineering and Technical Company for providing research conditions. Appendix A: The Average Surface Packing Factor and nS/nt Ratio of Nanocrystals Now, we calculate the number of the surface atoms of the nanocrystal. We assume that the surface of the nanocrystals is composed of Ne number of different crystalline edges, Nf number of different crystalline faces, and Nc number of corner sites. For each surface edge k having n(k) e number of atoms, we have (k) (k) (k) (k) n(k) e ) Pe Le /d where Pe and Le are, respectively, the atomicpacking fraction and the length of surface edge k, and d is the atomic diameter of the building atoms of the nanocrystal. P(k) e is defined as the ratio of the length of surface edge k occupied (k) by atoms (n(k) e d) to the total length of that edge (Le ), for example, for an edge with [110] direction in a face-centered cubic (FCC) lattice, being the intersect line of two (111) and (001) faces, we have Pe ) 1.0. Hence, the total number of atoms located at all of the surface edges of the nanocrystal (ne) is equal to Ne

ne )

∑

k)1

( ) (k) P(k) e Le d

)

j eLe P , Le ) d

Ne

∑ L(k)e ,

k)1

Ne

je ) P

∑ P(k)e L(k)e

k)1

Le

(1A)

j e are the total length of all edges located at the where Le and P nanocrystal surface and the average edge-packing fraction of the nanocrystal surface, respectively. Similarly, for nf(j), that is, the number of atoms located at each surface face j, we have nf(j) ) Pf(j)Af(j)/Aa with Pf(j) and Af(j) being the atomic packing factor and the area of that surface face j, respectively. Pf(j) is defined as the ratio of the area of the surface face j occupied by atoms (nf(j)Aa) to the total area of that face (Af(j)). Also, Aa ) πr2 is the occupied area of that surface face associated with each surface atom of the nanocrystal and r is the atomic radius (r ) d/2, where d is the atomic diameter) deduced from the molar atomic volume.4 Generally, Aa can be considered as the cross-section area of the nanocrystal surface associated with each surface atom. Hence, the total number of atoms located at all of the surface faces of the nanocrystal (nf) is

Nf

nf )

∑ j)1

( ) (j) P(j) f Af Aa

j fAf P ) ,A ) Aa f

Nf

∑ A(j)f , j)1

Nf

jf ) P

∑ P(j)f A(j)f j)1

Af

(2A)

j f and Af being the average face-packing factor of the with P surface of the nanocrystal and the total area of all of its surface faces, respectively. If we consider the outer surface of the nanocrystal as an atomic crust and Aa as its cross-section area associated with each surface atom, then the total occupied area of this assumed atomic crust by all of the surface atoms (face, edge, and corner atoms) is equal to nSAa. On the other hand, the average packing j SAS j S) can be defined by P fraction of the nanocrystal surface (P ) nSAa, where AS is the total surface area of the nanocrystal or equally the area of its outer atomic crust. Therefore, considering j S ) nSAa/AS and eqs 1A and 2A, we can write P

jS ) P

j fAf j e(Le /d)Aa P P ncAa + + AS AS AS

(3A)

where nc is the total number of atoms located at all of the surface j fAf is the occupied area of corners of the nanocrystal. Here, P j eLe/d)Aa is the the crystal surface by all of the face atoms, (P part of the crystal surface area filled by all of the edge atoms, and ncAa is that area filled by all of the corner atoms. We can define the area fraction (xarea) of each type of surface sites (faces, edges, or corners) as the ratio of the area amount of the crystal surface associated with that surface site, being occupied and/or unoccupied by atoms, to the total area of the crystal surface area ) (the total area of surface site: f, e, c)/AS. (AS), that is, xf/e/c Therefore, xfarea ) Af/AS is the area fraction of the surface faces, ) (Le/d)Aa/AS is the area fraction of the surface edges and xarea e ) ncAa/AS is the area fraction of the surface corners. The xarea c total areas of the surface faces, edges, and corners, that is, Af, (Le/d)Aa, and ncAa, are the portions of AS that can be potentially occupied by the face, edge, and corner atoms, respectively. The j fAf, actual filled area of the crystal surface by the face atoms is P j by the edge atoms is (PeLe/d)Aa, and by the corner atoms is ncAa. Here, we should notice that the corner packing fraction j c) defined as the ratio of ncAa to the total area belonging to (P area j corner sites (xarea c AS) is equal to Pc ) ncAa/(xc AS) ) 1.0. Now, by attention to these definitions and using eq 3A, we can write

jS ) P j fxarea j exarea j cxarea j P +P +P f e c , Pc ) 1.0

(4A)

In eq 4A, we present a well-formulated, new size-dependent definition of the average packing factor of the crystal surface including the packing effects of the surface faces, surface edges, and surface corners. This definition, which considers the averaged arrangement of atoms at different surface sites, was first introduced in ref 15. From the definition of the area + xarea ) 1. Also, for fractions, we can conclude xfarea + xarea e c area area the bulk crystal, we have xe f 0, xc f 0, hence xfarea f 1; j f, which implies that the only important jS f P consequently, P factor for the surface of a bulk crystal is its averaged surfaceface atomic arrangement. In this approach, we treat the surface edges and corners not as geometrical lines or dots, respectively, but as the parts of the surface area required for the placement

13494


Safaei

of the edge and corner atoms, namely, as the area domain to physically define the atoms and their arrangements. Defining the volume of the nanocrystal (Vc) by (nt - nS/2)Va ) PLVc,13 where PL is its lattice packing factor, defined as the ratio of the volume of the unit cell occupied by atoms to the j SAS/Aa, we can total volume of the unit cell, and using nS ) P calculate the nS/nt ratio of a disklike nanocrystal (nanodisk) having the height H and the diameter D as:

j Sd(1/H + 2/D) nS 2P ) nt j Sd(1/H + 2/D) (3PL /2) + P

(5A)

We obtained eq 5A for a nanodisk, but more importantly, we can easily extend it for all common types of nanocrystals, that is, for nano-wires, -particles, and -films, through defining the aspect ratio (AR) for the nanodisk as: AR ) H/D. The nS/nt ratio of a long nanowire of diameter D, a spherical nanoparticle of diameter D, and a nanofilm with thickness H can be, respectively, obtained from eq 5A through limARf∞ nS/nt, limARf1 nS/nt, and limARf0 nS/nt. This idea has been previously used as a shortcut to obtain the cohesive energy of nanowires and nanofilms from nanodisks,4,13 but it has never been considered for nanoparticles until now. This is a new view of different shapes in the nanoworld that a nanodisk may be considered as a fundamental nanocrystal from which the physical/chemical properties of other shapes of nanocrystals, especially a nanoparticle, can be easily obtained. Therefore, we can write a general equation for all nanocrystals. We introduce a critical nanodisk with the height H0 and the diameter D0 for which all of its atoms are located at its surface, that is, nS/nt ) 1. Defining X0 ) 1/H0 + 2/D0 and using nS/nt ) 1 and eq 5A, we can find

X0 )

3PL , X ) 1/H0 + 2/D0 j Sd 0 2P

(6A)

Therefore, defining X ) 1/H + 2/D, we can rewrite eq 5A as nS/nt ) 2X/(X0 + X). Hence, using eq 1, the cohesive energy (Ecn) and melting point (Tmn) of all nanocrystals can be formulated in a general form in terms of the reciprocal dimensions of the nanocrystal as below:

Ecn Tmn X ) ) 1 - 2(1 - Γ) , X ) 1/H + 2/D Ecb Tmb X0 + X (7A) This equation can be used for a nanodisk with X ) 1/H + 2/D, for nanowires with X ) 2/D, and, hence, X0 ) 2/D0 (because limARf∞ X ) 2/D), for spherical nanoparticles with X ) 3/D and X0 ) 3/D0 (because limARf1 X ) 3/D) and for nanofilms with X ) 1/H and, hence, X0 ) 1/H0 (because limARf0 X ) 1/H). Here, D0 is the diameter of the nanowire and/or spherical nanoparticle, and H0 is the thickness of the nanofilm for which all of its atoms are located at its surface. Appendix B: The Development of the Model j S/ With the use of the assumption that the average value of Z j V, as an adjustable parameter, for nanoparticles smaller than Z j V has been j S/Z 10 nm is 0.25, a size-dependent function for Z obtained in ref 14, which is applicable only for the case of

spherical nanoparticles. Here, using a different assumption on the lower boundary condition for the surface-to-volume energy contribution ratio of a nanocrystal (Γ), we are going to find a dimension-dependent function, free of any adjustable parameter, for Γ. As it has been well established, the cohesive energy (Ecn) and the melting point (Tmn) of free-standing nanosolids decrease with a decrease in their size,4–29 that is, d(Tmn/Tmb)/d(size) > 0 or, equivalently, d(Tmn/Tmb)/d(1/size) < 0 (size ) H, D). Because ∂X/∂(1/H) ) 1 > 0, ∂X/∂(1/D) ) 2 > 0, we have d(Tmn/Tmb)/dX < 0. Assuming that Γ is a differentiable function of X and using eq 7A, we can write d(Tmn/Tmb)/dX ) 2X(X0 + X)-1(dΓ(X)/dX) - 2(1 - Γ(X))X0(X0 + X)-2 < 0. Because the surface atoms have the lower thermal stability than the volume atoms, we can j V and so 1 - Γ > 0 (Γ ) E j S/E j V); therefore, jS < E conclude that E (1 - Γ(X))X/(X0 + X) > 0. Hence, dividing the above-mentioned equation by 2(1 - Γ)X/(X0 + X) > 0, we obtain

f(X) )

dΓ(X) /dX X0 0 and F(X) ) exp(∫(-f(X)) dX), we have

1 - Γ(X) ) C0

X0 + X F(X), F(X) > 0 X

(2B)

Equation 2B shows Γ as a function of the reciprocal dimensions of the nanocrystal. In the following, with the use of the appropriate boundary conditions, we discuss some possible functions of F(X) as follows. For adequately large and near to the bulk nanocrystals (X f 0), we should have limXf0 Γ(X) ) Γb. Hence, using eq 2B, we can obtain

lim

Xf0

(

)

1 - Γb X0 + X F(X) ) X C0

(3B)

where Γb is the surface-to-volume energy contribution ratio of the bulk crystal and can be obtained from eq 6. Now considering eq 3b as the first boundary condition, we can conclude that there are four possible situations for the value of limXf0 F(X), for any possible function of F(X), as follows. (I) limXf0 F(X) ) (∞, c, c * 0 (c is a nonzero constant): All of these three values ((∞, c) for limXf0 F(X) are in discrepancy with the condition of eq 3B, because for these cases we have limXf0((X0 + X)F(X)/X) ) (∞ * (1 - Γb)/C0. Besides this, considering eq 2B and also F(X) > 0, we could understand that limXf0 F(X) cannot have a negative value. Hence, all of these three values are not acceptable for limXf0 F(X). The only remaining value, which may be possible for limXf0 F(X), is zero, which is discussed below. (II) limXf0 F(X) ) 0: This leads to limXf0(1 - Γ(X)) ) 0/0, which is an ambiguous and indefinite value; therefore, limXf0((X0 + X)F(X)/X) ) limXf0((F(X) + (X0 + X)F˙(X))/1) ) X0 limXf0 F˙(X). From eq 2B, it can be found that F˙(X) is also a differentiable function like as Γ(X); hence, there exists a function like F˙(X) for which F˙(X) ) dF(X)/dX. Now, considering the condition of eq 3B, we can find limXf0 F˙(X) ) (1 - Γb)/X0C0. Therefore, F(X) should have the following conditions:


lim F(X) ) 0) and (Xf0

(

lim F˙(X) )

Xf0

1 - Γb X0C0

)


(4B)

Now, we find some possible functions of F(X) by considering the conditions of eq 4B.

nanocrystal, considering Smn ) 0 and eq 7B, we have (Tmn/ Tmb)min ) exp(-2Smb/3R); then, using eq 7A leads to

1 - Γmin )

(

[1 - exp(-2Smb /3R)] X0 1+ 2 Xmin

)

(8B)

i. A Polynomial Relation First, we assume F(X) ) c0 + c1X + c2X2 + ... + cIXI, where I and c0, c1, c2, etc. are constant values. Applying eq 4B to F(X), we can find c0 ) 0, c1 ) (1 - Γb)/X0C0. Hence, we have F(X) ) ((1 - Γb)/X0C0)X + c2X2 + ... + cIXI. Substituting F(X) into eq 2B and then simplifying leads to (1 - Γ(X))/(1 - Γb) ) (1 + (X/X0))(1 + c2C0X0(1 - Γb)-1X + ... + cIC0X0(1 - Γb)-1XI-1). Defining CJ-1 ) cJC0X0/(1 - Γb), where J ) 2, 3, ..., I, we can write

(

)

1 - Γ(X) X ) 1+ (1 + C1X + ... + CI-1XI-1) 1 - Γb X0

(5B)

Equation 5B is a nonlinear function in terms of the reciprocal dimensions of the nanocrystal. Assuming C1 * 0,C2 ) C3 ) ... ) CI-1 ) 0, we can find Γ(X) by only one another boundary condition as follows. The coordination numbers of the nanocrystal’s atoms and therefore its surface-to-volume energy contribution ratio (Γ) can be considered as a characteristic of its crystallinity order. The higher the coordination number and the bond energy, the higher the crystallinity order. Therefore, by attention to the sizedependent reduction of the average coordination number of atoms,31 we can conclude that the crystallinity order of nanocrystals decreases with a decrease in their size. This implies that there exists a nanocrystal size for which the crystallinity order of the solid is reduced to sufficiently small amounts to be equal to that of the liquid.10–12,34 Hence, this critical nanocrystal, whose structure is the same as the liquid, can be considered as the smallest crystalline nanosolid.10–12,34 Therefore, we can conclude that for this crystal the parameter Γ should have its minimum value (Γmin). Denoting the reciprocal dimensions of this smallest crystal as Xmin, considering eq 5B and C1 * 0, C2 ) C3 ) ... ) CI-1 ) 0, we can obtain

(

)

1 - ΓX X ) 1+ (1 + C1X), 1 - Γb X0 (1 - Γmin) / (1 - Γb) 1 -1 C1 ) Xmin 1 + (Xmin /X0)

[

]

( )

Tmn 3R ln 2 Tmb

Ecn Tmn X ) ) 1 - 2(1 - Γb) (1 + C1X) Ecb Tmb X0 1 [1 - exp(-2Smb /3R)] X0 -1 C1 ) Xmin 2(1 - Γb) Xmin

(

)

(9B)

To calculate C1 in eq 6B or 9B, we need to have the exact values of Xmin or Γmin, but these two unknown quantities are related together, with only one equation (eq 8B). Thus, we need another assumption to calculate Xmin or Γmin. Because of the lack of any reliable information about Γmin, we preferably use a rough approximation to calculate Xmin. Because a crystal is characterized by its long-range order, the smallest crystal may have approximately half of its atoms located at its surface, that is, nt ) 2nS.11,12 Therefore, using nS/nt ) 2X/(X0 + X), we can calculate XC for which nt ) 2ns, as below:

XC )

X0 3PL , X0 ) 3 j Sd 2P

(10B)

As mentioned above, we can approximately use Xmin ≈ XC, which may cause some deviations for Γmin from what actually is expected. Hence, assuming Xmin ≈ XC and using eq 9B, we can finally find the cohesive energy and the melting point of nanocrystals as:

(

)

Tmn Ecn X X ) ) 1 - 2(1 - Γb) 1 + C , Ecb Tmb X0 X0 9[1 - exp(-2Smb /3R)] C) -3 2(1 - Γb) j Sd X ) 1/H + 2/D, X0 ) 3PL /2P

(11B) (6B)

Because of having identical liquid and solid structures, the melting entropy of this nanocrystal is zero, i.e. Smn(Xmin) ) 0,10–12 leading to an estimation for its size (1/Xmin).10,12,34 Jiang et al. developed a well-known relation between the melting point of nanosolids and their size-dependent overall melting entropy (Smn) as below:12,32

Smn ) Smb +

Now, substituting eq 8B into eq 6B and then its result into eq 7A, we can have the following equation for the cohesive energy and melting point of nanocrystals:

(7B)

where Smb is the overall melting entropy of the bulk material and R is the ideal gas constant. Hence, for the smallest

Similar to eq 7A, eq 11B is applicable for nanodisks (X ) 1/H + 2/D), spherical nanoparticles (X ) 3/D), long nanowires (X ) 2/D), and nanofilms (X ) 1/H) and also other shapes of nanocrystals. ii. A Homographic Function for F(X) We can also assume a function like F(X) ) (c0 + c1X)/(c2 + c3X), where c0, c1, c2, and c3 are constants. Through considering the conditions of eq 4B, we can find c0 ) 0, c1/c2 ) (1 - Γb)/ X0C0, and then F(X) ) (1 - Γb)(X0C0)-1 X/(1 + C′X) where C′ ) c3/c2. Now, using eq 2B leads to

1 - Γ(X) 1 + (X/X0) ) 1 - Γb 1 + C'X

(12B)

13496


Safaei

To obtain the appropriate value of C′, we need another boundary condition. Hence, considering Γ(Xmin) ) Γmin in eq 12B, we can find 1 - Γ(X) 1 - Γb

)

1 + (X/X0) , 1 + C'X C' )

[

]

1 + (Xmin /X0) 1 -1 (1 - Γmin)/(1 - Γb) Xmin

(13B)

Now, using eqs 7A, 8B, and 13B, with a little algebra, leads to the following relation:

Ecn Tmn X/X0 ) ) 1 - 2(1 - Γb) , Ecb Tmb 1 + C'X 2(1 - Γb) Xmin 1 -1 C' ) [1 - exp(-2Smb /3R)] X0 Xmin

{

}

(14B)

The only unknown quantity in eq 14B is Xmin. If we had the exact value of Xmin, then, we could obtain Ecn/Ecb without any approximation. Because of the lack of decisive information on Xmin, here, we use the rough approximation of Xmin ≈ XC in eq 14B and then finally obtain

Ecn Tmn X/X0 ) ) 1 - 2(1 - Γb) , Ecb Tmb X 1+C X0 2(1 - Γb) C) - 3 (15B) 1 exp(-2S ( mb /3R)) Equation 15B is also another relation for the cohesive energy and the melting point of nanodisks (X ) 1/H + 2/D), spherical nanoparticles (X ) 3/D, X0 ) 3/D0), nanowires (X ) 2/D, X0 ) 2/D0), and nanofilms (X ) 1/H, X0 ) 1/H0) and other shapes of nanocrystals. References and Notes (1) Sun, C. Q. Prog. Solid State Chem. 2007, 35, 1. (2) Tateno, J. Solid State Commun. 1972, 10, 61. (3) Yang, C. C.; Li, S. Phys. ReV. B 2007, 75, 165413, and references therein. (4) Nanda, K. K.; Sahu, S. N.; Behera, S. N. Phys. ReV. A 2002, 66, 013208. (5) Buffat, P. A.; Borel, J. P. Phys. ReV. A 1976, 13, 2287. (6) Pawlow, P. Z. Phys. Chem. 1909, 65, 1.

(7) Hanszen, K.-J. Z. Phys. 1960, 157, 523. (8) Sakai, H. Surf. Sci. 1996, 351, 285. (9) Luo, W.; Hu, W.; Xiao, S. J. Chem. Phys. 2008, 128, 074710. (10) Jiang, Q.; Yang, C. C. Curr. Nanosci. 2008, 4, 179. (11) Jiang, Q.; Tong, H. Y.; Hsu, D. T.; Okuyama, K.; Shi, F. G. Thin Solid Films 1998, 312, 357. (12) Jiang, Q.; Shi, H. X.; Zhao, M. J. Chem. Phys. 1999, 111, 2176. (13) Safaei, A.; Shandiz, M. A.; Sanjabi, S.; Barber, Z. H. J. Phys.: Condens. Matter 2007, 19, 216216. (14) Safaei, A.; Shandiz, M. A.; Sanjabi, S.; Barber, Z. H. J. Phys. Chem. C 2008, 112, 99. (15) Safaei, A. J. Nanopart. Res. 2010, 12, 759. (16) Yang, C. C.; Li, S. J. Phys. Chem. C 2009, 113, 14207. (17) Kim, H. K.; Huh, S. H.; Park, J. W.; Jeong, J. W.; Lee, G. H. Chem. Phys. Lett. 2002, 354, 165. (18) Dick, K.; Dhanasekaran, T.; Zhang, Z.; Meisel, D. J. Am. Chem. Soc. 2002, 124, 2312. (19) Allen, G. L.; Bayles, R. A.; Gile, W. W.; Jesser, W. A. Thin Solid Films 1986, 144, 297. (20) Sambles, J. R. Proc. R. Soc. London A. 1971, 324, 339. (21) Skripov, V. P.; Koverda, V. P.; Skokov, V. N. Phys. Status Solidi A 1981, 66, 109. (22) Coombes, C. J. J. Phys. F: Met. Phys. 1972, 2, 441. (23) Peters, K. F.; Cohen, J. B.; Chung, Y.-W. Phys. ReV. B 1998, 57, 13430. (24) Kofman, R.; Cheyssac, P.; Aouaj, A.; Lereah, Y.; Deutscher, G.; Ben-David, T.; Penisson, J. M.; Bourret, A. Surf. Sci. 1994, 303, 231. (25) Lai, S. L.; Guo, J. Y.; Petrova, V.; Ramanath, G.; Allen, L. H. Phys. ReV. Lett. 1996, 77, 99. (26) Wronski, C. R. M. Br. J. Appl. Phys. 1967, 18, 1731. (27) Tsuboi, T.; Seguchi, Y.; Suzuki, T. J. Phys. Soc. Jpn. 1990, 59, 1314. (28) Qi, W. H.; Wang, M. P.; Chin., J. Nonferrous Met. 2006, 16, 1161. (29) Krausch, G.; Detzel, T.; Bielefeldt, H.; Fink, R.; Luckscheiter, B.; Platzer, R.; Wiihrmann, U.; Schatz, G. Appl. Phys. A: Mater. Sci. Process. 1991, 53, 324. (30) Chacko, S.; Kanhere, D. G.; Blundell, S. A. Phys. ReV. B 2005, 71, 155407. (31) Montejano-Carrizales, J. M.; Aguilera-Granja, F.; Moran-Lopez, J. L. Nanostruct. Mater. 1997, 8, 269. (32) Jiang, Q.; Shi, F. G. Mater. Lett. 1998, 37, 79. (33) Jiang, Q.; Lu, H. M.; Zhao, M. J. Phys.: Condens. Matter 2004, 16, 521. (34) Safaei, A.; Shandiz, M. A. Phys. E 2009, 41, 359. (35) Shim, J.-H.; Lee, B.-J.; Cho, Y. W. Surf. Sci. 2002, 512, 262. (36) Arcidiacono, S.; Bieri, N. R.; Poulikakos, D.; Grigoropoulos, C. P. Int. J. Multiphase Flow 2004, 30, 979. (37) Lee, J.; Nakamoto, M.; Tanaka, T. J. Mater. Sci. 2005, 40, 2167. (38) Lim, H. S.; Ong, C. K.; Ercolessi, F. Z. Phys. D 1993, 26, S45. (39) Weissmuller, J. Nanostructured Mater. 1993, 3, 261. (40) Jiang, Q.; Zhao, D. S.; Zhao, M. Acta Mater. 2001, 49, 3143. (41) Jiang, Q.; Shi, H. X.; Zhao, M. Acta Mater. 1999, 47, 2109. (42) Lu, H. M.; Wen, Z.; Jiang, Q. Colloids Surf., A 2006, 278, 160. (43) Lai, S. L.; Carlsson, J. R. A.; Allen, L. H. Appl. Phys. Lett. 1998, 72, 1098. (44) Sun, J.; Simon, S. L. Thermochim. Acta 2007, 463, 32. (45) Lide, D. R., Ed. Thermal and Physical Properties of Pure Metals. In CRC Handbook of Chemistry and Physics, Internet Version 2005; CRC Press: Boca Raton, FL, 2005; http://www.hbcpnetbase.com.

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Shape, Structural, and Energetic Effects on the Cohesive Energy and

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