Ferroelastic and ferroelectric phase transition in a molecular crystal : tanane - 3. — From ab initio computation of the intermolecular forces to statistical mechanics of the transition

On the basis of structural data of the disordered tetragonal phase of tanane, the Van der Waals intermolecular forces are computed for the 16 nearest neighbours of a given molecule. Considering first the molecular ordering in a « rigid » tetragonal lattice, the pair energies between neighbouring molecules are analysed within a pseudo-spin formalism which shows that the main ordering interactions define linear chains of molecules along the ferroelectric Z-axis. As there is no exact procedure for a statistical treatment of the 3-d Ising Hamiltonian, we just calculate the transition temperature To in the mean-field approximation. The predicted result : T0 ~ 200 K is in good agreement with the ordering temperature in the « rigid » lattice deduced from experimental investigations : To = 235 K. In a second step, we consider the « piezo-electric-like » coupling between the molecular ordering and the uxy shear of the lattice. Using a « Bragg-Williams-like » treatment of the strain-dependent Ising Hamiltonian, we build a free energy expansion whose bilinear terms agree well with those determined from experiments (shear modulus C66 and « piezo-electric-like » coefficient D), and which yields for the predicted transition temperature : T c ~ 240 K (actual value : Tc = 287 K). This analysis also reveals an important biquadratic coupling term which is supposed to enhance the instability of the structure in the vicinity of T c.