Ab Initio Modelling of the Structure and Properties of Crystalline Calcium Carbonate

Abstract

Many biominerals occur as crystalline materials, and their formation often involves steps where metastable crystalline phases appear. The latter can correspond either to intermediate species that then transform into more stable phases or to the final mineral crystal. Because of their instability and rare occurrence, the structure and properties of such intermediate metastable phases may not always be fully understood from experiment alone. Vaterite (CaCO₃) is one such phase, and recent advances in understanding its complex structure were achieved through ab initio modelling techniques. This chapter will highlight the importance of achieving a comprehensive understanding of the atomic details of the crystalline phases involved in biomineralisation. Examples will be focused on calcium carbonate, and especially on vaterite, and ab initio methods based on density functional theory (DFT) will be proposed as the main tool to undertake this kind of investigation, together with more traditional techniques such as spectroscopic methods, microscopies and X-ray and neutron diffraction.

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6.1 Introduction

During the past decade, the nucleation and crystal growth of biominerals, and calcium carbonate in particular, have attracted the attention and the imagination of the scientific community, due to the discovery of pathways, mechanisms, species and phases that differ from those expected through strictly classical nucleation theory (Demichelis et al. 2011; Gebauer et al. 2008; Wallace et al. 2013).

A full description of what is now called “alternative” or “non-classical” nucleation and crystal growth is presented in the previous chapters of this book, and at present it can be qualitatively summarised as follows:

• (i) ions interact in water solution (see De Yoreo et al., Chapter 2; Gebauer, Chapter 3; Lutsko, Chapter 4; Reichel and Faivre, Chapter 14; Penn et al., Chapter 15; Tobler et al., Chapter 17);

• (ii) stable pre-nucleation clusters (PNCs) form (Gebauer, Chapter 3);

• (iii) as the concentration increases these clusters aggregate with each other giving rise to a liquid-liquid phase separation (De Yoreo et al., Chapter 2; Gebauer, Chapter 3; Wolf and Gower, Chapter 5);

• (iv) dehydration and structural rearrangement lead to the formation of amorphous nanoparticles (De Yoreo et al., Chapter 2; Wolf and Gower, Chapter 5; Fernandez-Martinez and Wang, Chapter 6; Rodriguez-Blanco et al., Chapter 7; Birkedal, Chapter 12);

• (v) crystalline phases form following aggregation and structural rearrangement of amorphous nanoparticles (De Yoreo et al., Chapter 2; Fernandez-Martinez and Wang, Chapter 6; Rodriguez-Blanco et al., Chapter 7; Birkedal, Chapter 12; Delgado-Lopez and Guagliardi, Chapter 13; Penn et al., Chapter 15; Tobler et al., Chapter 17);

• (vi) the final crystalline polymorph grows, in some cases after transformation of one crystalline form to another (De Yoreo et al., Chapter 2; Andreassen and Lewis, Chapter 9; Rao and Calfen, Chapter 10; Penn et al., Chapter 15; Van Driessche et al., Chapter 16; Tobler et al., Chapter 17).

In the specific context of calcium carbonate, much recent research has been focused on investigating the details of the early stages of mineral nucleation, leading to the formation of stable pre-nucleation clusters and of amorphous nanoparticles, including their polyamorphism and size-dependent stability (Cartwright et al. 2012). However, in this chapter we will focus on the subsequent steps involving the appearance of intermediate crystalline phases and their eventual transformation into more stable mineral phases (Munemoto and Fukushi 2008; Pouget et al. 2009; Rodriguez-Blanco et al. 2011; Tang et al. 2009), which are equally challenging and important to understand.

Calcium carbonate is arguably the most studied biomineral (Falini and Fermani, Chapter 11). There are five known crystalline phases that are stable at room pressure; three of these correspond to the anhydrous CaCO₃ polymorphs calcite, aragonite and vaterite, while the remaining two are the metastable hydrated phases ikaite (CaCO₃⋅ 6H₂O) and monohydrocalcite (CaCO₃⋅ H₂O). Recently, nanosized crystals of two high-pressure calcium carbonate phases, namely, calcite-III and calcite-IIIb, have been detected for the first time in natural geological samples at ambient pressure (Schaebitz et al. 2015). If we consider magnesium and other divalent cations along with calcium, we obtain a variety of hydrated and hydroxylated species, many of which have structures and properties that are still undetermined. Many of these compounds are difficult to obtain as pure phases and are generally unstable unless in extreme conditions (e.g. high salinity, lower or higher temperatures than standard conditions, Munemoto and Fukushi 2008; Swainson 2008; Swainson and Hammond 2003) or under particular biogenic conditions (Kabalah-Amitai et al. 2013; Mikkelsen et al. 1999; Skinner et al. 1977). Nonetheless, they are important since they are likely to represent a link between the early stages of mineralisation and the formation and growth of the stable polymorphs. In particular, knowing the reasons why these intermediates form in the first instance, their formation mechanisms and the details of their structures and properties may provide insights that are relevant to the evolution of species during nucleation, as well as regarding the polymorphism of the final stable minerals.

In this chapter we will focus on the calcium carbonate crystalline phases that are most frequently observed in biomineralisation processes, i.e. calcite, aragonite, vaterite, monohydrocalcite and ikaite. In particular, we will devote most of our attention to vaterite, whose complex, disordered structure has been recently, at least partially, explained through applying ab initio methods. This represents evidence that computational tools can provide accurate results that are complimentary to those obtained through experimental techniques and therefore have the potential to play a significant role in advancing this field. In fact, ab initio methods allow the exploration of the atomic and electronic structure of a given compound, which are indeed responsible for defining its properties and reactivity. Moreover, computational techniques, in general, have almost unlimited freedom to explore conditions that may be difficult to access via experiment, such as high pressures and temperatures, or specific isotopic compositions, as well as hypothetical structures and transition states.

Despite the focus of this chapter being on calcium carbonate, the methods and the analysis shown here can serve as a guide for undertaking similar studies on the many other crystalline phases that can appear in biomineralisation processes.

6.2 Methodology

The aim of this chapter is to illustrate how ab initio techniques can be applied to the investigation of the structure and properties of biominerals. We begin by providing a brief summary of the theoretical framework underpinning such techniques. A more exhaustive description of these methods can be found in Sherman (2016), Erba and Dovesi (2016), De La Pierre et al. (2016), and references therein.

6.2.1 The Potential Energy Surface

The energy of a system can be defined as a function of its atomic coordinates. This function is called the potential energy surface (PES);

$$\displaystyle{ PES = E(\mathbf{x},\mathbf{H}) = E(x_{1},x_{2},\ldots,x_{3N},H_{11},H_{12},\ldots,H_{33}) }$$

(6.1)

where $\mathbf{x}$ represents the 3N atomic coordinates of an N-atom system and H is the matrix containing the Cartesian components of the three lattice vectors. More generally, at finite temperature one could include the momenta of the atoms to arrive at the free energy surface (FES), but for simplicity we will start by taking the approach of lattice dynamics in which the PES is explored first and then the effect of temperature is subsequently accounted for through consideration of the lattice vibrations (i.e. phonons).

Different conformations, configurations, polymorphs and transition states of a given system will therefore correspond to distinct points on the PES. In particular, stationary points of the PES (i.e. points where all the components of the gradient—the vector whose elements correspond to the partial first derivatives of the PES with respect to the atomic coordinates and lattice parameters—are zero) have a physical meaning: minima correspond to equilibrium geometries (minimum energy structures), whereas saddle points and maxima correspond to transition states.

The search for stationary points on the PES is known as geometry optimisation. Most of the optimisation techniques are able to locate the stationary point nearest to the starting geometry, through algorithms that minimise the energy and the components of the gradient. The search for the other stationary points requires the use of advanced optimisation tools, whose details are beyond the aim of this chapter. For most minerals, a reasonable guess for the atomic positions and for the lattice parameters is often available. As a consequence, the optimised structure is expected to correspond to either a realistic minimum energy structure or to a transition state close to it.

Once the stationary point has been found, it is of crucial importance to understand whether it corresponds to a minimum or to a saddle point. In order to do this, second derivatives must be calculated. In particular, for a 3N dimensional function, a 3$N\times 3N$ matrix can be defined, the Hessian matrix, that contains all the partial second derivatives of the energy with respect to the atomic coordinates. Through diagonalising this matrix, the eigenvalues are obtained, and if they are all positive, it means that the structure is a minimum energy structure. If one or more eigenvalues are negative, then the structure is a transition state.

The physical quantities directly linked to the second derivatives of the energy with respect to the atomic coordinates are the vibrational frequencies. In particular, the eigenvalues of the mass-weighted Hessian matrix, as defined in the previous paragraph, are related to the square of the vibrational frequencies, whereas the eigenvectors correspond to the atomic motions associated with the various modes.

Transition states are characterised by the presence of imaginary frequencies. In the presence of such modes, a successful scheme that allows the minimum energy structure to be reached is highlighted here. First, the nature of the imaginary mode can be examined through its corresponding eigenvector. Then, the system is allowed to relax following the direction of this eigenvector. This may require the removal of one or more symmetry constraints. Finally, vibrational frequencies (second derivatives) are computed for the new structure. This procedure is repeated until the final structure is free from the presence of imaginary modes. The application of this scheme has been used extensively in the search for valid possible structures of vaterite, as will be described in Sect. 6.3.

Aside from the vibrational frequencies, there are many physical properties that can be calculated through computing the derivatives of the PES. For example, elastic constants are related to the second derivatives of the PES with respect to strain being applied to the lattice parameters; the dipole moment (and then also the intensities of infrared active modes) corresponds to the first derivative of the PES with respect to an applied electric field and so on. Many thermodynamic properties can also be derived from vibrational frequencies as well. It is then clear the importance of ensuring that the system we are dealing with has been accurately optimised and that its final geometry corresponds to a minimum energy structure.

6.2.2 Ab Initio Methods: A Quick Summary

In the previous section, we have stated that the energy of a system is a function of its geometry (Eq. 6.1). However, we have not yet defined the functional form of the PES. In fact, one of the main differences between the many possible computational approaches is the definition of such a functional form.

The basic assumption of ab initio (Latin: “from the beginning”) or first-principles methods is that a system can be fully defined through its wave function, $\psi (\mathbf{x})$.¹ The energy of such system can be obtained through solving Schrödinger’s equation:

$$\displaystyle{ \widehat{H}\psi (\mathbf{x}) = E\psi (\mathbf{x}) }$$

(6.2)

where $\widehat{H}$ is the Hamiltonian operator and E the energy. In particular, we say that $\psi (\mathbf{x})$ is an eigenfunction of the Hamiltonian operator and E is its eigenvalue.

The main difference between the many types of ab initio techniques is in the definition of $\psi (\mathbf{x})$ (usually approximated through a linear combination of a set of either Gaussian-type or plane wave functions, the so-called basis set) and of $\widehat{H}$. In general, the Hamiltonian is expressed as a sum of the kinetic (T) and the potential (V ) energy operators:

$$\displaystyle{ \widehat{H} =\widehat{ T} +\widehat{ V } }$$

(6.3)

that in terms of the nuclei (n) and electrons (e) can be expressed as

$$\displaystyle{ \widehat{H} =\widehat{ T_{n}} +\widehat{ T_{e}} +\widehat{ V _{nn}} +\widehat{ V _{ne}} +\widehat{ V _{ee}} }$$

(6.4)

where we see the kinetic contributions of all nuclei ($\widehat{T_{n}}$, that within the Born-Oppenheimer approximation are separable and can be treated classically), the kinetic contribution $\widehat{T_{e}}$ of all electrons, the interactions between nuclei ($\widehat{V _{nn}}$), nuclei and electrons ($\widehat{V _{ne}}$), and electrons and electrons ($\widehat{V _{ee}}$).²

While the interaction potentials between nuclei-nuclei and nuclei-electrons can be fully described in terms of classical Coulomb interactions, $\widehat{V _{ee}}$ also has non-classical components, known as exchange and correlation. The exact formulation of the latter is unknown, though it plays a crucial role in defining the energy of a system, especially where weak interactions and van der Waals forces dominate.

In the Hartree-Fock (HF) approach, once the wave function has been defined, all terms apart from electron correlation can be calculated exactly and their accuracy depends only the quality of the basis set and numerical factors. Post HF methods are available to evaluate the electron correlation, which can be achieved with high accuracy using methods such as configuration interaction. However, such techniques are computationally expensive, and, up to very recently, implementations were only available for molecular cases. At present, these methods are rarely applied to solid-state system.

An alternative and popular method is density functional theory (DFT), which is based on the assumption that the energy of a system is a functional of its electron density, ρ(x, y, z), which corresponds to the product of the wavefunction with its complex conjugate (or just the square of the wave function if it is real) integrated over the electron coordinates. Within this approach, all non-classical terms are grouped into one contribution, namely, the exchange-correlation functional (V _xc ), whose exact formulation is unknown. Several kinds of approximations are available, the most widely adopted for minerals being the local density approximation (LDA), the generalised gradient approximation (GGA) and the hybrid DFT/HF approach (a percentage of exact HF exchange is included into V _xc ). Despite being quite extensively used in the past (e.g. Yu et al. 2010 and Li et al. 2007), predictions made at the LDA level often lack in accuracy, especially when dealing with systems that contain hydrogen or that are dominated by weak interactions.

In the following sections, we will make use of a Gaussian-type basis set specifically optimised for calcite (Valenzano et al. 2006) and of GGA and hybrid functionals, as implemented in the CRYSTAL code (Dovesi et al. 2014a). In particular, the PBE (Perdew et al. 1996) series of functionals will be considered, which comprises the pure GGA, PBE itself; its corresponding formulation devised for solids, PBEsol (Perdew et al. 2008); the 25 % hybrid, PBE0 (Adamo and Barone 1999); and PBE and PBE0 with inclusion of a long-range empirical correction terms, PBE-D2 (Grimme 2006) and PBE0-DC (Demichelis et al. 2013a), the latter with a dispersion contribution specifically fitted for carbonates. The popular hybrid functional B3LYP (Becke 1993) and the corresponding dispersion-corrected B3LYP-D2 (Grimme 2006) will also be used for further comparison. Specific details regarding the calculation set-up, parameters and algorithms adopted can be found in Demichelis et al. (2013a,b) and in De La Pierre et al. (2014b).

6.2.3 Calcite and Aragonite: Reference Systems to Assess the Accuracy of Methods and Parameters

One of the most delicate tasks, even for ab initio methods, is in ensuring that the model that we want to adopt is predictive for the system and properties that we would like to investigate. Therefore, we dedicate this section to validating the computational approach that we propose as a tool for the investigation of more complex calcium carbonate crystalline phases. Calcite and aragonite are relatively well-known and can be confidently used as reference systems to assess the accuracy of our methods. In particular, results obtained through our calculations will be compared to experimental data for a set of properties of one or both these two phases.

We will focus on testing and defining only those properties that we will use to characterise vaterite, without entering into the specific theoretical details of how they are computed. For a more comprehensive review of the many crystal properties that can be explored through ab initio methods and for further insights into the properties introduced in this section, we refer readers to Nye (1985), Dovesi et al. (2014b), Erba and Dovesi (2016) and De La Pierre et al. (2016).

6.2.3.1 Structural and Elastic Properties

The computed structure of calcite and some of its properties related to the elastic tensor are reported in Tables 6.1 and 6.2, where it is also compared against the available experimental data. Similar trends have been obtained for aragonite, but they are not reported here. The structure is obtained through minimising the energy with respect to the unit cell parameters and the atomic coordinates. Elastic properties can be calculated through evaluating the second derivatives of the energy with respect to the strain tensor components after the equilibrium structure is obtained (Dovesi et al. 2014b; Nye 1985).

Table 6.1 Relative % difference between calculated and experimental structures of calcite at 298 K (unit cell volume, Vol [ų]; lattice parameters, a, b and c [Å]; and bond lengths [Å]). Here the experimental structure is taken from Antao et al. (2009), Antao and Hassan (2010) and the values are given on the first line

Table 6.2 Bulk (K), shear (G) and Young’s (E) moduli, Poisson’s ratio ν, and symmetry independent components of the elastic tensor (C) [GPa] for calcite. Subscripts of V and R indicate whether the definitions of Voigt or Reuss were used, respectively (Experimental data are reproduced from Chen et al. 2001)

All DFT methods considered here provide a good description of the lattice parameters and of the atomic positions, reported in terms of bond lengths (Demichelis et al. 2013a). It is a known feature that PBE and B3LYP tend to overestimate the volume by 3–4 %, but the inclusion of empirical dispersion improves the results.

Table 6.2 is an example of how ab initio methods can be applied to realistically predict quantities that can be directly related to macroscopic properties of materials, such as elastic constants, bulk, shear and Young’s moduli and Poisson ratios. The table shows that relatively good agreement between experimental and calculated properties is achieved for calcite with the adopted methods.

6.2.3.2 Thermodynamic Properties

The energy that is obtained through solving Schrödinger’s equation (Eq. 6.1) corresponds to the electronic energy, U _el , and strongly depends on the formulation of the exchange-correlation functional, on the basis set and on the accuracy of other computational parameters (convergence thresholds, selected algorithms and approximations). As anticipated in a previous paragraph, the basis set adopted here was specifically optimised for calcite and therefore already shown to offer excellent predictions for several calcium carbonate crystal properties (Valenzano et al. 2006, 2007, Demichelis et al. 2014, 2012, 2013a,b, Carteret et al. 2013, De La Pierre et al. 2014a,b). The accuracy of the computational parameters was increased to a point that a further increase did not significantly affect the crystal properties. Therefore, in this chapter, any inaccuracy in predicting Δ U _el between two phases can be attributed to the approximation of the exchange-correlation functional.

To obtain quantities that can be compared to experimental values, i.e. the enthalpy (Δ H) and the free energy (Δ G) differences at a given temperature T, the vibrational contributions must also be computed. These are the zero-point energy, ZPE=$\frac{1} {2}h\sum _{\mathbf{k},i}\nu _{\mathbf{k},i}$ (where $\nu _{\mathbf{k},i}$ is the ith vibrational frequency at point $\mathbf{k}$ of the reciprocal space and h is Planck’s constant); the temperature-dependent constant-pressure specific heat, C _P (T); and the temperature-dependent entropy, S(T). In particular, the quantities that should be evaluated are H(T) = U _el (T) + PV (0) + ZPE + ∫ ₀ ^T C _P (T)dT and G(T) = H(T) − ∫ ₀ ^T S(T)dT, where V (T) is the volume at temperature T and P is the pressure. Here, our model is based on a few main assumptions that are considered valid for many ionic, covalent and semi-ionic crystalline systems at room temperature. Firstly, vibrations are assumed to be harmonic. Secondly, thermal expansion is disregarded, so that V (0) ≃ V (T) and C _P (T) ≃ C _V (T). Finally, only vibrational entropy is considered in the entropic term.

Computing the difference between these thermodynamic functions obtained at the DFT level for two polymorphs would, in principle, predict their relative stability. However, there are cases where this approach fails. In particular, the variation in relative stabilities associated with different DFT functionals is known to be larger than the actual values for phases that exhibit major structural differences and small energy differences. This is due to short- and long-range dispersion interactions not being properly accounted for Casassa and Demichelis (2012). Calcite and aragonite represent one of those cases: their structures are very different and their energetics very challenging, with Δ H and Δ G having opposite signs and being on the order of fractions of kJ/mol. On the contrary, for polymorphs having similar densities, such as calcite and vaterite, this error is fortuitously cancelled.

The relative energy differences between calcite and aragonite computed with different DFT functionals are compared to the experimental values in Table 6.3. A detailed description of the reasons behind the apparent discrepancies for many of the functionals is given in Demichelis et al. (2013a) and references therein, together with a discussion of the more complex case of hydrated polymorphs and water incorporation in the solid phase. Data reported in Table 6.3 show that DFT schemes that are able to provide an accurate description of many crystal properties (e.g. B3LYP, PBE0) are not necessarily able to reproduce the thermodynamics between polymorphs that are too different in structure and density. It is also shown that the choice of more appropriate dispersion coefficients (PBE0-DC) can at least partly redress this lack of accuracy.

Table 6.3 Values of the thermodynamic functions associated with the process aragonite → calcite at 298 K (S in J K⁻¹mol⁻¹, the rest in kJ/mol). Vibrational contributions were not calculated for B3LYP and PBE since they are expected to be of the order of fractions of kJ/mol and therefore would not be sufficient to lead to correct values of Δ H and Δ G (Experimental values from Wolf et al. 1996 and Königsberger et al. 1999)

The full energetics of vaterite with respect to calcite will be discussed in detail in Sect. 6.3. Table 6.4 reports only the values of the electronic energy differences between calcite and one of the stable phases of vaterite (viz. Vh in the table) obtained with a subset of DFT functionals, including those that give the largest deviations from experiment in Table 6.3. The differences in electronic energy between two forms of vaterite, namely, Vh and Vm, are also reported. Due to the more similar densities of the considered polymorphs, all DFT methods are able to reproduce the right stability order with small variations in the electronic energy difference.

Table 6.4 Differences in electronic energy [kJ/mol per fu] between hexagonal vaterite (Vh, P3₂21) and calcite (C) and between hexagonal and monoclinic vaterite (Vm, C2). For the different structures of vaterite, see Sect. 6.3

6.2.3.3 Vibrational Spectroscopy

Raman and IR spectra of calcite, aragonite and related phases have been extensively investigated from both a computational and an experimental point of view (e.g. Donoghue et al. 1971, Frech et al. 1980, Gillet et al. 1996, Alía et al. 1997, Prencipe et al. 2004, Valenzano et al. 2006, 2007, Wehrmeister et al. 2011, 2010, Carteret et al. 2013, De La Pierre et al. 2014a,b). The method that we are applying here is the same as the one adopted in many of the aforementioned publications (see, e.g. De La Pierre et al. 2014a,b), with a demonstrated ability to provide accurate predictions of both vibrational frequencies and intensities. Here we simply highlight that the accuracy in computing vibrational properties is important for three main reasons. Firstly, a predictive method, like the one we apply here, can be successfully used to investigate phases whose properties and structure are still a matter of debate. Secondly, as mentioned above, many thermodynamic functions can be obtained via computing the vibrational frequencies of the system; the more accurate the vibrational frequencies, the more accurate the quantities that depend on them. Thirdly, since computed vibrational frequencies contain information about the topography of the PES, they allow us to discriminate between stable structures and transition states. Crucially, the third of these points has been used to investigate the crystallography of vaterite and provide a model for its complex structure.

Table 6.5 shows that the B3LYP hybrid functional is, in principle, one of the best choices for reproducing vibrational properties.³ The maximum ( | Δ _max  | ) and the mean absolute difference ($\vert \bar{\varDelta }\vert$) between computed and experimental frequencies are the lowest, and the mean difference ($\bar{\varDelta }$) is around 0 cm⁻¹, meaning that there is no systematic shift towards higher or lower values with respect to experiment. However, Table 6.3 shows that B3LYP is not accurate in predicting the electronic energy difference between different phases. As Δ U _el is the contribution that dominates the energetics, we should look for a compromise since ideally we want a model that is able to reasonably reproduce vibrational spectra and thermodynamics simultaneously. PBEsol can be considered a good compromise, as it provides reasonable results for carbonate properties, including vibrational spectra, and for their energetics. Despite the data in Table 6.5 suggesting that PBEsol is significantly worse than B3LYP for predicting vibrational frequencies, if we consider the five main regions of the spectrum separately (i.e. lattice modes, L; symmetric and asymmetric stretching of CO₃ ²⁻, ν ₁ and ν ₃; in-plane and out-of-plane bending of CO₃ ²⁻, ν ₄ and ν ₂), all frequencies within a single region are shifted with respect to the actual value by nearly the same quantity (Δ L = +3.1; Δ ν ₁ = −11. 4; Δ ν ₂ = −44. 4; Δ ν ₃ = −7. 8; Δ ν ₄ = −24. 3; all in cm⁻¹). To obtain a more accurate prediction, it is therefore sufficient to subtract this quantity from each frequency of the region (De La Pierre et al. 2014b).

Table 6.5 Statistical analysis of the differences between calculated and experimental Raman active modes of aragonite. | Δ _max  | and $\vert \bar{\varDelta }\vert$ are the maximum and the mean absolute difference, respectively, and $\bar{\varDelta }$ is the mean differences. All data are in cm⁻¹ (Experimental data from De La Pierre et al. 2014a and Carteret et al. 2013)

6.3 The Multiple Structures of Vaterite

As mentioned in Sect. 6.1, vaterite is one of the three crystalline polymorphs of anhydrous calcium carbonate. Though it is metastable with respect to calcite and aragonite, it is found in nature as a result of biomineralisation (Cartwright et al. 2012; Hasse et al. 2000; Qiao and Feng 2007). For example, vaterite is crystallised in bivalve organisms and mussels to repair damage in their hard tissues (Spann et al. 2010; Wilbur and Watabe 1963). Vaterite is also often observed as an intermediate phase during the transformation that leads from amorphous calcium carbonate nanoparticles (ACC) to calcite and aragonite (Cartwright et al. 2012; Gebauer et al. 2010; Rodriguez-Blanco et al. 2011). For these reasons, vaterite is now attracting much interest as further insight into its features could reveal new details about non-classical nucleation pathways, polymorphism and crystallisation under biogenic conditions.

The structure of vaterite has been debated for over 50 years, due to what is often referred to as “disorder” of the carbonate anion sites. Table 6.6 presents a summary of the most relevant studies from 1959 to 2012 aimed at resolving the structure of vaterite, showing a clear disagreement as to the crystal system (orthorhombic, hexagonal, monoclinic and triclinic), the crystal symmetry and site occupancy and even as to the unit cell size. We refer the readers to the original literature for more details about these studies. Here we show how the approach described in this chapter has contributed a new perspective through proposing a model which is in agreement with most of the experimental findings.

Table 6.6 Structures proposed for vaterite up to 2012: space group (SG) and lattice parameters (a, b, c [Å]). Cell angles from Mugnaioli et al. (2012): β=118.94^∘ (C2∕c); α=γ=90.0^∘, β=99.22^∘ ($\overline{C}1$); α = 82.1^∘, β=97.9^∘, γ=119.3^∘ ($\overline{P}1$, in parentheses, corresponds to the primitive unit cell of the structure with $\overline{C}1$ symmetry). Cell angles from Wang and Becker (2012): β=115.9^∘ (C2∕c)

In Demichelis et al. (2012, 2013b), electronic structure calculations were performed for the orthorhombic models (Pbnm, Ama2), for the hexagonal model (P6₅22) that was obtained by Wang and Becker (2009) as a result of re-examining previously proposed models through DFT and force-field-based calculations and for the monoclinic and triclinic models (C2∕c and $\overline{C}1$) proposed by Mugnaioli et al. (2012) as a result of combined automated diffraction tomography (ADT) and precession electron diffraction (PED). Note that Wang and Becker (2012) predicted the presence of a monoclinic C2∕c basin slightly less stable than their hexagonal P6₅22 one through applying molecular dynamics techniques.

The calculation of second derivatives, which was not performed in previous computational studies, showed that all of these structures but one (Ama2) correspond to transition states rather than to minimum energy structures and therefore cannot be a true representation of vaterite.

As mentioned in Sect. 6.2, transition states are characterised by the presence of one or more imaginary modes (i.e. the Hessian matrix has one or more negative eigenvalues). The structure with Pbnm symmetry exhibits one imaginary mode, the P6₅22 hexagonal model four and the monoclinic (C2∕c) and triclinic ($\overline{C}1$) structures two. Following the displacement along the eigenvectors related to the imaginary modes, it was found that they all correspond to symmetry forbidden rotations of carbonate anions. After removing the corresponding symmetry constraints, the true minimum energy structures were found.

As a result of this procedure, it was found that the structure with Pbnm symmetry relaxes to a P2₁2₁2₁ configuration that is 1–2 kJ/mol per formula unit more stable than the original one, depending on the DFT functional. The P6₅22 arrangement relaxes towards three minimum energy structures, P3₂21 (−3.5 kJ/mol), P6₅ (−2.5 kJ/mol) and P112₁ (−3.0 kJ/mol),⁴ and one transition state with two imaginary modes. The latter was not fully analysed, but through removing the symmetry constraint related to the imaginary modes and reoptimising, P3₂21 and a P112₁ structures were obtained. The C2∕c structure assumes either a C2 or a Cc configuration, which are 1.4 and 0.5 kJ/mol more stable, whereas the $\overline{C}1$ structure falls into two very similar structures, both with symmetry C1 and about 0.1 and 0.2 kJ/mol more stable.⁵

A more accurate analysis of the structure and relative energies show that both of the orthorhombic models considered (Ama2 and P2₁2₁2₁) do not fit with the experimental expectations. Apart from being too dense, due to the optimised b lattice parameter being about 10 % shorter than that suggested by Meyer (1959) and Le Bail et al. (2011), P2₁2₁2₁ and Ama2 are about 1 kJ/mol and 18 kJ/mol less stable, respectively, than the lowest energy structure derived from Wang and Becker’s model (P3₂21). A second point against the most stable orthorhombic model comes from a study performed by Balan et al. (2014) through simulation of the mixing energy of sulphate into vaterite and reanalysis of the anomalous data presented by Fernández-Díaz et al. (2010). Notably, this study also ruled out the hexagonal P6₃∕mmc structure in favour of the P3₂21 model proposed by Demichelis et al. (2012), whereas the monoclinic and the triclinic models were not considered.

The seven structures that were obtained through the computational procedure highlighted in this chapter and that have structural and energetic features that are in agreement with the experimental observations were divided into three groups: six-layer hexagonal, referring to the structures derived from the hexagonal model that has six layers of carbonate anions in the unit cell (P3₂21, P6₅, P112₁); two-layer monoclinic, referring to those derived from the monoclinic structure that has two layers of carbonate anions in the unit cell (C2, Cc); and six-layer triclinic, referring to those obtained from Mugnaioli’s triclinic structure (C1, C1). Figure 6.1 shows that these groups of structures can be considered different polytypes of vaterite, differing only in the sequence of the carbonate layers stacked along the z Cartesian axis. In particular, one can go from the monoclinic to the hexagonal model by switching two planes, whereas the triclinic model has two missing planes with respect to the other two models, causing its γ lattice parameter to deviate from 90^∘. Starting from the hexagonal model, by performing all the possible physically allowed permutations of the six different planes along the c axis, the only structures that are found correspond to one of those in the hexagonal or monoclinic groups (including their chiral images).

Fig. 6.1

Top: Ball and stick representation of the structures for vaterite belonging to the three different families of polytypes, viewed along the a lattice vector. O, Ca and C are represented in red, green and grey colours, respectively. From left to right: the six-layer hexagonal model, the six-layer triclinic model and the two-layer monoclinic model. The latter has a smaller unit cell (bold black line), so that a six-layer supercell (thin black line) is built to show the structural similarities and differences with respect to the other models. The stacking sequence of CO₃ ²⁻ layers refers to the orientation of the CO₃ ²⁻ units that have a C-O bond parallel to a: planes with the same shift with respect to b are labelled with the same letter (A for 0; B for 1/3; C for 2/3); planes with the C-O bond pointing towards + a and − a are labelled with and without a prime, respectively. Bottom: The chiral image of a portion of the hexagonal structure is represented (left), as well as the meaning of rotational disorder of CO₃ ²⁻, which is allowed at room temperature (Reprinted with permission from Demichelis, Raiteri, Gale, Dovesi, The Multiple Structures of Vaterite, Cryst. Growth Des., 13 (6), 2247–2251. Copyright 2013 American Chemical Society)

Notably, the hypothesis of stacking faults in vaterite was originally suggested by Meyer (1969), and twin stacking faults were detected by Qiao and Feng (2007) using HRTEM and SAED techniques on natural samples of vaterite. Wehrmeister et al. (2010) suggested the existence of different polytypes of vaterite, though there was not enough evidence in their data to fully support the claim. On the contrary, chirality was never suggested prior to the studies of Demichelis et al. (2013b). At this point, it is reasonable to hypothesise that a certain number of other structures may exist, having missing planes and different sequences with respect to the three groups of structures reported here.

Symmetry and thermodynamic data obtained for the aforementioned hexagonal, monoclinic and triclinic models are summarised in Table 6.7. All structures present similar free energies and are equally likely to represent a good model for vaterite, with C2 and P3₂21 models being the most stable. Data in the table also show that switching two carbonate layers with each other has no or little cost in energy. Figure 6.2 shows a plot of Δ U _el between a given structure and P3₂21. While the data presented here are unable to predict whether it is possible to transform a structure belonging to one basin into a structure belonging to another one, the energy barrier for rotating CO₃ ²⁻ anions is so low that within the same basin we can suppose that structures can interconvert between each other at 298 K.

Table 6.7 Symmetry and stability of the seven structures of vaterite found by Demichelis et al. (2012, 2013b). For each structure, the corresponding chiral image, crystal system and number of independent carbonate anions in the unit cell ($N_{CO_{3}}$) are indicated. The thermodynamic quantities (PBEsol) associated with the process vaterite → calcite at 298 K are calculated (all in kJ/mol except for Δ S in J K⁻¹ mol⁻¹). The Boltzmann relative populations p at 298 K are also calculated for each structure. The Boltzmann-weighted average of the thermodynamic quantities and the corresponding experimental data from Königsberger et al. (1999) are shown

Fig. 6.2

Representation of Δ U _el for the three polytypes of vaterite, separated by vertical lines. Space groups having a chiral image that belong to another space group are indicated. The structure labelled as TS4 is a transition state obtained after the analysis of one of the four imaginary modes found for P6₅22; see text for more details (Reprinted with permission from Demichelis, Raiteri, Gale, Dovesi, The Multiple Structures of Vaterite, Cryst. Growth Des., 13 (6), 2247–2251. Copyright 2013 American Chemical Society)

From the above discussion, we can conclude that from a theoretical point of view, the hypothesis of vaterite existing in several forms rather than assuming one particular structure is very likely to be correct. In particular, our analysis shows that there are three levels of complexity in the structure of vaterite: the intrinsic rotational disorder of carbonate anions, the possibility of having different carbonate layer sequences and the presence of chirality. The claim of vaterite assuming multiple structures was confirmed for the first time in the study by Kabalah-Amitai et al. (2013), where HRTEM images show that there are at least two interdispersed crystal structures in a vaterite sample. However, this work does not investigate the details of the structural features of the two different phases. It was found that their main difference is related to carbonate anions, whereas calcium layers are nearly the same. While one of the phases is claimed to be hexagonal, the structure of the other one could not be determined.

As a further attempt to definitively solve the problem of the structure of vaterite, two independent studies have been recently undertaken, using experimental and computed NMR, XRD and Raman spectra, all leading to the same result.

One of these works De La Pierre et al. (2014b) re-examined existing and new experimental Raman data in the light of the new structural information from Demichelis et al. (2013b) and Kabalah-Amitai et al. (2013). The rich collection of Raman spectra recorded on a variety of vaterite samples of geological, biological and synthetic origin from Wehrmeister and co-workers (2011; 2010) was used. New samples considered in this study included Herdmania momus spicules from the Great Barrier Reef (Queensland, Australia), the same kind of samples where Kabalah-Amitai et al. (2013) identified two distinct structures.

One of the most significant points arising in this study was Wehrmeister et al.’s (2010) observation, based on the analysis of their Raman spectra, that vaterite has at least three independent carbonate units. Notably, all structures proposed in the literature prior to Demichelis et al. (2012, 2013b) have no more than two such groups. This means that either the new structures are a better choice or that multiple structures within the same sample are present or both.

The Raman spectra for the seven structures proposed here as possible candidates were computed by De La Pierre et al. (2014b) and compared to the experimental data. Overall, they all look very similar. However, all the computed spectra but two have peaks that do not appear in the most intense region of the experimental spectra, namely, ν ₁ (symmetric stretching of CO₃ ²⁻, 1020–1160 cm⁻¹) and ν ₄ (in-plane bending of CO₃ ²⁻, 660–750 cm⁻¹). Figure 6.3 shows that C2 and P3₂21, the most stable two-layer monoclinic and six-layer hexagonal structures, respectively, have Raman spectra that match with those experimentally recorded. The latter structure is in agreement with Kabalah-Amitai et al. (2013), who found that one of the structures in their sample has a hexagonal unit cell.

Fig. 6.3

Calculated Raman spectra for the seven structures of vaterite (red) and comparison with one experimental spectrum recorded on a geological sample from Wehrmeister et al. (2010) (black). Other samples exhibit very similar features. In the latter, signals resulting from calcite and aragonite impurities are highlighted

On the basis of Raman spectroscopy alone, it is not possible to tell whether the first, the second or both structures are present in the experimental samples. Also, the remaining five theoretical structures seem to be ruled out according to this study. These results then seem to provide evidence that opposes the room temperature intra-basin structural interconversion via carbonate rotations suggested by Demichelis et al. (2012). This would result in different domains being present in the same crystal and thus in a spectra that is a Boltzmann-weighted average of all seven spectra. A possibility is that the system becomes kinetically trapped in a subset of the possible stable structures. To verify this hypothesis, further investigation is required in order to determine the activation barriers for the interconversion between all of the different energy minima.

A second study, by Burgess and Bryce (2015), has recently shown that only C2 and P3₂21 structures are compatible with⁴³Ca NMR spectra recorded on samples of vaterite, after considering most of the structures that have been proposed in the literature. C2 and P3₂21 were also shown to be compatible with the measured XRD diffraction pattern. As with the Raman spectroscopy, also⁴³Ca NMR seems to be unable to provide a definitive answer on whether the former, the latter or both structures are present in the real sample.

6.4 Conclusions and Future Challenges

The theoretical approach highlighted in this chapter is highly predictive in determining the structures and properties of crystalline phases. These tools are therefore particularly useful when studying phases and processes that are challenging to investigate with experimental techniques. The case of vaterite represents one of the best examples of how ab initio methods can be applied to unravel the complexities of metastable biomineral phases and more generally of minerals that are not easily accessible as large single crystals for diffraction. It also represents a successful example of how computational schemes can be combined with experimental techniques to investigate complex mineral structures.

Future challenges in this field are related to the development of algorithms able to address the non-generality of the many approximations on which ab initio methods rely. In the particular case of biominerals, one of the most important features that would open the way for an appropriate investigation of the composition, the stability and the stoichiometry of the many hydrated and hydroxylated phases would be the ability to properly account for weak interactions in a fast and accurate way. Methods that are currently available are either too approximate or not available in solid-state codes or computationally too expensive for systems that contain tens of atoms in their unit cell (Del Ben et al. 2012; Pisani et al. 2008).

In the particular case of vaterite, future challenges are mostly related to finding or developing experimental techniques that are able to distinguish the different structures and confirm their features. Moreover, the average size of a vaterite single crystal (tens of nm in the best case scenario) is too small to undertake any investigation as to its chirality with available optical techniques (Bilotti et al. 2002). From a biomineralisation perspective, chirality would be an extremely interesting property to investigate, since vaterite often grows under biogenic conditions, therefore in the presence of biological molecules that are often optically active. In this context, also other biominerals may exhibit optical activities, such as monohydrocalcite, which like vaterite belong to one of those space groups that are intrinsically chiral (Demichelis et al. 2014).