# Size effects on melting temperature of nanosolids

Raviraj Mandalia

St. Xavier's College (Autonomous), Navrangpura, Ahmedabad, Gujarat 380009

Prof. D Arivuoli

Anna University, Guindy, Chennai, Tamil Nadu 600025

## Abstract

In the present study, the size and shape effects are mentioned using semi-empirical relations. There are many models describing the change in physical, electrical, magnetic, optical, and thermal properties of nanomaterials below a certain size, different environment and of different shapes. All of the above properties not only depend upon the chemical composition of the crystal but also depend upon size and shape of the crystal, a good example of this is gold, a particle of few nanometres size of gold will be paramagnetic, will absorb green light and appear red which is totally different from its bulk counterpart. Here, different models considering the variation in melting points are discussed, as melting temperature is an important property of a crystal. Debye temperature, melting enthalpy etc. can be evaluated from melting temperature. W.H Qi proposed a theoretical model of size effects on melting temperature using the linear relation between cohesive energy and melting temperature. The model is verified for gold and silver nanospheres and after that the same model is implemented for the case of molecule (GaN). In that case the molecules are assumed to be spherical and the approximate molecular radius is estimated using experimental values molar volume. The result suggests a depression of 648 K in the value of melting temperature of bulk GaN.

Keywords: semi empirical relation, size and shape effects, cohesive energy, variation in melting temperature

## Abbreviations

Abbereviations
 Msd mean square displacement Cohesive energy per coordination Cohesive energy per atom / Coordiantion number per atom

## Background

In the present report, size and shape effects in nanoscale are discussed. Decreasing size can change the properties of a crystal with particle size to few nanometres. Even with the same chemical composition the nanocrystal behaves totally different from the bulk, many properties are seen to diverge. The well-known example of this is gold. A gold crystal with size in few nanometres will absorb green light and appear red, bulk gold is diamagnetic but on decreasing size the particle show magnetism ​Emil Roduner, 2006​, another example is that the elastic constant of Pb and Ag nanowires becomes twice the value of it bulk metal ​Dakshanamoorthy Arivuoli, 2012​. Small particles were seen to melt at a temperature below the melting point of the bulk. In nanotechnology a better understanding of melting temperature is important so, one can make new devices.

## History

The study of size and shape dependence melting temperature can be dated back to the start of 20th century, many models were proposed which were based on empirical or semi-empirical relations. In 1909, Pawlow predicted that particles with few nanometres size melts at a temperature below the melting point of their counter bulk part. After a year, Lindemann gave a criterion for melting which will be discussed in the next section. Melting point of a nanoparticle is a function of particle size, and can be used to evaluate Debye temperature, specific heat, melting enthalpy ​​Y.N Stepanov, 1994​​ etc. Many models based on empirical formulas were proposed in last several decades. Some were based on the model Lindemann’s criterion of melting dealing with the msd. Other models were based on the linear relation of melting point of a particle and inverse of the particle size,

Where, Tm is the melting point of the particle, Tmb is the melting point of the bulk counterpart, C is the material constant and R is the particle size. The value of material constant is not easy to evaluate.

## Lindemann’s Criterion

Lindemann’s theory talks about the simplest assemblies of closed packed atoms, the average vibrations of the atoms describe melting. Atoms are vibrating about their equilibrium position. To quantify these vibrations, the value of msd (σ) is used. In Lindemann’s criterion it is assumed that the atoms vibrate with same frequency υE (Einstein’s approximation), and the relation of average thermal energy and equipartition theorem is as follows,

$E=4{\pi }^{2}m×\upsilon {E}^{2}×{\sigma }^{2}=kb×T$

The value of the σ2 is equated by using the equation given below,

${\sigma }^{2}=cl×{a}^{2}$

where, cl is a threshold value and a is the lattice constant and using ​ Equation 2​​ and ​ Equation 3​ the value of the melting temperature is given below,

${T}_{m}=\left(\frac{4{\pi }^{2}m}{kb}\right)×\upsilon {E}^{2}×cl×{a}^{2}$

## LITERATURE REVIEW

2.1 In 1994, F.G Shi proposed a theoretical model on melting of nanoparticles based on Lindemann criterion. In which the melting point was a function of known parameter h (atomic diameter) and directly experimentally measurable parameter σs2v2. Here, α is the ratio of mean square displacement of surface atoms and interior atoms. The values of α for monoatomic crystal clean surfaces lies between 2.5 and 4. For such values of α there will be a depression in melting point with decreasing the particle size, but under different conditions the surface might undergo adsorption of impurities then the value of α will change if the values is less than 1 than the crystal shows superheating with decreasing size. The relation is given below,

${T}_{m}={T}_{mb}×exp\left[\left(-\left(\alpha -1\right)\right)×\left(\frac{r}{3h}-1{\right)}^{-1}\right]$

As mentioned above we can see that the value of melting point is in linear relation with inverse of particle size r in ​​Equation 5​​ and h is the atomic diameter. In this model the value of α is to be perfected with the experimental data then one can predict the curve and the value at even smaller size. The surface of the particle which is in contact with the substrate will wet it and then the substrate will affect the msd of the atoms, but the msd of free surfaces is not affected by the substrate. Here, σs and σv are assumed to be size dependent. But the ratio is taken as a size independent parameter i.e. σsv = σs(ꝏ)/σv(ꝏ) = α where, σs(ꝏ) and σv(ꝏ) corresponding to the bulk values. For interfacial surfaces, the value of α depends upon the tendency of the substrate atoms to hold the atoms. If the substrate atoms supress (α < 1) the msd then the crystal will undergo superheating for e.g. superheating of 50K was observed for small In and Pb nanoparticles embedded in Al matrix. Total opposite of this can also happen for α > 1 depression of 200K of Kr was observed in Ni ​Frank G. Shi, 1994​​.

2.2 The earlier model is extended for different shapes by F.G. Shi, 1997. By adjusting a parameter r0 and the results predict the melting temperature decreases not only with particle size but also with shapes. For different shapes, the melting temperature changes for same particle size. The size and shape dependent melting temperature is as follows,

${T}_{m}={T}_{mb}×exp\left[\left(-\left(\alpha -1\right)\right)×\left(\frac{r}{{r}_{0}}-1{\right)}^{-1}\right]$

where, r0 is the adjustable shape dependent parameter defined as,

r0 = (3-d) × h

for different values of d r0 is tabulated in table 1.

Different Values of r0
 Shape d r0 Nanocrystals 0 3h Crystals in nanotubes 1 2h Thin films 2 h

2.3 In 2002, ​S. N. Behera, 2002​ studied the relation of melting temperature with particle size based on Liquid drop model. In which, the total cohesive energy (Eb) of n atoms is defined as the difference of volume energy aυ and surface energy 4πr2γn2/3 (r is the atomic radius and γ is the coefficient of surface energy of the material). So the cohesive energy aυ,d (Eb/n) per atom will be as follows,

${a}_{\upsilon ,d}={a}_{\upsilon }-\frac{4\pi \left({r}^{2}\right)\gamma }{{n}^{1/3}}={a}_{\upsilon }-{a}_{s}×{n}^{1/3}$

And the total number of atoms in the volume with diameter d is given by,

Substitute the value of n in ​ Equation 7​​ and ​ Equation 8​, then the expression for cohesive energy per atom is as follows,

From ​​Equation 9​​, the cohesive energy is a function of atomic volume V, particle size d and the coefficient of surface energy γ. Also, the cohesive energy per atom decreases as the particle size decreases. Then using the expression derived by ​​Jun Tateno, 1972​​ for melting temperature of the bulk material and the expression for cohesive energy per coordination, a linear relation between cohesive energy per coordination and melting temperature was noted,

${a}_{\upsilon }=0.0005736{T}_{mb}+c$

Similarly, for the melting temperature of nanoparticles,

${a}_{\upsilon ,d}=0.0005736{T}_{m}+c$

Now using ​ Equation 9​​, ​ Equation 10​​ and ​ Equation 11​ the relation for melting temperature of nanoparticles in terms melting temperature of bulk material and other known parameters is obtained,

$Tm=Tmb×\left\{1-\left[\frac{6V}{0.0005736d}×\left(\frac{\gamma }{{T}_{mb}}\right)\right]\right\}$
${T}_{m}={T}_{mb}×\left(1-\frac{\beta }{d}\right)$

In this model, by slight modification of ​ Equation 12 b​​ the values can be obtained for different shapes like nanosphere, nanowire and nanofilm and a comparison with above model can be done. The shape dependent expression is as follows,

For different shapes only the value of z will change, and the values are given below,

Different values of z
 Shape z Nanospheres 1 Nanowires 3/2 Nanofilms 3

Here, the values of the d in the case of nanosphere and nanowire the diameter of the solid and for nanofilm it is the thickness. For the same value of d and different shape, the change in melting temperature is 3:2:1, maximum for nanospheres and minimum for nanofilms.

2.4 Kim and Lee (2009) proposed a thermodynamic model for size dependent melting temperature ​​Byeong-Joo Lee, 2009​​. As the melting temperature changes other thermodynamic properties also changes like heat enthalpy and entropy. They considered Gibbs free energy for the melting of bulk material is given as follows,

∆Gbulk is the Gibbs energy of melting for bulk and it can be written in terms of ∆Hm enthalpy and ∆Sm entropy of fusion. Enthalpy and entropy are assumed to be constant near melting temperature,

∆Gsurface stands for the surface effect known as the capillarity effect and it depends on the surface energy γ and molar volume Vm. For a spherical particle of radius r the surface term will be as follows,

$∆{G}_{surface}=\frac{2}{r}\left({\gamma }^{L}{{V}^{L}}_{m}–{\gamma }^{S}{{V}^{S}}_{m}\right)$

To derive the relation between the melting temperature and size ∆Gparticle for solid to liquid is assumed to be 0 and the expression is as follows,

$∆{H}_{m}×\left({T}_{m}–T\right)={T}_{m}×\frac{2}{r}\left({\gamma }^{L}{{V}^{L}}_{m}–{\gamma }^{S}{{V}^{S}}_{m}\right)$

Given the values of necessary physical quantities, enthalpy of fusion, surface energy and molar volumes of liquid and solid phase one can calculate the change in melting temperature. The above quantities VLm, VSm, γL and γS) also have size and temperature dependencies and are taken from available literature. ∆Hm and Tm values are from​A.T. Dinsdale, 1991​, the values of γS its relation (γSL = (1.18 ± 0.03) with γL is taken from​W.A. Miller, 1977​​. This thermodynamic approach and the experimental data suggest that the ​ Equation 17​​ should be changed for better results. The molar volumes for solid and liquid phase are taken same VLm ≈ γSVSm≈ Vs. From the work of ​​L. H. Allen, 1996​. to measure the size dependencies of enthalpy of fusion it is found that the results are good for r > 5 nm but for r < 5 nm the above eq. can be written as follows,

In the above expression γS− γL is the surface energy difference and it can be written as ∆γ. The surface energy is due to the presence of broken bonds on the surface of the nanoparticles. The no. of broken bonds depends on the curvature of the radius. The number of broken bonds per surface atoms increases as the particles size decreases. Therefore, the surface energy increases as the particle size decreases.

For spherical nanoparticles, the surface energy is proportional to the ratio of areas of the upper layer A1 and the surface area of the particle A2 such that,

Here, δ is the interlayer distance between the two surface layers and is equal to R i.e. first neighbour nearest neighbour. The solid surface energy can be expressed in a functional form of the particle size r by multiplying ​ Equation 19​​ so, ​ Equation 18​​ will be as follows,

The depression of melting temperature of Au is calculated. The experimental results by ​​J-P. Borel, 1976​ coincides the curve by this model.

For nanowires, the capillary effect term will be γV/r instead of 2γV/r and the multiplying factor to the solid surface energy will now be proportional to r instead of r2 so for nanoparticles ​ Equation 20​​ will be as follows,

$∆Tm=\frac{1}{r}×{V}_{S}×{T}_{m}×\left(∆{H}_{m}{\right)}^{-1}×∆\gamma ×\left(1+\frac{R}{r}\right)$

2.5 Qi proposed a semi empirical relation between particle size and its melting temperature based on size dependent atomic cohesive energy ​​W.H. Qi, 2005​​. This model can be applied to shapes like nanoparticles, nanowires, and nanofilms with free surface. The surface atoms contribute E0/2 to the total cohesive energy because of the dangling bonds where, E0 is the cohesive energy per atom,

${E}_{total}={E}_{0}×\left(n-N\right)+{E}_{0}×\left(\frac{N}{2}\right)$

Then if n is the total no. of atoms and N is the number of surface atoms, Qi suggested the relation of cohesive energy of the nanosolid and the corresponding bulk is as follows,

${E}_{p}={E}_{b}×\left(1-\frac{N}{2n}\right)$

Here by using the linear relation of the cohesive energy and melting temperature the relation of melting temperature is as follows,

${T}_{p}={T}_{b}×\left(1-\frac{N}{2n}\right)$

where, Tp is the melting temperature of the nanoparticle and Tb is the melting temperature of the bulk particle. The ratio (N/n) is dependent on the shape and size of the nanocrystal. Qi evaluate this ratio for nanosphere and disk like nanosolids as a function of atomic diameter and an adjustable parameter.

For nanosphere with diameter D, the total no. of atoms n can be obtained by diving the total volume of the nanosphere with the volume of the single atom having diameter d as,

Now for N, one can divide the surface area of the nanosphere πD2 by the area of the atom using its great radius πd2/4 so, N = 4D2/d2. Therefore, the value of N/n can be written as,

$\frac{N}{n}=\frac{4d}{D}$

For disk like nanosolids with diameter l and height h the volume and n can be written as,

And the surface area and N can be written as,

Therefore, the value of the N/n for disk like nanosolid is as follows,

$\frac{N}{n}=\frac{4}{3}d×\left(\frac{1}{h}+\frac{2}{l}\right)$

The above equation can be reformed for nanowire and nanofilm,

Different values of N/n
 Nanosolids N/n Nanoshpere 4d/D Nanowire 8d/3l Nanofilms 4d/3l

In ​ Table 3​​, the value of N/n for nanosphere, nanowire and nanofilm is mentioned. From the expressions of N/n, the change in melting point will be maximum for nanosphere then nanowire and nanofilm and the for equal values of particle size (D=l=h) the change will be in the ratio of 3:2:1.

## Concepts

In this report, the model proposed by W.H. Qi is used to verify the change in melting temperature for single atom gold and silver nanocrystals. The reasons for this particular model is that it is simple and only requires two known parameters namely melting temperature of bulk and diameter of atom. Also, the model can be used for molecules if, molecules are assumed to be spherical. We can compute an approximate value for molecular diameter using its molar volume. The model is based on size and shape dependent atomic cohesive energy. Cohesive energy is the amount of energy require to breakdown a crystal into its individual atoms. And melting phenomenon is related to breakdown of crystal, a crystal will have high melting temperature if the cohesive energy is high. There is a linear relation between melting temperature and cohesive energy ​​W.H. Qi, 2005​​,

And the size dependent cohesive energy is as follows,

${E}_{p}={E}_{b}×\left(1-\frac{N}{2n}\right)$

So, the size and shape dependent melting temperature will be,

${T}_{p}={T}_{b}×\left(1-\frac{N}{2n}\right)$

And the necessary input parameters are melting temperature of the bulk material and the values of N/n. The value of N/n will be according to shape of the nanosolid as mentioned in the ​​Table 3​​.

## Input Parameters

The values of melting temperature with variable particle size is computed using this model. Gold and silver nanocrystals are considered for different shape and size. The model shows depression in the melting temperature.

## Melting temperature of bulk materials and diameter

The value of melting temperature of the bulk materials is taken from the available data. And for the value of the atomic diameters for gold and silver is calculated from the available data of the atomic volume. Diameters are calculated by assuming the atoms and molecules to be spherical. The melting temperatures and the calculated diameters are as follows,

Input Parameters
 Name Melting temperature of bulk materials (K) Diameters (nm) Gold (Au) 1337.6 0.31872 Silver (Ag) 1234 0.31934 Gallium Nitride (GaN) 2770 0.35088

## Particle size

The values of the particle size are taken as variable, for nanosphere and nanowire their diameters and for nanofilm its thickness is in the range of [3,20] nm.

## Gold and Silver

A depression in melting temperature of Au and Ag nanospheres is seen by using the theoretical model which match the trend. Specifically, the depression obtained in nanospheres of Au is around 300 K which agrees with available data ​​J-P. Borel, 1976​​ but, for nanospheres of Ag prediction by the model (is also 300 K) is quite less compared to available data ​S. Marsillac, 2012​. The shape dependence can also predict that the variation will be more in nanosphere and least in nanofilm. The graph given below shows the results obtained by the model.

Melting temperature (K) against Particle size (nm)
Melting temperatures with particle size
 size (nm) Au Nanosphere Au Nanowire Au Nanofilm Ag Nanosphere Ag Nanowire Ag Nanofilm 3 1053.38 1148.12 1242.86 971.28 1058.85 1146.43 5 1167.07 1223.91 1280.76 1076.37 1128.91 1181.46 10 1252.33 1280.76 1309.18 1155.18 1181.46 1207.73 15 1280.76 1299.70 1318.65 1181.46 1198.97 1216.49 20 1294.97 1309.18 1323.39 1194.59 1207.73 1220.86

The shift of melting temperature in nanosphere is 284.22 K and 262.72 K, in nanowire is 189.48 K and 175.15 K and in nanofilm is 94.74 K and 87.57 K for 3 nm for gold and silver respectively.

Change in melting temperature (K) against Particle size (nm)

Change in melting temperature is tabulated below,

Change in melting temperature with particle size
 Particle Size (nm) Au Nanosphere (K) Au Nanowire (K) Au Nanofilm (K) Ag Nanosphere (K) Ag Nanowire (K) Ag Nanofilm (K) 3 284.22 189.48 94.74 262.72 175.15 87.57 5 170.53 94.74 47.37 157.63 105.09 52.54 10 85.27 56.84 28.42 78.82 52.54 26.27 15 56.84 37.90 18.95 52.54 35.03 17.51 20 42.63 28.42 14.21 39.41 26.27 13.14

## Gallium Nitride

Gallium nitride melting temperature is 2770K for bulk and its approximated diameter is calculated to be 0.35088 nm ​ Table 4​​, using these values the depression in melting temperature is estimated to be 648 K. The results agree with another model ​​Dakshanamoorthy Arivuoli, 2012​​ and they are shown below,

Melting temperature (K) against particle size (nm)

Melting temperatures corresponding to the particle size for Gallium Nitride are tabulated below,

Melting temperature with particle size
 size(nm) Nanosphere (K) Nanowire (K) Nanofilm (K) 3 2121.89 2338.03 2554.01 5 2381.14 2510.82 2640.41 10 2575.57 2640.41 2705.20 15 2640.41 2683.61 2726.80 20 2672.78 2705.20 2737.60

The shift of melting temperature in nanosphere is 648.11 K, in nanowire is 431.97 K and in nanofilm is 215.99 K at 3 nm for GaN.

Change in Melting temperature (K) with Particle size (nm)

Change in melting temperature is tabulated below,

Change in melting temperature with particle size
 size(nm) Nanosphere (K) Nanowire (K) Nanofilm (K) 3 648.11 431.97 215.99 5 388.86 259.18 129.59 10 194.43 129.59 64.80 15 129.59 86.39 43.20 20 97.22 64.80 32.40

## Summary

Summary of the literature reviewed in this report

## Conclusions

• Theoretical models describing the size and shape dependence were studied.
• Model by F.G Shi was based on the Lindemann’s criterion, and it was able to explain the size dependent suppression and superheating of the nanocrystals.
• Model by Jiang et al. was an extension of the above model and it explained the same with a shape dependence adjustable parameter r0.
• K.K Nanda et al. studied the Liquid drop model and applied it to the melting phenomenon.
• From the Liquid drop model, the expression of cohesive energy and its linear relation with melting temperature was used.
• Model by W.H Qi is based on a semi empirical relation of total cohesive energy and melting temperature for simple single atom nanoparticles.
• Assuming the molecules to be spherical the above model proves results which are in agreement with the available results.
• Shortcoming of these models is that they are based on empirical relations but, at nanoscale the quantum effects dominate.
• None of these models deals about the quantum size and shape effects.
• We need a criterion for melting phenomenon in terms of quantum mechanics.
• Lindemann’s criterion deal with msd, if we can link the change in wavefunction due to the change in msd then we can formulate the new criterion.

## ACKNOWLEDGEMENTS

I would like to thank Prof. D. Arivuoli for giving me this wonderful opportunity and guide me through tough yet knowledgeable and new experience which I will cherish forever. I would like to extend my heartiest thanks to Crystal Growth Centre and all the lab members for supporting me.

I would like to thank Physics Department, St. Xavier’s College Ahmedabad, HOD Dr. Rajesh Iyer for granting me exemption and my mentor Dr. Tushar Pandya for supporting me and encouraging me for this fellowship. I would like to thank my friends and family to support me.

And I would like to thank Indian Academy of Science for this opportunity and Authorscafe for providing a wonderful platform for my report.

#### References

• Emil Roduner, 2006, Size matters: why nanomaterials are different, Chemical Society Reviews, vol. 35, no. 7, pp. 583

• Paneerselvam Antoniammal, Dakshanamoorthy Arivuoli, 2012, Size and Shape Dependence on Melting Temperature of Gallium Nitride Nanoparticles, Journal of Nanomaterials, vol. 2012, pp. 1-11

• M.I Alymov, E.I Maltina, Y.N Stepanov, 1994, Model of initial stage of ultrafine metal powder sintering, Nanostructured Materials, vol. 4, no. 6, pp. 737-742

• Frank G. Shi, 1994, Size dependent thermal vibrations and melting in nanocrystals, Journal of Materials Research, vol. 9, no. 5, pp. 1307-1314

• Q. Jiang, N. Aya, F.G. Shi, 1997, Nanotube size-dependent melting of single crystals in carbon nanotubes, Applied Physics A: Materials Science & Processing, vol. 64, no. 6, pp. 627-629

• K. K. Nanda, S. N. Sahu, S. N. Behera, 2002, Liquid-drop model for the size-dependent melting of low-dimensional systems, Physical Review A, vol. 66, no. 1

• Jun Tateno, 1972, An empirical relation on the melting temperature of some ionic crystals, Solid State Communications, vol. 10, no. 1, pp. 61-62

• Eun-Ha Kim, Byeong-Joo Lee, 2009, Size dependency of melting point of crystalline nano particles and nano wires: A thermodynamic modeling, Metals and Materials International, vol. 15, no. 4, pp. 531-537

• A.T. Dinsdale, 1991, SGTE data for pure elements, Calphad, vol. 15, no. 4, pp. 317-425

• W.R. Tyson, W.A. Miller, 1977, Surface free energies of solid metals: Estimation from liquid surface tension measurements, Surface Science, vol. 62, no. 1, pp. 267-276

• S. L. Lai, J. Y. Guo, V. Petrova, G. Ramanath, L. H. Allen, 1996, Size-Dependent Melting Properties of Small Tin Particles: Nanocalorimetric Measurements, Physical Review Letters, vol. 77, no. 1, pp. 99-102

• Ph. Buffat, J-P. Borel, 1976, Size effect on the melting temperature of gold particles, Physical Review A, vol. 13, no. 6, pp. 2287-2298

• W.H. Qi, 2005, Size effect on melting temperature of nanosolids, Physica B: Condensed Matter, vol. 368, no. 1-4, pp. 46-50

• S. A. Little, T. Begou, R. W. Collins, S. Marsillac, 2012, Optical detection of melting point depression for silver nanoparticles via in situ real time spectroscopic ellipsometry, Applied Physics Letters, vol. 100, no. 5, pp. 051107

#### Source

• Table 1: F. G. Shi, 1997
• Table 2: S. N. Behera, 2002
• Table 3: W. H. Qi (2005)
• Table 4: K. K. Nanda et al., 2002, Antoniammaland Arivuoli, 2012
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