Configurational Entropy of High-Entropy Ceramics

Under the framework of equilibrium statistical mechanics, the concept of entropy for a macroscopic variable could be interpreted as measuring the extent to which the probability of the system is spread out over different possible microstates. In this study, the (RE^I_0.25RE^II_0.25RE^III_0.25RE^IV_0.25)₂Si₂O₇ solid solution with random occupations of RE³⁺ cations on its sublattice sites could be understood as the macrostate; while the possible metastable configurations could be understood as the microstates. The thermodynamics of the macrostate could be derived from the partition function, which encodes how the probabilities are partitioned among different microstates based on their individual configurational energies. The partition function could be written in the formula for the canonical ensemble^{38 (link),50} : $Z=\sum _{i=1}^N\exp \left(-\frac{E_i}{k_{\mathrm{B}}T}\right)$ Herein, i is the index for the microstates of the system; N is the total number of the microstates; k_B is the Boltzmann constant; and T is the absolute temperature. E_i is the energy for the ith microstate, derived from the standard DFT total energy calculations (at temperature of T = 0 K): $E_i=E_{4\mathrm{mix}}-\frac{1}{4}{E}_{{\mathrm{RE}}^{\mathrm{I}}}-\frac{1}{4}{E}_{{\mathrm{RE}}^{\mathrm{II}}}-\frac{1}{4}{E}_{{\mathrm{RE}}^{\mathrm{III}}}-\frac{1}{4}{E}_{{\mathrm{RE}}^{\mathrm{IV}}}$ where E_4mix is the total energy of the (RE^I_0.25RE^II_0.25RE^III_0.25RE^IV_0.25)₂Si₂O₇ unit cell; E_RE (with superscript I−IV) is the total energy of the corresponding single-RE-principal-component RE₂Si₂O₇, in the same polymorphic structure as the multicomponent material. Herein, the calculated E_i could be understood as the energy of mixing, which interprets the procedure of several single-RE-principal-component RE₂Si₂O₇ mixed into a multi-RE-principal-component system through random assignment of the RE atoms onto the RE lattice sites. Then, the configurational energy for the macrostate could be written as the ensemble average of E_i, which is the sum of energies for the microstates weighted by their probabilities: $⟨E⟩=\frac{1}{Z}\sum _{i=1}^N{E}_i\exp \left(-\frac{E_i}{k_{\mathrm{B}}T}\right)$ And, the free energy raised from configurational disorders is given by: $F=-k_{\mathrm{B}}T\ln Z$
In practice, it is computationally challenging to model the ensemble of all possible configurations for the multicomponent materials. Instead, the ensemble average could be performed over a representative subset of the whole microstate populations, provided that the subset extensively samples the energy landscape of all the microstates, and thus lead to statistically converged thermodynamic properties with respect to those of the whole population. This restriction is expressed by: $\frac{Z}{N}=\frac{Z^{\prime }}{N^{\prime }}$ Herein, N’ is the total number of the sampled microstates; and Z′ is their partition function. Imposing the Eq. (5) into the Eq. (4) gives: $F=-k_{\mathrm{B}}T\ln \left(N\cdot \frac{Z^{\prime }}{N^{\prime }}\right)=-k_{\mathrm{B}}T\ln N-k_{\mathrm{B}}T\ln \frac{{\sum }_{i=1}^{N^{\prime }}\exp \left(-\frac{E_i}{k_{\mathrm{B}}T}\right)}{N^{\prime }}$
Finally, the configurational entropy is derived from: $S_{\mathrm{config}}=\left(⟨E⟩-F\right)/T$
It should be noticed that, in general, the entropy of materials comprises contributions from configuration, vibration, electronic excitation, magnetism, etc. This model mainly deals with the configurational entropy of mixing, as it is expected to have significant effect on the formation of multicomponent ceramics. Such theoretical framework has been successfully adopted in Anand et al.’s work to interpret the thermodynamics of high-entropy oxides.^{38 (link)} Alternatively, one could also examine the energy spread of a system by calculating the formation energy for every metastable configuration in the ensemble, such as using the RE₂O₃ and SiO₂ as reference states in the Eq. (2). Results are discussed in the Supplementary Note 5 and Supplementary Fig. 9. Such modification will not change the calculated S_config, as it is derived from the dispersive features of the energy spread.
The available configurations for the multicomponent solid solution are generated by employing the special quasirandom structures (SQS) generation code implemented in the Alloy Theoretic Automated Toolkit (ATAT) package^{51 (link)}, which allows for the simulation of disordered crystallines by sampling on supercells with varied shapes and randomized occupations on the RE cationic sites (Supplementary Fig. 6a). Supercells with 88 lattice sites are used to construct the β-type and γ-type (RE^I_0.25RE^II_0.25RE^III_0.25RE^IV_0.25)₂Si₂O₇ systems, corresponding to four times of the minimum cell size necessary to reproduce the required stoichiometry. The configuration ensembles are constructed to include 558 unique configurations for the β-type (RE^I_0.25RE^II_0.25RE^III_0.25RE^IV_0.25)₂Si₂O₇, and 319 for the γ-type structures. Details on the stochastic generation of configuration ensembles, as well as the convergence test on the size of configuration ensembles, are presented in the Supplementary Note 6 and Supplementary Fig. 10.