Comparison of different models for melting point change of metallic nanocrystals
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Our phenomenological model without adjustable parameters for the size dependence and dimension dependence of melting point depression and enhancement of nanocrystals is introduced. The predictions of our models are consistent with both of experimental results and other thermodynamic models for metallic nanocrystals while the difference between our model and other theoretical considerations in mesoscopic size range is discussed.
I. INTRODUCTION
Both melting point depression and enhancement of nanocrystals have been found to depend on size, dimension, and surface conditions of the nanocrystals.1–19 To illustrate the phenomena, thermodynamic considerations have been carried out. A well-known equation to predict size-dependent melting point Tr of a particle with a radius of r was first derived by Pawlow20 and then reevaluated by Hanszen:21 T r /T0 ⳱ 1 − 2Vs[␥sv − ␥lv(s/1)2/3]/[Hmr]
, (1)
where T0 denotes bulk melting temperature, Vs is molar volume of the crystal, ␥ and are interfacial energy per unit area and density, respectively, subscripts of s, l, v denote crystal, liquid and vapor phases, and Hm is molar melting enthalpy at Tr , being temperature dependent12,13 and size dependent.13,16 It will be observed in the following function that Hm in Eq. (1) disappears when the value of interface energy of Eq. (10) is substituted into Eq. (1) as shown in Eq. (11). Thus, the detailed form of Hm is not considered further. For most cubic metals,14 ␥sv − ␥lv ≈ ␥sl
.
(2)
With s ≈ 1, (s /1)2/3 ≈ 1 and in terms of Eq. (2), Eq. (1) can be expressed as Tr /T0 ⳱ 1 − 2Vs␥sl/[Hmr]
.
(3)
In fact, Eq. (3) is identical to the Gibbs–Thomson equation13,22 Tr /T0 ⳱ 1 − (1/rl + 1/r2)Vs␥sl/Hm ,
a)
(4)
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J. Mater. Res., Vol. 16, No. 11, Nov 2001 Downloaded: 08 Jan 2015
where r1 and r2 are principal radii of curvature of the interface that bound a solid. For a spherical particle, 1/r1 ⳱ 1/r2 ⳱ 1/r, Eq. (4) ⳱ Eq. (3). When nanocrystals are embedded in a matrix and if interfaces between the nanocrystals and the matrix are coherent or semi-coherent, the melting point is enhanced.23 –34 From a thermodynamic point of view, the corresponding Tr is described by4 Tr /T0 ⳱ 1 − [3V(␥sm − ␥lm)/r − ⌬E]/Hm ,
(5)
where V ⳱ (Vs + Vl)/2 and Vl denotes liquid volume, ␥sm and ␥lm are interface energies between the nanocrystals and the matrix and between the corresponding liquid particles and the matrix, respectively, and ⌬E shows energy density difference between the solid and the liquid particle. If ⌬E is negligible, Tr of the particle can be either larger or smaller than T0, depending on the sign of ␥sm − ␥lm, which is closely related to nature of the interface between the particles and the matrix. For a coherent or semi-coherent interface, ␥sm − ␥lm < 0 and superheating may arise. Note that if the matrix is gas, in terms of Eq. (2) and the condition of ⌬E ⳱ 0, Eq. (5) is again equal to Eq. (3). Both Eqs. (3) and (5) are in good agreement with experimental data for melting poi
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