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Theory of positional disordering

Numerous X-ray studies have made evident that regular repetition of lattice positions of atoms over long sequences in 3D disappears on changing to a melt. The first microscopic theory of melting based on positional disordering was proposed by Lennard-Jones and Devonshire [19]. A simple model of a rare-gas crystal was considered. In order to describe order-disorder transition the system was modeled to be consisted of two inter-penetrating cubic lattices with fcc structure(namely $A$ and $B$ lattices. If all atoms of the solid are located at the sites of the first A-lattice, than the system is perfectly ordered. However, if an atom can jump from the first lattice to the second, then a vacancy-interstitial pair is formed, i.e. atoms belonging to the B-lattice considered as self-interstitials. To place a single atom from the A-lattice on a site of the B-lattice when all the rest atoms on the A-lattice sites requires a considerable increase of energy, owing to the repulsive field of the neighboring atoms. But this energy does decrease if several atoms of the A-lattice jump to the B-lattice or some sites of the B-lattice are already occupied. This energy would decrease to zero in a state of complete disorder. This is an example of cooperative effect, which decreases considerably the defect formation energy.

The partition function for an assembly of $N$ atoms in a state of perfect order is calculated, than it is modified to account for positional disordering. The partition function for a perfect crystal $F$ is given by $F=f^N$ , where $f$ is contribution of each atom:

\begin{displaymath}
f=\left( \frac{2mk_BT}{h^2} \right) ^{3/2} v_T \times exp{ \left( -\frac{\Phi_0}{Nk_BT} \right)}
\end{displaymath} (2.10)

here $v_T$ is the volume per atom, $\Phi_0$ the potential energy of the system with all the atoms in their position of equilibrium. At higher temperatures positional disorder sets in, defects are formed, and all atoms distributed randomly between the two lattices. The order parameter $Q$ is introduced as $Q=N_A/N $, where $N_A$ is number of sites of the A-lattice occupied by the atoms, and $N=N_A+N_B$ is the total number of the atoms. The $Q=1$ corresponds to an ordered state, while $Q=1/2$ to a fully disordered state. Knowledge of the fraction of atoms in each lattice $N_A/N=Q $ and $N_B/N=1-Q $ allows us to estimate the total energy of interactions between the defects. Thus in order to modify the partition function of the perfect crystal to account for the positional disordering, one has to multiply the total partition function by the disorder factor:
\begin{displaymath}
F=f^N \rightarrow F=f^N Y(Q)exp\left(\frac{zWNQ(1-Q)}{k_BT} \right)
\end{displaymath} (2.11)

where $W$ is defect interaction energy, $Y(Q)$ is a combinatorial factor. The partition function has a maximum at some values of $Q$, which are the roots of the equation:
\begin{displaymath}
\frac{zW(2Q-1) }{2k_BT}=ln(Q)-ln(1-Q)
\end{displaymath} (2.12)

This equation has a solution with $Q=1/2$ (a disordered state) which is always satisfied, yet when $T \le T_c = zW/4k_B$ there is another root with $Q>1/2$ (an ordered state), at which the free energy has a deeper minimum. The critical point $T_c$ is interpreted as the melting point $T_m=T_c$. The phase transition is found to be first-order. This simple theory of ``order-disorder'' transition was later generalized to the case of more realistic interaction potential between atoms, and numerical methods were applied to investigate the more complicated models [19].

The Lennard-Jones and Devonshire theory based on positional disordering is capable of predicting some properties of the solid and its melt at the melting transition in rather good agreement with experiment. Nevertheless, there are some flaws in this theory, for example, the theory predicts a continuous transition between the solid phase and the liquid phase at very high pressure, yet experimentally no critical melting point was found even at a very high pressure.


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Next: Melting theories based on Up: Bulk melting Previous: The Born criterion
2003-01-15