The effect of monovacancies on the elastic properties of rigid disk solids

Abstract

The isothermal elastic stiffness constants of a system of N rigid disks on a two-dimensional hexagonal lattice confined to an area A are calculated analytically in the high-density limit using one-particle cell cluster theory. The constants are calculated using the following equation: C ijkl = 1 A ( ∂, 2F ∂η ij∂η kl ) η=0 , where C ijkl is the isothermal elastic constant, F is the Helmholtz free energy of the system, and the η ij 's are the strains (η represents all the strains). The isothermal elastic constants are then calculated in the high density limit for the rigid disk system into which a small concentration of monovacancies has been introduced, using the same approach appropriately modified. All the elastic constants are calculated to order 1 t 2 where t = a σ − 1 , a is the smallest distance between lattice sites, and a is the diameter of the disks. All the nonzero isothermal elastic constants are found to be proportional to the temperature. The elastic constant C 1122 = C 2211 which is zero in the perfect system becomes nonzero upon the introduction of monovacancies. However, the fact that C 1122 is found to be zero is an artifact of the one-particle cell cluster approximation.