q6 Order Parameter

In chapter 4, a way was required to detect the overall degree of crystallinity in a system between a FCC LJ crystal and its melt. In that context, q6 was used, which is widely known to be able to reliably distinguish between the two phases[81]. However, it is not clear if it could be suitably applied to any or all of the mW ice-water systems of interest, and so this must be checked.

The bond correlation distribution for ice-Ic, over 3.5 ˚A, is shown in figure 5.2 (left), which clearly shows that the liquid displays a broad range of correlations about a peak correlation of 0.26, while the ice-Ic is more strongly positively corre- lated, peaked at 0.88; the overlap of the distributions is only 11%. Correlations of

c(i, j)≥0.62 are also more strongly favoured by ice-Ic, meaning correlations match- ing this criteria are likely solid-like. This threshold would correctly identify 94.1% of ice-Ic and incorrectly identify 5.6% of the water bond correlations as solid-like. Alternatively, a threshold ofc(i, j)≥0.5 could be used, which identifies 98.4% and 14.7% of the ice-Ic and water bond correlations, respectively, as solid-like. This demonstrates it is possible to correctly identify a greater proportion of the ice-Ic correlations as solid-like by lowering the correlation threshold, while increasing the proportion of incorrectly identified water bond correlations as solid-like. This is

0 0.01 0.02 0.03 0.04 0.05 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 P ( c ( i, j )) c(i, j) Liquid Ice-Ih

Figure 5.3: Theq6 bond correlation distribution in ice-Ih and water over 3.5 ˚A. why a second piece of criteria is required to improve upon the identification of solid particles; the number of neighbouring particles over 3.5 ˚A.

Using this additional criteria, as shown in figure 5.2 (centre), 99.6% of the solid has 4 nearest neighbours, compared to 25.3% of the liquid. When combining these two constraints, a solid-like bond can be defined if an atom has a bond correla- tion ofc(i, j)≥0.5 with its neighbour and has a total of 4 nearest neighbours. The effect of these combined constraints in both the liquid and solid ice-Ic on the number of solid-like bonds, is shown in figure 5.2 (right). Now, a particle can be classified as being solid when having 3 or more solid-like bonds. This correctly identifies on average over the 5 ns bulk simulations, 98.9% of the ice-Ic as solid and 0.37% of the water as solid. This is a significant improvement on using only the criteria of c(i, j) ≥ 0.5, since comparable correct solid identification is achieved while the amount of liquid being incorrectly identified as solid is now reduced by 97.5%. This combined criteria is consequently used for distinguishing between ice-Ic and water systems using q6, where particles are assigned the number of their solid-like bonds as their intrinsic order parameter value.

Whileq6 successfully distinguishes between ice-Ic and water, it suffers from limitations when applied to ice-Ih. As shown in figure 5.1, the distribution of the number density over 3.5 ˚A is the same in both ice-Ic and ice-Ih, which indicates that there must be the same number of neighbours in the first coordination shell for these two polytypes. This means the same criteria for the number of neighbours can be used in ice-Ih as ice-Ic. However, the bond correlation distribution for ice-Ih

0 0.2 0.4 0.6 0.8 1 0 1 2 3 P (B on d s) Bonds c(i, j) = 0.25 0 1 2 3 4 Bonds c(i, j) = 0.5 Liquid Ice-Ih

Figure 5.4: The number of solid-like bonds in ice-Ih and water over 3.5 ˚A when nearest neighbours = 4 andc(i, j)≥0.25 (left) orc(i, j)≥0.5 (right).

is different to that of ice-Ic, as shown in figure 5.3, where now a second lesser peak is clearly present atc(i, j) = 0.36.

The presence of two distinct peaks in the ice-Ih correlation distribution is indicative of the two types of bonds that form in ice-Ih: staggered bonds (s-bonds); and eclipsed bonds (e-bonds)[82]. In ice-Ic, only the peak atc(i, j) = 0.88 is present, which is indicative of the 4 nearest neighbour s-bonds formed within the crystal. In the ice-Ih crystal structure, each atom forms 3 s-bonds and 1 e-bond between its 4 nearest neighbours[82, 83]. In ice-Ih at least a quarter of the particle correlations have an e-bond with a correlation peak atc(i, j) = 0.36, increasing the distribution overlap of ice-Ih with water to 37% compared to 11% in ice-Ic. Now, if solid-like correlations are defined whenc(i, j)≥0.5, only 75.3% of ice-Ih is identified as solid, compared previously to 98.4% in ice-Ic at the same cutoff. However, if using a threshold of c(i, j) ≥ 0.25, enclosing the second peak, 97% of ice-Ih is correctly identified, while 46.3% of the water is misidentified as solid. The effect of using these cutoffs, combined with the requirement of 4 neighbour particles, is shown in figure 5.4 for the number of solid-like bonds in ice-Ih and water.

Analysis of these cutoffs show, that if 3 or more solid-like bonds are used to identify a particle as solid, then 93.6% (99.4%) of ice-Ih and 0.36% (7%) of water atoms are identified as solid when usingc(i, j) ≥0.5 (c(i, j) ≥0.25). However, if 4 solid-like bonds are required, 8.2% (88%) of ice-Ih and 0.02% (1.3%) of water parti- cles are registered as being solid when usingc(i, j)≥0.5 (c(i, j)≥0.25). Therefore,

0 0.005 0.01 0.015 0.02 -1 -0.5 0 0.5 P ( c ( i, j )) c(i, j) 3.5 ˚A -1 -0.5 0 0.5 1 c(i, j) 5.2 ˚A Liquid Ice-0

Figure 5.5: Theq6 bond correlation distributions in ice-0 and water over the first coordination shell (left) and the second coordination shell (right).

the best criteria that can be used to correctly identify solid ice-Ih particles, is if a particle has 3 or more solid-like bonds; where a bond is only recognised if an atom has 4 nearest neighbours and the correlation with a neighbour isc(i, j) ≥0.5. On average over the 5 ns bulk simulations of ice-Ih and water, 93.6% of ice-Ih and 0.36% of water is identified as being solid ice-Ih.

Since only a maximum of 93.6% of ice-Ih can, on average, be correctly iden- tified as solid, this is not a suitable way to accurately measure the presence of ice-Ih in a system. In systems where an ice-Ih/water interface exists, the local position of the interface cannot be reliably known, compared to the accuracy of that of ice- Ic/water interface systems. q6 can therefore not be used to compare the interfacial free energies between ice-Ih and ice-Ic systems when in contact with water.

Ice-0 is also of interest to this research and hence an order parameter that can distinguish between ice-0 and water is of necessity. The correlation distributions forq6 in bulk simulations of ice-0 and water are shown in figure 5.5 over the first two coordination shells of ice-0. As can be seen, there is no way to bound the bond correlation distributions of ice-0 to distinguish between the solid and the water. The coordination shell cutoffs used result in correlation distribution overlaps of the ice-0 and water of 70% and 76% for the 3.5 ˚A and 5.2 ˚A cutoff, respectively. Hence, q6 can clearly not be used to distinguish between ice-0 and water.

Type s-bonds e-bonds Neighbours Ice-Ic 4 0 4 Ice-Ih 3 1 4 Interface 3 or 2 0 4 Water Any Any Any

Table 5.1: Atom type identification criteria using q3.