Biomedical Engineering Reference
In-Depth Information
computational time. Our calculations are performed either with the pure GGA PBE
functional (Perdew et al., 1996) or with the hybrid B3LYP (Becke, 1993), both well-known
functionals. Two
ab initio
approaches are possible within DFT and they differ for the type of
basis set functions. Indeed, a localized Gaussian basis set can be considered, as in the
present case, or a plane waves one, also extremely diffuse. An
excursus
of advantages and
disadvantages of these approaches is not useful in this context and will be omitted, by
focusing exclusively on Gaussian type functions.
2. Hydroxyapatite and Bioglass® as computational case study
All the calculations mentioned in this Chapter have been performed using the CRYSTAL
code in its latest release (Dovesi et al., 2005a; Dovesi et al., 2005b; Dovesi et al., 2009). This
periodic quantum-mechanical software has been developed by the Theoretical Chemistry
group of the University of Turin (Italy) together with the Daresbury Laboratory (UK) since
1988. CRYSTAL is capable of computing systems with every dimensionality, from molecules
to real infinite crystals and it supports massive parallel calculations. This code uses local
Gaussian basis sets and can deal with many electronic structure methods, from Hartree-
Fock to Kohn-Sham Hamiltonians. Structural, electrostatic and vibrational properties of the
studied materials have been characterized with the program. Another crucial aspect in
modeling is the graphical visualization and representation of structures. For all the images
displayed in this Chapter, MOLDRAW (Ugliengo et al., 1993), J-ICE (Canepa et al., 2011b)
and VMD (Humphrey et al., 1996) programs were used. Further more precise computational
details can be read in a number of our recent papers on both HA (Corno et al., 2009; Corno
et al., 2006; Corno et al., 2007; Corno et al., 2010) and bioactive glasses (Corno & Pedone,
2009; Corno et al., 2008).
2.1 Defects in hydroxyapatite bulk and surfaces
Hydroxyapatite (HA) is a mineral which occurs in nature in two polymorphs, a monoclinic
form, thermodynamically stable at low temperatures, and an hexagonal form, which can be
easily stabilized by substitution of the OH
-
ions (Suda et al., 1995). These ions are aligned
along the c axis (the [001] direction), as highlighted in Fig. 1. The single crystal structure of
the hexagonal form of HA is characterized by the
P6
3
/m
space group. The mirror plane,
perpendicular to the [001] direction, is compatible with the column of OH
-
ions because of
an intrinsic static disorder of these ions, which can point, with no preference, in one of the
two opposite directions ([001] or [00-1]). The result is a fractional occupation of the sites in
the solved crystallographic structure (50% probability for each direction). As
ab initio
simulation cannot take into account the structural disorder, we reduced the symmetry to
P6
3
, removing the mirror plane and fixing the directions of the OH
-
ions. In the most stable
configuration found, both the OH
-
ions point in the same direction, as reported in Fig. 1. The
oxygen atom of the OH
-
ion is close to three Ca ions, which form an equilateral triangle in
the
ab
plane. Moreover, there are six phosphate ions inside the crystallographic cell, all
symmetry equivalent.
The bulk structure of crystalline HA, fully characterized in the literature (Corno et al., 2006),
has been considered as a starting point to model the surfaces which are experimentally
found to be the most important: (001) in terms of reactivity, and (010) in terms of exposure
in the crystal habit (Wierzbicki & Cheung, 2000). Those surfaces have already been fully
characterized at an
ab initio
level, and all the structural, geometrical and electronic properties
