The Born criterion

Another version of a ``one-phase'' theory of bulk melting suggested by Born [9] in 1937. His theory is based on the fact that a liquid differs from a crystal in having zero resistance to the shear stress. The distances between the atoms are increased due to thermal expansion, hence the restoring forces between the atoms are reduced, and therefore the shear elastic moduli decrease with rising temperature. The softening of the shear moduli leads to a mechanical instability of the solid structure and finally to a collapse of the crystal lattice at some temperature. The general conditions for stability of a crystal lattice were derived by Born. He analyzed the free energy of a solid with a cubic crystalline lattice. For a lattice to be stable, the free energy must be represented by a positive defined quadratic form, and this is fulfilled if the following inequalities for the shear elastic coefficients $C_{11},C_{12},C_{44}$ are satisfied:

$$2C'=C_{11}-C_{12}>0$$ (2.4)

$$C_{44}>0$$ (2.5)

$$C_{11}+2C_{12}>0$$ (2.6)

According to Born, $C_{44}$ goes to zero first and the melting temperature can be found from the condition: $C_{44}(T_m)=0$. Hunter and Siegal [10] measured the elastic coefficients of single crystal rods of NaCl over the temperature range from $20^{\circ} C$ to $804^{\circ} C$, up to the melting point. It was found that the shear elastic moduli $C_{44}$ and $C'$ decrease nearly linearly with temperature, but reach non-zero values at the melting point.(See Fig.1.4)

Figure 1.4: The variation of $C_{44}$ near the melting point, from ref. [10].

The shear elastic moduli for a series of fcc metals were measured by Varishni [11], and was found that $C_{44}$ decreases to $55\%$ of its value at zero-temperature. The discrepancy between the Born theory and experimental results was partly explained by enhanced thermal generation of defects in the crystal bulk as the melting temperature is approached [12,13]. Recently new calculations [14,15] have shown that Born did not take the contribution of the external stress, $P$, into account. The new generalized stability criteria are:

$$C_{11}+2C_{12}-P>0$$ (2.7)

$$C_{11}-C_{12}-P>0$$ (2.8)

$$C_{44}-P>0$$ (2.9)

In addition, on the basis of numerous experiments it has been concluded that the shear modulus $C'$ can become unstable first, $C'(T_m)=0$ at the melting point in some cases.

The applicability of the Born criterion at zero and non-zero external stress could be directly tested in computer simulations. Bulk melting (the surface of the solid is eliminated by means of periodic boundary conditions) of copper under condition of zero external stress was studied in molecular dynamics (MD) simulations [15]. It was found that the shear modulus $C'$ vanishes at some temperature $T_f$, but this temperature is large than thermodynamic melting temperature $T_m$, i.e $T_f > T_m$, measured experimentally. Thus the elastic coefficients remain finite at $T_m$. The conclusion was drawn that mechanical bulk melting, proposed by Born, is valid only for a perfect infinite crystal, but a real crystal with boundaries and defects undergoes thermodynamic melting at $T_m$, before it reaches the mechanical melting point.

The Born criterion was modified by Tallon to reach better agreement with experiment [16]. Tallon measured the shear moduli of various substances (metallic, organic, molecular and ionic crystals) as a function of molar volume. He confirmed that the shear moduli do not vanish at the melting point, though the volume of the melt at the melting point can be predicted by a continuous extrapolation one of shear modulus to the zero value at zero external stress. (See Fig. 1.5)

Figure 1.5: The variation of $C'$ with molar volume for various metals [16].

Tallon suggested that the shear modulus decreases with an increase of the volume of the solid and one of the shear moduli vanishes as a critical volume is reached. At that point each atom can access the entire substance volume due to the enhanced diffusion. The entropy of the system increases and the Gibbs free energy is lowered. At $T=T_m$ the system transforms discontinuously from the solid to the liquid state. In another words, the Born scenario would work if the crystal could be superheated until it will reach the molar volume of the liquid! Unfortunately another mechanism of melting is triggered at the surface of the real solid at temperature which lower than the melting temperature predicted by the Born model, preventing from us to observe mechanical collapse of the crystal lattice.