Computational Evidence

Force Fields

Any atomistic level study of calcium carbonate nucleation requires the use of a force field which accurately captures the structure of emerging phases, and also the correct energetics of solvation and ion binding equilibria. A number of force fields accurately reproduce the cell parameters of stable anhydrous phases, such as those

of Dove et al. [Dove et al., 1992] and Pavese et al. [Pavese et al., 1992]. Both of these force fields were shown to match well with experimental measurements for calcite elastic constants and lattice frequencies. Carbonates were treated as a single molecule with angle bending and improper dihedrals.

Unfortunately, these early models failed to accurately reproduce interfacial structures and energetics of crystalline phases with water, which is crucial for studies of biomineralisation. Parker and co–workers [Leeuw and Parker, 2001; Kerisit and Parker, 2004] used a shell model for water, combined with elements of the Pavese potential to model the free energies and structure of water adsorption onto the surface of calcite. Freemanet al. took the most successful elements from the Pavese potential and that of de Leeuw and Parker [Leeuw and Parker, 2001], combined with a TIP3P water model [Jorgensenet al., 1983] to derive a force field specifically for biomineralisation studies [Freeman et al., 2007]. They developed a generic method to produce force fields which combine off-the-shelf potentials for organic molecules with pre-existing mineral potentials, in order to enable efficient studies of organic– mineral interfaces.

Raiteri et al. highlighted that while previous force fields perform well when modelling crystalline phases, they fail to reproduce the thermodynamics of com- ponent ions in solution as well as the free energies differences between crystalline phases of calcium carbonate [Raiteriet al., 2010]. They, therefore, developed a new force field based on rigid carbonates with a number of Buckingham interatomic po- tentials, and with the rigid four–site TIP4P–Ew water model [Horn et al., 2004]. As well as reproducing the cell parameters and properties of crystalline phases, the force field accurately modelled the solvation enthalpies of ions in solution. The use of rigid carbonates may not be ideal when investigating amorphous phases, and so the force field was later adapted [Raiteri and Gale, 2010] to include flexible carbonate and water, making use of the SPC/Fw potential for water [Wuet al., 2006]. Further adaptation was made by Demichelis et al. when bicarbonate ions were included and cross–terms were added to molecular anions in an effort to improve vibrational spectra modelling [Demicheliset al., 2011].

DOLLOP

Demicheliset al. performed large free ion solution simulations at concentrations of 0.06, 0.26 and 0.5 M in the pH range 8.5–11.5 [Demicheliset al., 2011]. The pH was defined by the ratio of bicarbonate to carbonate ions in solution, and simulations were conducted for up to 70 ns. Spontaneous aggregation of ions was observed, be- yond the expected ion pairs. Large numbers of ions clustered to form chains, rings

and a multitude of cluster shapes. A typical example of a cluster is shown inset in Figure 1.10. At high pH, where carbonate dominates the equilibrium, clusters grew to large sizes and resembled a branched polymer with average Ca–C coordination of two. At lower pH, smaller clusters formed as bicarbonate tended to bind to one cation only, resulting in termination of chains. Cluster size distributions suggest that maximum cluster sizes were_≈25 and 220 ions at the extremes of low and high pH. Clusters at all pH levels showed a large amount of conformational freedom and were liquid–like. Furthermore, the cluster sizes both decreased and increased contin- ually throughout the simulations as ions dissolved and aggregated, suggesting that a dynamic (dis)ordering was taking place (where order in this description refers to short range atomic order). The authors therefore described clusters as dynamically ordered liquid–like oxyanion polymers (DOLLOPs).

Figure 1.10: Probability distribution for a 36 formula unit calcium carbonate cluster in water, highlighting radius of gyration,Rgyr, regions for three states. Inset is an example of calcium carbonate DOLLOP, and here cyan, red and green atoms show carbon, oxygen and calcium, respectively, while purple lines highlight connections between ions. Images were taken from Demicheliset al. [Demicheliset al., 2011].

Free energy sampling of a six formula unit DOLLOP as a function of radius of gyration showed that a dense, high coordination cluster was unstable compared with the highly disordered DOLLOP, for which there was no energy barrier to changing conformation or even partially dissolving. The probability distribution for a 36 formula unit cluster as a function of radius of gyration is shown in Figure 1.10. Three regions representing an anhydrous (Dry NP) and hydrous nanoparticle (Wet NP) as well as DOLLOP are clearly identifiable in the distribution. DOLLOP has a much wider distribution of accessible gyration radius, suggesting that the entropy of the calcium carbonate in this state is higher than in solid nanoparticles.

A speciation model was fitted to the data to estimate equilibrium constants and free energies of ion binding in solution. Using the model, the authors were able to show the fraction of calcium bound in ion pairs and DOLLOP (_∼70%) was close to those found by Gebaueret al. [Gebaueret al., 2008], leading the authors to conclude that PNCs are DOLLOP. The speciation model was fitted to the simulation data at concentrations hundreds of times larger than those of experiment, so to verify the stability of the liquid phase, four small DOLLOPs containing four ion pairs were simulated at 0.4 mM for one nanosecond. Partial disassembly was observed, but as the clusters remained largely unbroken on a time scale much longer than the time for water exchange in the calcium coordination sphere (_∼ 80 ps), the stability of DOLLOP was confirmed.

Liquid–Liquid Separation

A separate study of the nucleation of calcium carbonate from constituent ions in solution was made by Wallaceet al. [Wallaceet al., 2013]. As opposed to previous simulations, the authors ensured the concentration of ions was comparable to exper- iment, with [Ca2+]=15 mM. Calcium carbonate cluster growth was simulated using the method of Kawska and Zahn [Kawska et al., 2006]. From an initial solvated cluster (water molecules within 4.5 ˚A of ions were considered to be solvating), an additional solvated ion pair was placed on a surrounding sphere. With the cluster fixed, relaxation of the solvated ion pair was performed. The new cluster coordi- nates were fixed in a pre-equilibrated water box to relax solvent molecules for 50 ps. Finally, all atoms were mobilised and replica exchange molecular dynamics (REMD) was performed; eight replicas were simulated in the temperature range 300–400 K for 0.5 ns. The 300 K cluster was extracted, from which further growth was initiated. Clusters were grown up to a size of around 2 nm and comprised of a max- imum of 40 ions. For small clusters, low density configurations were observed, with coordination numbers comparable to those found for DOLLOP. As cluster size

increased, higher densities were observed and compact clusters were found. The dif- fusion of ions decreased as a function of size, approaching a bulk liquid phase, and the diffusion remained much higher than the diffusion of ions in ACC. The average coordination was found to change from two to three at a size of_∼26 ions. The free energy of clusters decreased monotonically as a function of increasing size. Wallace

et al. note that this is indicative of a solution undergoing a spontaneous phase separation by spinodal decomposition, and enables the formation of crystal either by ion–ion or cluster mediated pathways. Making use of an Ising lattice gas model of phase separation, they show that in the dense liquid phase _∼ 100 nm clusters are likely to form, while smaller clusters are likely to evolve in undersaturated solu- tions. The coexistence of small and large clusters seems to fit well with experimental observations [Pougetet al., 2009].

Wallace et al. proposed the phase diagram shown in Figure 1.11 [Wallace

et al., 2013]. In the diagram, the solubility of all solid polymorphs is shown by the solubility line, SL, and the blue region represents undersaturated solution. Following the addition of ions at constant temperature (green line), a stable homogeneous system crosses the binodal line (L–L) with increasing IP and becomes metastable. On further increase in IP, the system crosses the spinodal curve (SP) and phase separation is spontaneous. Tc marks the lower critical solution temperature, below which no liquid–liquid phase separation occurs.

Figure 1.11: A schematic of a suggested phase diagram for hydrated calcium car- bonate, as suggested by Wallace et al. [Wallace et al., 2013]. For a full description see the text.