Thermal expansion of Carbonate-Nosean: a phase-transition kinetics model

The description of a Landau phase-transition considers group-subgroup relations with respect to the initial phase at thermodynamic equilibrium. Accordingly, the Gibbs free energy is modified in terms of an order parameter Q [1]. Data collection on a particular phase at equilibrium is challenging in case of associated kinetic effects. Thus describing a phase-transition using the Landau theory leads quite often to a marked discrepancy between the observed and calculated data close to T_c. Evaluating an accurate order parameter consequently must be a hurdle. In some cases the order parameter is proportional to the unit-cell volume of the phase. Similar effect could be seen while evaluating the lattice volume of a phase using Debye-Einstein-Anharmonicity model [2] where the internal energy of a crystalline lattice volume is considered to be sum of the Einstein harmonic, Debye quasi-harmonic and anharmonic vibrational potentials. The DEA-model gives possibility to compare the experimental thermal expansion coefficient (TEC) with that of the theoretical one. For example, in dehydrated carbonate-nosean [3] the kinetic effect close to the P23-P23 phase-transition could be seen in terms of the decrease of the TEC over several data points around 800 K (Figure). In this case, a pure Landau transition would have sharply dropped the TEC leaving no intermediate values. These deviations in the experimental data could be theoretically modeled by overlapping the TEC's of the low-temperature (TEC_LT) and the high-temperature (TEC_HT) phase using a sigmoidal term f_s(T) {0 < f_s(T) < 1}: TEC_total(T) = f_s(T)·TEC_low(T) + (1- f_s(T))·TEC_high(T). The slope of this sigmoidal function corresponds to the deviation of the model used for the volume calculation close to the phase-transition and therefore carries information about the reaction kinetics. The same could be calculated for the dehydration process of the hydrated carbonate-nosean |Na₈(CO₃)(H₂O)₄|[AlSiO₄]₆ around 400 K. Figure: Experimental temperature-dependent volume change of hydrated carbonate-nosean and respective DEA-model calculation results. The individual DEA-models as well as TEC changes including kinetic overlaps are given inserted.