Dimension of the Gibbs function topological manifold: 3. Configuration entropy determined by the isotopic composition of binary quasicrystals

Abstract

Even though the graph representation of thermodynamic states rules out the existence of stable binary quasicrystals, Tsai’s research team was able to discover two such systems in 2000. Our previous article showed that in terms of isotopic composition, Tsai’s two quasicrystals, as well as several other ones that have been discovered since 2000, are pseudobinary. Such systems owe their stability to high configuration entropy, which is determined by their isotopic composition. In this article, we shall use the basic methods of statistical thermodynamics to show that configuration entropy for systems without symmetry (including translational symmetry) is always higher than the entropy of their symmetrical counterparts. Thermodynamic calculations for a model of a 1-D binary Fibonacci quasicrystal, $\hbox {AB}_{\uptau }$, indicate that the internal energy of this system is the same as the internal energy of its approximant, $\hbox {AB}_{\mathrm{t}}(t\approx \uptau$), i.e., the crystalline counterpart of the quasicrystal. We shall show that assuming Element A in both systems is composed of two isotopes, the configuration entropy corresponding to a single atom of the quasicrystal is about 20 % higher than the entropy of its crystalline approximant. This article also includes calculations of the configuration entropies of the few pseudobinary quasicrystals that have been synthesised so far. These entropies are exceptionally high, most often amounting to about 1.5 $k_{\mathrm{B}}$, due to a large number of isotopic components of these quasicrystals. It in unknown whether such a high value is sufficient to ensure the stability of the quasicrystalline phase compared to the crystalline phase, as the number of configurations of the approximant required for calculations is astronomical.