Nanoscale thermodynamics of multicomponent, elastic, crystalline solids: diamond, silicon, and silicon carbide

Abstract

This paper extends the thermodynamic behaviour of two-dimensional and simple three-dimensional crystalline solids developed by Oh et al. and Slattery and Lagoudas to more complex, multicomponent, three-dimensional, elastic, crystalline solids. The analysis recognizes that the Helmholtz free energy is an explicit function of the lattice vectors defining the crystalline structure. From this theory, we obtain the stress-deformation behaviour and the elastic properties of diamond, silicon, and silicon carbide, which are face-centred, cubic, crystal structure. These are compared with available experimental values.

Notes

†It is important to recognize that the approach of Slattery and Lagoudas 3, in which the mass density is included as one of independent variables, and our use of equation (11) are fundamentally equivalent as the result of equation (10). The results derived by these two approaches can be used somewhat interchangeably, so long as care is taken to properly recognize the independent variables. Equation (11) has the advantage of being consistent with Green and Adkins 15, who further discussed the particular functional dependence upon the six E _(m) · (C − I) E _(n) to be expected for various crystal classes. The approach of Slattery and Lagoudas 3 has the advantage of explicitly recognizing the role of thermodynamic pressure in the description of stress–deformation behaviour, in the Euler equation, and in the Gibbs phase rule.

†For a simple three-dimensional crystal, Slattery and Lagoudas 3 have obtained a stress–deformation behaviour in terms of the mass density and five invariants. For a simple crystal, the stress–deformation behaviour is fundamentally equivalent to equation (28), as mentioned in the footnote of equation (11). The only difference is values of the coefficient μ_(I,mn). The internal lattice vectors e ₍₄₎, … e _(k) introduced to explain internal structures of the complicated three-dimensional solids will affect the coefficients with equation (30).