Chapter 5: Calcium Carbonate Clusters

5.1 Introduction

“In eighteenth-century Newtonian mechanics, the three-body problem was insoluble. With the birth of general relativity around 1910 and quantum electrodynamics in 1930, the two-and three-body problems became insoluble. And within modern quantum field theory, the problem of zero bodies (vacuum) is insoluble. So, if we are out after exact solutions, no bodies at all is already too many.” – Prof. G. E. Brown, quoted in Computer Simulation in Material Science, 1996, p.77.

Calcite is the most abundant mineral found, comprising approximately 4% by weight of the Earth’s crust and is formed in many different geological environments. The crystals of calcite can have thousands of different crystal forms by building on the hundreds of different basic shapes, like the positive rhombohedron, negative rhombohedron, steeply, moderately and slightly inclined rhombohedrons, various scalahedrons, prism and pinacoid.

The most common polymorphs of calcium carbonate are aragonite and calcite. That is, they contain the same elements but they differ in crystal structure. Aragonite is an orthorhombic structure, similar to a rectangle, except that the angles at the two opposing corners are about 75 degrees and 105 degrees. Calcite has a trigonal structure. Calcite and Aragonite are polymorphous to each other. Sometimes, the crystals of Calcite and Aragonite are too small to be detected by the eye, and it is only possible to distinguish these two minerals by optical magnification. The polymorphs are simply labeled as "Calcium Carbonate". The polymorph vaterite, which has a hexagonal structure, is extremely rare. The mechanism responsible for the formation of the various polymorphs of calcium carbonate (CaCO₃); calcite, aragonite and vaterite, is still the subject of much research.

Calcite, the polymorph of CaCO₃, is used in a wide range of applications, like the construction industry (in the interiors of cathedrals, temples, and public buildings), in the manufacturing of fertilizers, and in the metal, glass, rubber and paint industries. It is also used in some optical devices, like the Nicol prisms, which is used in polarizing microscopes. Fig. 5.1 shows an image of dolomite and calcite crystallites taken through a polarizing microscope [61]. The color photograph is from the dinosaur bone collection and shows the fossilized teeth in a dinosaur. Over time the spaces between the bones in the mouth begin to mineralize, forming calcite crystallites, dolomite, and other minerals.

The specific gravity of calcite is approximately 2.71 and the molecular weight is 100.09 amu. The work in the past has been focused on determining physical properties of calcite by developing an empirical model that is based on the properties of bulk material. A. Pavese et al. [18] used computational methods based on the quasi-harmonic approximation to calculate the temperature dependence of elastic constants and structural features of the crystal. A. Skimmer [62] and R. K. Singh et al. [63] investigated the lattice dynamics of several types of molecular crystals, including CaCO₃. Their approach was to use the Rigid Ion Model (RIM) as the interaction potential [18,64].

There has also been interest in the mechanisms responsible for the growth of the observed crystalline structure. A few researchers have begun using ab-initio calculations in the study of small clusters of calcite. Y. Mao et al. [65] modeled the properties of the monomer and dimer of CaCO₃ by using ab-initio Hartee-Fock calculations. The structures they achieved for the monomer and dimer agreed with the structures I found. M. Catti et al. [66] determined the static crystal energy of calcite and its structure as functions of pressure by using all-electron periodic HF calculations for the bulk material.

Figure 5.1: A photomicrograph of Dolomite and Calcite Crystallites taken through an optical microscope. The photograph is from the dinosaur bone collection [61].

My approach will be to perform ab-initio calculations on small clusters (N = 1à 4) of CaCO₃ and then perform a fit of the parameters of the RIM potential to the results of the ab-initio calculation. Once the parameters of the RIM potential are refined, I will use the simulated annealing method to determine the structures for larger clusters. I will investigate the relationship between binding energy per molecule and the cluster size and investigate the rates of diffusion for the systems.

5.2 Rigid Ion Model Potential

The interatomic potentials for CaCO₃ polymorphs have been modeled using the Rigid Ion Model (RIM), or two-body Born-Meyer type potential, supplemented by coulomb interactions between point charges and angular and torsional terms in the CO₃ group [18]. In the RIM potential the atoms are assumed to be point charges based on the electronic properties of the atom when participating in ionic bonds. The RIM is a sum over the pair potentials between atomic pairs and is defined as follows:

(5.1)

where

(5.2)

(5.3)

(5.4)

(5.5)

Here ‘Z_i’ (i=Ca, C, O) are the point charges on the various atoms, A_ij, r_ij, and c_ij are the parameters. The potential energy terms defined above are the different interactions that exist between atoms within the molecular structure. The coulomb interaction between the ions of different charge signs is the major contributing factor to the stability of the crystal lattice in the bulk material. The exponential term is always a positive quantity and represents the short-range repulsive force caused by the overlap between the outer-shell electrons of the atom. This is the main factor that counteracts the attractive coulomb term and contributes to the stability of the lattice. The 1/r⁶ term is a dispersive attractive force in the O-O interactions and results from the instantaneous dipole-induced dipole interaction. Additionally, two equations governing the angular restoring forces associated with the CO₃ group are added to the RIM potential: