Comparing Open and Compact Clusters

To investigate whether larger cluster sizes are likely to become dynamically ordered structures when immersed into water, clusters of nCaCO₃, but with lower initial nCa, have also been simulated. As discussed in section 3.3.2, asnCadecreased over the five sampled clusters (from gas phase optimisations) per formula unit, more open structures were found, and the dissipation of a well defined ionic core was observed. Figure 4.3 shows the potential energy of clusters containing 20–29 formula units, after relaxation in water. It is clear that lower initial nCa does lead to a solvated cluster with lower potential energy, with the difference being_∼0.1 eV per CaCO₃ unit. This difference is comparable to the potential energy barrier evident in Figure 4.1 for compact clusters.

The mobility of ions around the centre of mass of clusters with low coordi- nation is higher than for equivalently sized clusters with more dense packing. The final configurations for a 28CaCO₃ cluster have been included in Figure 4.3; these show a much less ordered structure after 5 ns of simulation for a cluster with low initialnCa, than for the corresponding high coordination structure (simulated from the minimum energy structure in vacuum) after 10 ns. The average U of all five samples considered appears to converge over the range of cluster sizes considered.

Longer simulations (50 ns) were conducted for the most open and compact clusters that were sampled forn=10, 20, 30 and 40. The potential energies measured across this range from the final 5 ns of simulation (see Figure C.3), showed the same energy difference between open and compact cluster states. Forn= 10, the energy of clusters in the limits of low and high initial coordination converged to -32.5 eV.

Figure 4.3: Potential energy,U, per formula unit,n, of clusters in aqueous solution for (20–29)CaCO₃ with maximum (black) and minimum (blue) initial nCa. The red–dashed line gives the average U for the five samples simulated, where these samples span the nCa distribution, as described in Chapter 3. Data points were calculated from the final 1 ns of respective 10 or 5 ns trajectories, with uncertainty of one standard error of the mean indicated. Final configurations for clusters of 28CaCO₃ are provided. Atom and bond colours are as for Figure 4.2. Bulk water energies have been subtracted.

There does appear to be a slight positive gradient in the potential energy as a function of cluster size for the most compact clusters, and this is likely to be due to increased levels of ionic coordination, and therefore a concomitant decrease in ionic solvation.

Figure C.4 shows representative structures for clusters with initial low (open) and high (compact) coordination after relaxation in water. As expected from simu- lations of compact clusters,nCa (Figure 4.4) and CSDs for both open and compact clusters converged very quickly in the simulations. However, for larger clusters, while there was an evident but slight decrease in the averagenCa, as shown in Figure 4.4, there was a consistent difference in the level ofnCa as a function of time.

The average coordination of larger compact clusters by the end of long simu- lations was consistent with ACC, and that for open clusters was closer to DOLLOP (1.9_±0.2₋2.5_±0.1), albeit with a slight difference in the pattern of bond ordering. As shown by Demichelis et al., when DOLLOP was formed with clusters contain- ing large amounts of carbonate, the probability of calcium binding to two carbons

Figure 4.4: Calcium carbonate coordination number,nCa, as a function of time for simulations of nCaCO₃ where n =10, 20, 30, and 40, in water. Blue and black curves represent clusters with initial low and high density respectively.

was around 0.3, with binding to three and four carbons being _∼ 0.2 and 0.05 re- spectively. [Demichelis et al., 2011] Even for 10 CaCO₃, the binding probabilities measured, as shown in Table 4.1, suggest an increase in the mass density of clusters due to an increased amount of two-fold coordination and a decrease in the level of calcium binding to a single anion. For larger open clusters, a similar binding pattern was found, albeit with a decrease in the probability of Ca–C, as Ca–4C and Ca–5C increased. It is not clear how the coordination in DOLLOP will evolve over long sim- ulation times, and whether DOLLOP with increased “branching” will become more likely. The difference in bond ordering could be due to increased concentrations in these simulations compared with those of Demicheliset al.. For the most compact clusters, the binding probabilities showed a greater likelihood of binding to more than three carbons (see Table 4.1). This is further suggestive that in large compact clusters, the packing of ions in the core remained high, and that these clusters were not indicative of a DOLLOP structure.

Table 4.1: Coordination probabilities for calcium binding toN carbons, measured as a mean average from the final 10 ns of a 50 ns simulation of open and closed clusters containing 10 and 40 CaCO₃ units in water.

N 10 CaCO3 open 40 CaCO3 open 40 CaCO3 compact

1 0.30±0.11 0.15±0.04 0.10±0.03 2 0.52±0.06 0.45±0.09 0.18±0.02 3 0.16±0.06 0.24±0.05 0.25±0.02 4 0.01±0.02 0.10±0.02 0.19±0.01 5 0 0.06±0.01 0.18±0.01 6 0 0 0.10±0.01

The behaviour of the larger, open clusters in water were more comparable with those of smaller sizes, in that they displayed dynamic coordination of ionic species throughout the duration of the trajectories. Calcium–carbonate bond life- time probability densities were measured for clusters over the final 10 ns of a 50 ns simulation. A bond was defined by a distance criteria of rCa−_C < 4 ˚A. As the

RDFs of Ca–C (Figure A.1) and Ca-Owat(Figure A.4) show, this distance includes the first Ca–C coordination shell, but is also large enough such that solvation of calcium is likely if ionic coordination is broken. This ensured that large spatial fluctuations in the coordinating ions did not lead to inaccurate estimation of short bond lifetimes. In larger clusters, a fraction of the connections were unbroken over the 10 ns sampling window: 2.4% and 1.8% of connections were fixed in compact clusters containing 30 and 40 units, while 0.6% of connections were fixed in the low density 40 CaCO cluster.

Figure 4.5 shows the bond lifetime probability densities for simulations con- taining ten formula units of calcium carbonate, as well as those for open and compact clusters of the largest size studied. The plot shows that ionic connections are re- tained for long times in both open and compact clusters. Compact clusters showed much longer Ca–C binding, with probability density up to around 4 ns. The most open cluster, that of 10 CaCO₃, had the highest probability density for lifetimes <0.5 ns, highlighting that this cluster showed the most dynamic (dis)ordering. The dynamic ordering in the largest open cluster was on a similar time-scale to that of 10 CaCO₃, but the probability density of bond ordering at short times is slightly re- duced. Figure C.5 provides the bond lifetime probability densities for open clusters across the cluster size range studied, which further shows that as cluster size was increased, calcium carbonate connections persisted for longer times. This can be explained by considering the dynamics of the breaking and reforming ionic connec- tions within the clusters. For very small clusters, when a connection is broken, there is a limited, small number of possible connections that can be made to reproduce a cluster of equal size to the initial state. However, for larger clusters, there are many possible ionic connections apparent upon cluster dissociation. Furthermore, the life- time of an insufficiently coordinated ion in larger clusters is reduced, compared with smaller clusters of comparable density, as ionic concentration in the local proximity of cluster constituents increases.

Figure 4.5: Calcium carbonate bond lifetime probability densities measured from the final 10 ns of 50 ns simulations of 40 CaCO₃ with low (orange) and high (red) density, and 10 CaCO₃ with low density (blue) in water. Bonds were defined by a distance criteria between calcium and carbon atoms of<4 ˚A. The average lifetime of the bond is given on thex axis. Inset is the probability densities for bonds with a lifetime under 0.5 ns.