syntesis.articulo.melting.temperatute.nanoparticles

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Materials Chemistry and Physics 88 (2004) 280284

Size and shape dependent melting temperatureof metallic nanoparticles

W.H. Qi , M.P. WangSchool of Materials Science and Engineering, Central South University, Changsha 410083, China

Received 12 December 2003; received in revised form 27 March 2004; accepted 8 April 2004

Abstract

A new model accounting for the particle size and shape dependent melting temperature of metallic nanoparticles is proposed in this paper,where the particle shape is considered by introducing a shape factor. It is shown that the particle shape can affect the melting temperatureof nanoparticles, and the particle shape effect on the melting temperature become larger with decreasing of the particle size. The presentcalculation results on the melting temperature of Sn, Pb, In and Bi nanoparticles are well consistent with the corresponding experimentalvalues and better than these given by liquid drop model. 2004 Elsevier B.V. All rights reserved.

Keywords: Melting temperature; Nanoparticles; Shape effect

1. Introduction

Crystal melting is a very complicated process. Since Tak-agi rst reported the size dependent melting temperature of small particlesby meansof transmission electron microscope[1], researchers have paid more attention to this basic but stillunclear phenomenon [25]. Now it is found that the meltingtemperature of metallic [2,3], organic [4] and semiconduc-tors [5] nanoparticles decreaseswith decreasing theirparticlesize. In other words, their melting temperature is lower thanthe corresponding bulk materials.

It is known that the melting temperature depression re-sults from the high surface-to-volume ratio, and the surfacesubstantially affects the interior bulk properties of these

materials. A lot of models try to explain the size dependentmelting temperature [6,7], such as liquid drop model [6] andJiangs model [7], etc. However, one common characteris-tic of these models is that the nanoparticles are regarded asideal spheres. Since the melting temperature depression re-sults from the large surface-to-volume ratio, the surface areasof nanoparticles in different shape will be different even in

Corresponding author. Fax: +86 731 8876692. E-mail address: [email protected] (W.H. Qi).

the identical volume, and the area difference is large espe-cially in small particle size. Therefore, it is needed to takethe particle shape into consideration when developed modelsfor the melting temperature of nanoparticles.

In this contribution, we will introduce a shape factor to ac-count for theparticle shape difference, anddevelopthemodelfor the size andshape dependent cohesive energyof nanopar-ticles. According to the relation between melting temperatureand cohesive energy, the expression for the size and shapedependent melting temperature of nanoparticles will be de-veloped, and the theoretical predictions of this expression forthe melting temperature of Sn, Pb, In and Bi nanoaparticleswill be compared with the available experimental values andthe theoretical results given by liquid drop model.

2. Model

To account for the particle shape difference, we have in-troduced a new parameter [8], i.e., the shape factor , whichis dened by the following equation

=S S

(1)

0254-0584/$ see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.matchemphys.2004.04.026


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W.H. Qi, M.P. Wang / Materials Chemistry and Physics 88 (2004) 280284 281

where S is the surface area of the spherical nanoparticleand S = 4R 2 ( R is its radius). S is the surface area of thenanoparticlein any shape,whosevolume is thesameasspher-ical nanoparticle. It should be mentioned that this denitionof the shape factor is dimensionless, and which is easier tobe introduced in the theoretical models of spherical nanopar-

ticles as a modied parameter to generalize these models.According to Eq. (1), the surface area of a nanoparticle inany shape can be written as

S = 4R 2 (2)

If the atoms of the nanoparticle are regarded as idealspheres, then the contribution to the particle surface area of each surface atom is r 2 (r is the atomic radius). The numberof the surface atoms N is the ratio of the particle surface areato r 2, which is simplied as

N = 4R2

r 2 (3)

Thevolume of thenanoparticle V is the same asthe spheri-calnanoparticle,which equals to (4 / 3)R 3. Then the numberof the total atoms of the nanoparticle is the ratio of the particlevolume to the atomic volume ((4 / 3)r 3), which leads to

n =R3

r 3 (4)

If the surface atoms denote the atoms in the rst layer of thesurface of thenanoparticle, then thenumberof the interioratoms is n N .

It is known that cohesive energy is an importantparameterto estimate the metallic bond, which equals to the energy thatcan divide the metal into isolated atoms by destroying allmetallic bond. For simplicity, we can also regard the metallicbond as the interactions between different atoms. In otherwords, the metallic bonds of each atom equal to the sum of the interaction energiesbetween theatom andtheother atoms(in most cases, only the nearest interactions are considered).Each interior atom forms bonds with its surrounding atoms,and we denote the number of its bonds as .

It is reported that the distance between the surface atomsand the nearest interior atoms is larger than the distance be-tween interior atoms [9]. Therefore, less than half of the vol-

ume of each surface atom in the lattice, which means morethan half of the bonds of the surface atom are dangling bonds,then we approximately regard the number of the bonds of a surface atom as (1 / 4) . The cohesive energy of metallicnanoparticle is the sum of the bond energies of all the atoms.Considering Eq. (4), the cohesive energy of metallic crystalin any shape ( E p) can be written as

E p =12

14

4R2

r 2 +

R3

r 3 4

R2

r 2E bond (5)

where E bond is bond energy. The value 1/2 results from thefact that the each bond belongs to two atoms. For simplicity,

we can rewrite Eq. (5) as

E p =12

nE bond 1 6rD

(6)

where D is the size of the crystal and D = 2 R.For bulk solids, D 6 r , Eq. (6) is reduced to E 0 =

(1/ 2)nE bond , where E 0 is the cohesive energy of solids. Wecan rewritten Eq. (6) as

E p = E 0 1 6rD

(7)

It should be mentioned that the size D is the diameter of the spherical crystal. For a non-spherical crystal, its size isdened as the diameter of the spherical crystal which has theidentical volume with the non-spherical crystal. If D denotesthe diameter of the metallic nanoparticles, Eq. (7) can beused to predict size and shape dependent cohesive energy of metallic nanoparticles.

In Eq. (7), we have used E p and E 0 . The most difference

between E p and E 0 is that E p is taken the surface effect onthe cohesive energy into consideration. For bulk crystal, E pequals E 0; formetallicnanoparticles, thesurfaceeffect cannotbe neglected, E p and E 0 are different. Therefore, E p can alsodescribe the cohesive energy of nanoparticles, but E 0 cannot.According to these discussions, Eq. (7) can be regarded as amore general relation for the cohesive energy of crystals.

Rose et al. [1013] proposed a universal model for solidsfrom thebinding theoryof solid.Combining their theorywithDebye model, they theoretically derived the well-known em-pirical relation of the melting temperature and the cohesiveenergy for pure metals

T mb = 0.032kBE 0 (8)

where T mb is themelting temperature of bulk pure metals, andk b the Boltzmanns constant. Similar to the cohesive energy,the melting temperature is also a parameter to describe thestrength of metallic bond. Therefore, Eq. (8) can be regardedas the mathematical conversion of both parameters. We re-place the cohesive energy of solids E 0 by the more generalform E p , then

T m =0.032

kBE 0 1 6

rD

(9)

The main difference between Eqs. (8) and (9) is that Eq.(9) has taken the crystal size and shape (surface effect) intoaccount. For convenience, we denote the size and shape de-pendent melting temperature as T m . According to Eq. (8), wecan rewrite Eq. (9) as

T m = T mb 1 6rD

(10)

Eq. (10) is the more general relation for the size and shapedependent melting temperature of crystals. The relation be-tween T m and T mb are similar to the relation between E p and E 0, i.e., for bulk crystal, T m and T mb are the same, but T m canalso describe the melting temperature of nanoparticles. Here


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we mainly deal with the melting temperature of nanoparti-cles, therefore, we assume that T m is the melting temperatureof nanoparticles and T mb denotes the melting temperature of the corresponding bulk materials.

3. Results and discussion

The main difference between Eq. (10) and other expres-sions [6,7] for the size dependent melting temperature is thatthe particle shape is considered in Eq. (8), where the particleshape is well describedby the shape factor . To calculate themelting temperature of metallic nanoparticles by Eq. (8), itis needed to determine the shape factor for different particleshapes. According to the denition of shape factor, we havecalculated the shape factor of different particle shapes, andwhich is listed in Table 1 .

For the particle shape is different from each other, theshape factor is only approximate description of the particleshape difference. In most experiments, the nanoparticles areclose to regular polyhedral shape [14], so itis neededfor ustodiscuss the shape factor of regular polyhedralnanoparticles inmore details. Thesimplest polyhedral particle is regular tetra-hedral nanoparticles, and its shape factor equals 1.49. Theshape factors of other regular polyhedral particlesare smallerthan 1.49 according its denition (Eq. (1)), i.e., the value of 1.49 is the up limitation for regular polyhedral nanoparticles.Furthermore, the down limitation of the shape factor of theregular polyhedral nanoparticles is 1, i.e., the shape factorof the spherical nanoparticles. In the present calculation, wewill give the calculation results of the melting temperatures

of metallic nanoparticles at the two limitations.It is shown in Eq. (10) that the melting temperature is the

function of the particle size and the shape factor. Therefore,we can discuss the melting temperature variation with theshape factor in a specic particle size, and can also discussthe melting temperature variation with the particle size in aspecic particle shape. The input values in our calculationare listed in Table 2 .

The variation tendency of the relative melting tempera-ture with respect to the shape factor calculated by Eq. (10) isshown in Figs.1and2 , where themelting temperatureof 5,10and 20 nm Sn and Pb nanoparticles are calculated. It is shownthat the melting temperature of Sn and Pb nanoparticles de-creases with increasing of the shape factor. According to thedenition of the shape factor, the surface area increases with

Table 1The calculated shape factor for different particle shapes

Particle shape Shape factor ( )

Spherical 1Regular tetrahedral 1 .49Regular hexahedral 1 .24Regular octahedral 1 .18Disk-like >1 .15Regular quadrangular >1 .24

Table 2The input values of present model and liquid drop model

Elements Atomic radius[15] (nm)

[6] (nm) Melting temperature of bulk materials [16] (K)

Sn 0.140 2.2784 505.1Pb 0.175 1.7957 600.6In 0.162 2.6500 429.8Bi 0.174 2.1273 544.5

Fig. 1. Variation of the relative melting temperature of Sn nanoparticles asa function of shape factor. The solid lines are the results calculated from Eq.(10).

increasing of theshapefactor ina specic particle size.There-fore, thesurfaceeffect on themelting temperatureof nanopar-ticles may be strengthened in large shape factor, which leadsto the decreasing of the melting temperature in wide range.Furthermore, it is found that the particle shape have largereffect on small particles than on large particles. For example,the relative melting temperature variation of Sn nanoparti-

cles in 5 nm is 0.15, and is 0.02 in 30nm, which suggests thatthe particle shape should be taken into consideration whenstudied the melting properties of nanoparticles in small size.For In and Bi nanoparticles, the variations of their meltingtemperature with respect to the shape factor are similar tothese of Sn and Pb nanoparticles, which are not plotted.

From Figs. 36 , the theoretical results on the melting tem-perature of Sn, Pb, In and Bi nanoparticles calculated bypresent model and liquid drop model are presented, and the

Fig. 2. Variation of the relative melting temperature of Pb nanoparticles asa function of shape factor. The solid lines are the results calculated from Eq.(10).


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W.H. Qi, M.P. Wang / Materials Chemistry and Physics 88 (2004) 280284 283

Fig. 3. Variation of the melting temperature as the function of the inversediameterof Snnanoparticle.The symbols( ) denote theexperimentalvalues[17].

Fig. 4. Variation of the melting temperature as the function of the inversediameterof Pbnanoparticle.The symbols( ) denote theexperimentalvalues[18].

available experimental results [1720] are also shown. In liq-uid drop model, the relation for the size dependent meltingtemperature is T m = T mb(1 /D ), where values of the pa-rameter for different particles are listed in Table 2 [6] .

It is shown that the melting temperature of these nanopar-ticles decreases with decreasing of the particle size, the vari-ation tendency of our results and these of liquid drop modelis consistent with the experimental values. It is also shownthat the results given by our present model are more close

Fig. 5. Variation of the melting temperature as the function of the inversediameterof In nanoparticle.The symbols( ) denote theexperimentalvalues[19].

Fig. 6. Variation of the melting temperature as the function of the inversediameterof Bi nanoparticle.The symbols( ) denote theexperimentalvalues[20] .

to the experimental values, and these given by liquid dropmodel are lower than theexperimental values. In other words,our model is better than the liquid drop model in predictingthe melting temperature of nanoparticles. For Sn, Pb and Binanoparticles, the experimental results are close to our the-oretical results of = 1 when the particle size D is largerthan about 10 nm, and in the middle of our theoretical resultsbetween = 1 and 1.49 when the particle size is smaller thanabout 10 nm, which suggests that the shape of the nanoparti-cles may be in spherical when the particle size is large, andin polyhedral shape when the particle size is small. For Innanoparticles, theexperimental values lie in themiddleof thetwo curves = 1 and 1.49, which may suggest that the shapeof the nanoparticles may be in polyhedral. The shape of thenanoparticles is mainly determined by the preparation meth-

ods [21], i.e., the spherical nanoparticles may be prepared bythe chemical methods, and the non-spherical nanoparticlesmay be prepared by physical method. Thepresent theoreticalpredictions on the particle shape effect on the melting tem-perature of metallic nanoparticles may be tested by furtherexperiments.

The melting temperature depression of nanoparticles isapparent only when the particle size is smaller than 100 nm.If the particle size is larger than 100 nm, the melting tem-perature of the particles approximately equals to the corre-sponding bulk materials, in other words, the melting tem-perature of nanoparticles is independent of the particle size.This fact is supported by experimental results [1720] andcan be explained by the present model. If the particle sizeis large enough (larger than 100nm), the percentage of thesurface atoms is fairly small. According to thepresent model,the melting temperature variation results from the effect of surface atoms and the effect of small percentage of surfaceatoms on the melting temperature can be neglected. In Eq.(10), if the particle size is fairly large, we have 6 r/D 1and T m T mb .

It should be mentioned that the shape factor in thepresent work only approximately describes the shape differ-ence between the spherical nanoparticles and the polyhedralnanoparticles.Theapproximatelyis stressedheredue to the


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284 W.H. Qi, M.P. Wang / Materials Chemistry and Physics 88 (2004) 280284

fact that some different polyhedral nanoparticles may havethe identical shape factor, which results from the fact thatthe shape factor is dened by the surface area. However, thepresent calculation result shows that the present denitionof shape factor is enough for predicting the shape dependentmelting temperature of nanoparticles.Furthermore, the shape

factor is a new parameter to characterize the nanoparticles,and it can be experimentally determined by measuring theparticle shape, which is the subject of further experiments.

It is reported that the atomic radius of metallic nanopar-ticles contracts with decreasing their particle size [22,23] ,which means that the atomic radius will change a little if theparticle size fairly small. However, for most metal particle,the ratio of contraction is less than 1% [23] and which isignored in our model.

The expression for the size dependent melting tempera-ture of nanoparticles in present work is derived from theirsize dependent cohesive energy. It should be mentioned thatthe present model for the size dependent cohesive energy isonly for the free surface nanoparticles, i.e., the matrix has noeffect on the surface of the nanoparticle. If the nanoparticlesare embedded in a matrix with high melting point, and thesurface of the nanoparticles have coherent or semi-coherentinterface with the matrix, the matrix may affect the cohesiveproperties of nanoparticle. The melting temperature of thenanoparticles may increase with decreasing of the particlesize (superheating) [24]. In our further work, we will gener-alize the present model to explain this special phenomenon.

4. Conclusions

We have studied the size and shape dependent meltingtemperature of nanoparticles by our new model, where theshape of thenanoparticles is considered by introducing a newparameter, i.e., the shape factor. It is shown that the presentcalculated results of the melting temperature of Sn, Pb, Inand Bi nanoparticles are well consistent with the experimen-tal values and better than these given by liquid drop model.Furthermore, it is found that the particle shape can affect themelting temperature of nanoparticles, and this effect on themelting temperature become larger with decreasing of the

particle size. For the melting temperature is a very importantquantity, we are condent that the model developed in thispaper may have potential application in the research of thetemperature-related phenomena of nanoparticles.

Acknowledgement

One of the authors (W.H. Qi) thanks Professor Q. Jiangfor sharing important literature.

References

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[19] V.P. Skripov, V.P. Koverda, V.N. Skokov, Phys. Status Solidi A 66(1981) 109.

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