First Principles Calculations and Thermodynamic Modeling of Carbon's Solubility Limits in Zirconium Diboride

Abstract

First principles calculations were implemented to determine the carbon solubility limits in zirconium diboride forming a Zr(B1-x,Cx)2 solid solution phase. Density functional theory coupled with the generalized gradient approximation and projector augmented wave basis set provided enthalpies of formation for ZrB2 and solid solution compositions. Special Quasirandom Structures of 150-atom 5x5x2 supercells were generated for Zr(B1-xCx)2 solid solutions for x = 0.50, 0.25, 0.18, 0.14, 0.10, 0.04, and 0.02 fractions of carbon substitution within the boron sublattice. Structures were relaxed, and symmetry was analyzed by determining each spacegroup as well as with the integration of the radial distribution function. The calculated enthalpy of formation for pure ZrB2 was -291.074 kJ/mol, with this value becoming more negative with an increasing composition of carbon for the enthalpies of mixing. Structures were volumetrically deformed up to 5%, and total energies were calculated again to determine entropic contributions to the vibrational free energy utilizing the Debye-Grüneisen approach. From this 0 K energy data, a Debye temperature of 874.4 K, linear coefficient of thermal expansion of 5.71x10-6K-1, and room temperature specific heat capacity of 51.6 J/mol·K for ZrB2 were calculated, which aligns well with experimental literature. Equations modeling the free energy of mixing are supplied for select solution compositions, which show a trend of increasing free energy with increasing carbon contents within ZrB2. A future work section details the remaining steps to incorporate this data within a Zr-B-C ternary model and to define the solubility limits of carbon in ZrB2.