Structure and single-phase regime of boron carbides

The boron carbides are composed of twelve-atom icosahedral clusters which are linked by direct covalent bonds and through three-atom intericosahedral chains. The boron carbides are known to exist as a single phase with carbon concentrations from about 8 to about 20 at. %. This range of carbon concentrations is made possible by the substitution of boron and carbon atoms for one another within both the icosahedra and intericosahedral chains. The most widely accepted structural model for ${\mathrm{B}}_{4}$C (the boron carbide with nominally 20% carbon) has ${\mathrm{B}}_{11}$C icosahedra with C-B-C intericosahedral chains. Here, the free energy of the boron carbides is studied as a function of carbon concentration by considering the effects of replacing carbon atoms within ${\mathrm{B}}_{4}$C with boron atoms. It is concluded that entropic and energetic considerations both favor the replacement of carbon atoms with boron atoms within the intericosahedral chains, C-B-C\ensuremath{\rightarrow}C-B-B. Once the carbon concentration is so low that the vast majority of the chains are C-B-B chains, near ${\mathrm{B}}_{13}$${\mathrm{C}}_{2}$, subsequent substitutions of carbon atoms with boron atoms occur within the icosahedra, ${\mathrm{B}}_{11}$C\ensuremath{\rightarrow}${\mathrm{B}}_{12}$. Maxima of the free energy occur at the most ordered compositions: ${\mathrm{B}}_{4}$C,${\mathrm{B}}_{13}$${\mathrm{C}}_{2}$,${\mathrm{B}}_{14}$C. This structural model, determined by studying the free energy, agrees with that previously suggested by analysis of electronic and thermal transport data. These considerations also provide an explanation for the wide single-phase regime found for boron carbides. The significant entropies associated with compositional disorder within the boron carbides, the high temperatures at which boron carbides are formed (>2000 K), and the relatively modest energies associated with replacing carbon atoms with boron atoms enable the material's entropy to be usually important in determining its composition. As a result, boron carbides are able to exist in a wide range of compositions.