Size effects on melting temperature of nanosolids
Abstract
In the present study, the size and shape effects are mentioned using semiempirical 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
Msd  mean square displacement 
Cohesive energy per coordination  Cohesive 
INTRODUCTION
Background
History
The study of size and shape dependence melting temperature can be dated back to the start of 20^{th} century, many models were proposed which were based on empirical or semiempirical 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, T_{m} is the melting point of the particle, T_{mb} 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,
The value of the σ^{2} is equated by using the equation given below,
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,
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 σ_{s}^{2}/σ_{v}^{2}. 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,
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, σ_{s }and σ_{v} are assumed to be size dependent. But the ratio is taken as a size independent parameter i.e. σ_{s}/σ_{v }= σ_{s}(ꝏ)/σ_{v}(ꝏ) = α where, σ_{s}(ꝏ) and σ_{v}(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 r_{0} 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,
where, r_{0} is the adjustable shape dependent parameter defined as,
r_{0 }= (3d) × h
for different values of d r_{0} is tabulated in table 1.
Shape  d  r_{0} 
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 (E_{b}) of n atoms is defined as the difference of volume energy a_{υ} and surface energy 4πr^{2}γn^{2/3} (r is the atomic radius and γ is the coefficient of surface energy of the material). So the cohesive energy a_{υ,d} (E_{b}/n) per atom will be as follows,
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,
Similarly, for the melting temperature of nanoparticles,
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,
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,
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.
∆G_{bulk }is the Gibbs energy of melting for bulk and it can be written in terms of ∆H_{m} enthalpy and ∆S_{m} entropy of fusion. Enthalpy and entropy are assumed to be constant near melting temperature,
∆G_{surface} stands for the surface effect known as the capillarity effect and it depends on the surface energy γ and molar volume V_{m.} For a spherical particle of radius r the surface term will be as follows,
To derive the relation between the melting temperature and size ∆G_{particle} for solid to liquid is assumed to be 0 and the expression is as follows,
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 V^{L}_{m}, V^{S}_{m}, γ_{L }and γ_{S}) also have size and temperature dependencies and are taken from available literature. ∆H_{m} and_{ }T_{m} values are from A.T. Dinsdale, 1991, the values of γ_{S }its relation (γ_{S}/γ_{L }= (1.18 ± 0.03) with γ_{L} 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 V^{L}_{m} ≈ γ_{S}V^{S}_{m}≈ V_{s}. 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 γ_{S}− γ_{L} 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 JP. 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 r^{2} so for nanoparticles Equation 20 will be as follows,
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 E_{0}/2 to the total cohesive energy because of the dangling bonds where, E_{0} is the cohesive energy per atom,
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,
Here by using the linear relation of the cohesive energy and melting temperature the relation of melting temperature is as follows,
where, T_{p} is the melting temperature of the nanoparticle and T_{b} 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 πD^{2 }by the area of the atom using its great radius πd^{2}/4 so, N = 4D^{2}/d^{2}. Therefore, the value of N/n can be written as,
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,
The above equation can be reformed for nanowire and nanofilm,
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.
METHODOLOGY
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,
So, the size and shape dependent melting temperature will be,
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,
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.
RESULTS AND DISCUSSIONS
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 JP. 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.
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 is tabulated below,
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 temperatures corresponding to the particle size for Gallium Nitride are tabulated below,
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 
Change in melting temperature is tabulated below,
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 AND CONCLUSIONS
Summary
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 r_{0}.
 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

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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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