Lindemann criterion

The first theory explaining mechanism of melting in the bulk was proposed by Lindemann [1], who used vibration of atoms in the crystal to explain the melting transition. The average amplitude of thermal vibrations increases when the temperature of the solid increases. At some point the amplitude of vibration becomes so large that the atoms start to invade the space of their nearest neighbors and disturb them and the melting process initiates. Quantitative calculations based on the model are not easy, hence Lindemann offered a simple criterion: melting might be expected when the root mean vibration amplitude $\sqrt{<u^2>}$ exceeds a certain threshold value (namely when the amplitude reaches at least $10\%$ of the nearest neighbor distance).

Assuming that all atoms vibrate about their equilibrium positions with the same frequency $\nu_E$ (the Einstein approximation) the average thermal vibration energy can be estimated relying on the equipartition theorem as:

$$E=m4\pi^2\nu_E^2<u^2>=k_BT$$ (2.2)

where $m$ is the atomic mass, $\nu_E$ is the Einstein frequency, $<u^2>$ is the mean square thermal average amplitude of vibration, and $T$ is absolute temperature. Using the Lindemann criterion for the threshold $<u^2>=c_la^2$, where $c_l$ is Lindemann's constant one can estimate the melting point

$$T_m=4\pi^2mc_La^2/k_B$$ (2.3)

Lindemann's constant $c_l$ was assumed to be the same for crystals with similar structure, hence it could be calculated from the melting temperature of one particular crystal. A detailed experimental examination showed that $c_l$ is not strictly a constant and the correlation is only fair (See Fig. 1.2).

Figure 1.2: The measured melting temperature versus the melting temperature estimated using the Lindemann rule, from ref. [95]

Recently, the validity of the Lindemann instability criterion have been tested in computer simulations of bulk melting of Lennard-Jones fcc crystals [6]. It has been found that melting occurs when a sufficiently large number spatially correlated destabilized atoms of the crystal ( e.g. cluster of quasiliquid) are generated (See Fig. 1.3). These clusters are distributed homogeneously thruout the solid. The Lindemann criterion of the lattice instability is found to be valid for these clusters. The accumulation, growth and coalescence of the clusters of the liquid phase constitute, according to Jin et al. [6], the mechanism of homogeneous bulk melting.

Figure 1.3: 3D visualization of the collective appearance of the Lindemann particles at $T/T_m=0.79$. (a) a few clusters with 20-200 particles (larger black circles) against other Lindemann particles (smaller gray circles) which do not form such clusters (b) four large clusters with 219, 214, 187, and 117 particles colored with red, blue, black and gray, respectively. From ref. [6]

It should be stressed that the original Lindemann model for vibrational melting, like many of its more sophisticated successors, refers only to a crystal with the simplest possible structure, i.e. assemblies of closed packed atoms. Crystals containing more complex molecules as unit of structure exhibit a vibrational complexity which rules out any simple rule of lattice stability, determined merely by vibrational amplitudes of the molecular centers of mass. Futhermore, the Lindemann model is based on harmonic forces, which never give way, whereas melting must involve bond breaking. This is another serious defect of the model. Furthermore numerous experiments carried out at high pressures indicate that the Lindemann model does not estimate adequately the pressure dependence of the melting temperature [7]. The most serious defect of the model is that melting is described in term individual atomic property, i.e. mean square amplitude of vibration, while a phase transition is a cooperative process. In addition, the Lindemann model describes melting in terms of the solid alone, although the melting transition must involve both solid and liquid phases. Nevertheless the predictive success of the Lindemann melting criterion lent support to the belief that melting could be a gradual process, beginning within the solid at temperatures below the melting point. Subsequent theories and numerous experiments helped to bolster the idea.