Melting theories based on point defects

Point defects may play a substational rôle in bulk melting transition. The theory proposed by Weber and Stillinger [20] is an example of a statistical model of bulk melting transition of crystals in which the influence of point defects on the shear moduli is taken into account. According to the model, at temperatures close to the melting point a cooperative formation of point defects takes place. The first excitation in a crystal is a displacement of an atom from its lattice site and hence a vacancy-interstitial pair is formed. The presence of the defect softens the solid, and this effect is amplified while the concentration of defects increases, thereby easing the insertion of more defects and introducing new local modes of vibration. At some critical concentration of defects the lattice of the solid becomes unstable and collapses, i.e. a mechanical melting transition occurs. Minimizing the free energy $F$ with respect to the concentration of point defects, Weber and Stillinger obtained the temperature dependence of the defect concentration $c_{def}=l/N$ (See Fig.1.6).

Figure 1.6: Defect concentration, solid line corresponds to stable equilibrium values, dashed line to meta-stable and unstable branches. From ref. [20].

As is seen from the Figure 1.6 at some temperature the defect concentration jumps from $c_{def}=0.000117$ (as $T \rightarrow T_{m}-0$) to $c_{def}=0.99882$ (as $T \rightarrow T_{m}+0$). This temperature is identified as the melting temperature.

As a further development of the theory of bulk melting based on point defects, Granato [21] derived an interstitial-concentration-dependent Gibbs free energy, appropriate for calculation of all the thermodynamic properties of crystalline, liquid (melt) and amorphous (or glassy) states of metals. The Helmholtz free energy of the perfect crystal $F_p$ is given by:

$$F_p=F^p_0(V)+F^p_{vib}=F_0(V)+3Nk_BTln\left(\frac{\hbar \omega_E}{k_BT} \right)$$ (2.13)

where $F_0(V)$ is the free energy of the static lattice, i.e. without thermal or zero-point vibrations, and the vibrational free energy $F_{vib}$, which at high-temperatures is calculated in the framework of a single frequency Einstein approximation. According to Granato, in order to take interstitials into account one has to add to the free energy the following terms:

$$F_{def}=F_{w}+F_{vib}+F_{conf}$$ (2.14)

where $F_{w}$ is an energy necessary to create a concentration $c$ of interstitials, where $c=n/N$ ($n$ is the number of interstitials and $N$ is the number of atoms). $F_{vib}$ is the change of the vibrational free energy resulting from the change in frequency spectrum, and $F_{conf}$ is the configurational free energy. A salient feature of the model is that the interstitialcy configuration is extended, strongly coupled to the shear stress, with low-frequency resonance modes providing an unusually large entropy per defect. The shear modulus $G$ carries the burden of providing the volume, shear strain, and concentration dependence needed for thermodynamic treatment. The theory predicts that the shear elastic modulus decreases when the concentration of the self-interstitials increases (for example as a result of increase in temperature), which leads to instability of the solid and eventually to a melting transition at $T_m$.

When the Gibbs free energy is obtained from the Helmholtz free energy we add a term $pV$ to $F$. Figure 1.7 shows variation of the Gibbs free energy with interstitial concentration.

Figure 1.7: The Gibbs free energy of copper as a function of the interstitial concentration for different temperatures $t=T/T_m$, from ref. [21]

Three different regimes are found by minimizing the Gibbs free energy. For low temperatures (the first regime), when $t=T/T_m<0.85$ the only stable configuration is the solid with the equilibrium concentration of interstitials about $c \sim 10^{-5}$ defects/atoms. In the second regime, when $0.8 < t <1.15$, the two minima coexist, the first one corresponds to the solid state with a very low defect concentration, and the second one to the liquid state (regarded as crystals containing a few percent of interstitials, for example for copper it is about $9\%$ at $T_m$). In the third regime $t >1.15$ the only stable state is the liquid state. The model predicts the possibility of supercooling of a liquid which is possible experimentally, but also predicts the possibility of superheating of a solid which has never been observed. However, this asymmetric nature of melting is to be understood from the fact that interstitials are produced thermally at surfaces, dislocations, or other shear strain centers.