A new calculation strategy to analyze the vibrational free energy of a slab and calculate the vibrational contribution of the surface free energy

Abstract

This is an author version of the contribution published on:Questa è la versione dell'autore dell'opera: BRUNO M., PRENCIPE M. (2013) Summary: We developed a new calculation strategy for determining the vibrational contribution of each layer or atom forming a slab, where the latter is a two-dimensional periodic structure of given thickness, generated by separating the bulk structure along the hkl plane of interest and used to simulate the crystal surfaces. By means of this new calculation methodology, it is now possible to estimate how the surface free energy of a crystal face changes with temperature by only taking into account the entropic contribution due to the vibrational motion of atoms in the slab. Furthermore, the model is extended to the calculation of the vibrational contribution to the free energy of the interface between (ii) two identical crystals in twinning relationship and (ii) two different crystals in epitaxial relationship.Our model uses the frequencies of the vibrational modes of a slab and it is based on the construction of a weight function taking into account how the vibrational amplitude of the atoms involved in the vibrational mode is modified by the presence of the surface. We applied the model to the following systems: (i) 28-layer (100) slab of LiF and (ii) 10-layer (10.4) slab of calcite (CaCO 3 ). In both cases, the vibrational energy, vibrational entropy and vibrational free energy of the optimized slab, and the contribution to these quantities of each atom and layer forming the slab were calculated.

The free energy density of a crystal: calcite (CaCO3) as a case of study

“…1 Therefore, in the present work I only detail the operating procedure to follow for calculating A vib j . A two-step procedure can be envisaged:…”

“…(iv) Instead, as detailed in a later paragraph, the estimate of A vib j requires the use of two computational codes in sequence: (a) initially, a code (GULP or CRYSTAL) for the calculation of the frequencies of the vibrational modes of a slab; (b) successively, the SLAB code, 1 which reads the output files generated by GULP or CRYSTAL and computes A vib j by means of the relations reported in section 2.2.…”

“…The strategy and equations for determining the vibrational contribution of each layer or atom forming a slab were described in a recent paper by Bruno and Prencipe. 1 Therefore, in the present work I only detail the operating procedure to follow for calculating A vib j . A two-step procedure can be envisaged:…”

“…Furthermore, always according to the work by Bruno and Prencipe, 1 it is licit to write γ 298. 15 (10.4) as the sum of two terms:…”

“…As concerns point (i) I am not able to intervene, while a contribution to the solution of the problem (ii) is furnished in this paper. To date, a first contribution to this item was recently given by Bruno and Prencipe, 1 which conceived a new calculation methodology for determining the vibrational contribution of each layer forming the slab (i.e., how the vibrational free energy density at the temperature T varies within the slab), where the latter is a slice of crystal delimited by two parallel (hkl) faces. Their model uses the frequencies of the vibrational modes of a slab and it is based on the construction of a weight function taking into account how the vibrational amplitude of the atoms involved in the vibrational mode is modified by the presence of the surface.…”