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As mentioned in Section 2.2.1, the decrease in bonding energy due to enhanced surface and grain boundary area in nanocrystalline materials can reduce the enthalpy of fusion and the melting temperature. At the melting point of a material, the solid and liquid phases are in equilibrium. For this to be thermodynamically possible, it is necessary that the chemical potential of a component (μ_i) is the same in both solid and liquid phases (μ_il_{= μ}

is). However,

due to inherent internal stresses, the pressure of the solid is greater than that of the liquid.

P_s = Pl + 2γ

r

where γ is the surface energy and r is the radius of the solid sphere. The chemical potential is related to pressure by the following equation:

dμ = –SdT + VdP

By substituting the pressure relationship into the above equation and recalling that at equilibrium the chemical potential of the liquid is equal to that of the solid, assuming the overall pressure change in the liquid is negligible, and noting that S_l – S_s = ΔS_m, the following equation is derived:

ΔS_mdT = –2Vγ dr

r2

It should be recognised that the bulk melting temperature is related by:

ΔS_m = ΔHm

Combining the above relations yields a relationship that describes the decrease in the melting point of a substance as the radius decreases.

ΔT = 2VγTm

rΔH_m

It can be seen that the change in melting temperature is inversely proportional to the radius of the sphere. In other words, as the grain size is reduced, the melting point is lowered.

It has been observed that nano-CdS of diameter ~ 2.5 nm melts at 600 K, much lower than the bulk melting point (1675 K). The single-walled carbon nanotube melts at ~ 1600 K, 0.42 times its bulk melting point (3800 K). A similar decrease in the melting point has also been made in case of nano-Si with a concurrent increase in hardness. At the. At theAt the surface of a nanosolid, the atomic coordination number (CN) is much lower than the standard atomic CN inside a bulk.

It is known that atoms in a solid vibrate about their mean position. The amplitude of the vibrations increases with increasing temperature. When the vibration amplitude exceeds a certain percentage of the bond length, melting begins at the surface and propagates through the solid. Atoms at the surface and grain boundary are less constrained to vibrate compared to atoms inside the crystal lattice. As the grain size decreases, the percentage of atoms residing at surfaces and grain boundaries increases significantly. Hence, free- standing nanoparticles may show a lower melting point compared to bulk. A similar effect has been reported on zinc nanowires embedded in holes in an anodic alumina membrane. The melting point of zinc nanowires was found to decrease with decreasing diameter of the nanowire.

In contrast to nanoclusters and nano-agglomerates, nanoparticles within a matrix may, in fact, experience an enhancement in the melting temperature. The matrix exerts a pressure,

p, which can affect the melting temperature of the particles. This is described by:

p = 2μκΔV 3V_o

where μ is the shear modulus of the matrix, κ is a dimensionless factor that takes into account the presence of other particles, ΔV is the change in particle volume due to thermal expansion, and V_o is the initial particle volume. From the Clausius–Clapeyron equation (dT/dP = ΔV/ΔS, where T is the transition temperature, P is the pressure, ΔV and ΔS are the change in volume and entropy during transition), it is understood that the transition temperature will increase with increase in pressure, if change in volume on melting is positive. Hence, the melting point of nanoparticles embedded in a bulk matrix increases with decreasing size of particulate as the pressure increases with decrease in particle size (Fig. 2.4). Researchers at the University of California, Berkeley, USA, Lawrence Berkeley National Laboratory, USA and Australian National University, Australia, have found that Ge nanocrystals embedded in silica glass do not melt until temperatures are almost 200°C above the melting point of bulk

Ge, and resolidify only when the temperature is more than 200°C below its bulk melting point. Lead nanoparticles embedded in an Al matrix have also exhibited superheating.

It is fascinating to observe that the melting temperature does not continuously decrease with decreasing grain size in nano-dimensions. In fact, as the cluster size is reduced below a critical limit, the melting point of clusters is seen to increase above the bulk melting temperature of the material, at least in some cases. It has been found that a solid containing about 10 atoms of Ga or IV A elements (C, Si, Ge, Sn and Pb) melts at temperatures that are higher than the melting point of the corresponding bulk solid (T_m,b). This is in contrast to the behaviour of nanograined solids with grain size in the range of 1–100 nm, where auniversal decrease in melting point is observed with the grain size. Compared to the melting point of bulk Ga, which is about 303 K, clusters of Ga_39–40 melt at 550 K, while smaller clusters of Ga₁₇ do not melt up to 700 K. Consistent insight into the phenomenon of melting pointoscillation (suppression followed by elevation as the particle size is reduced from bulk to sub-nanometre size) over the whole range of sizes remains a scientific challenge.