q3 Order Parameter

The failings ofq6 motivate the use of another order parameter that is able to work in all the structures of ice investigated and can also distinguish between the ice-I polytypes. Usingq3, it is possible to identify between ice-Ic, ice-Ih, water and even interfacial atoms.

q3 is implemented through a variation of the CHILL algorithm[83]; known as CHILL+[82]. The CHILL algorithm is able to identify the crystal types of interest, however CHILL+ improves upon the percentage of ice-Ih detected. CHILL+ makes use of the different numbers of s-bonds and e-bonds formed in ice-Ic and ice-Ih[82]. While the difference in the bond correlation distributions forq6, between ice-Ic and ice-Ih, clearly shows the different bond types, the signal from the single e-bond in ice-Ih is not intense enough to be resolved against the bond correlation distribution of water. Comparatively, these distributions are much clearer in the case ofq3, as shown for bulk ice-Ic and ice-Ih in figure 5.6, where the correlation distributions are measured over particles within 3.5 ˚A.

Following the CHILL+ algorithm, s-bonds have a correlation valuec(i, j)≤ −0.8, where e-bonds have a correlation value −0.35 ≤c(i, j) ≤0.25. Particles are defined as being interfacial (between the solid phases and water), if they have 4 nearest neighbour atoms within 3.5 ˚A, but only form 2 or 3 s-bonds with any of its neighbours[82]. Water may form any number of neighbour bonds of any type and may have any number of neighbour atoms. The work carried out by Nguyen and Molinero[82] also indicates that q3 can identify clathrates; however this is not of importance to this research and so is not discussed here. The criteria to identify an atom usingq3 is summarised in table 5.1.

Figure 5.6 shows the bond correlation distributions of ice-Ic and ice-Ih when using q3, clearly indicating the presence of the respective s-bonds and e-bonds in both polytypes according to the CHILL+ algorithm. In ice-Ic, when using the full identification criteria of CHILL+, the bulk simulations identified on average, over 5 ns, 99.4% of the bulk ice-Ic and 0.006% of the bulk water as being ice-Ic particles; a significant improvement overq6. There is still a significant degree of overlap between

0 0.05 0.1 0.15 0.2 -1 -0.5 0 0.5 1 P ( c ( i, j )) c(i, j) -1 -0.5 0 0.5 1 c(i, j) Liquid

Ice-Ic LiquidIce-Ih

Figure 5.6: Theq3 bond correlation distribution in ice-Ic and water (left) and ice-Ih and water (right), over an order parameter cutoff of 3.5 ˚A.

the water and ice-Ih correlation distribution functions of 23.8%, which compares to 37% when usingq6. However, this difference significantly improves the probability of identifying ice-Ih e-bonds compared to using q6. The success of the algorithm is demonstrated in figure 5.7, where the frequency of s-bonds and e-bonds in both polytypes is clear, over a well defined narrow range. Indeed, when using the complete criteria of CHILL+ to identify ice-Ih, it is found on average over 5 ns from the bulk simulations that 98.8% of ice-Ih atoms and 0.03% of water atoms are identified as ice-Ih; again, a significant improvement overq6. It should be noted here, that 0.03% misidentification of water as ice-Ih compares to the previously calculated value of 0.006% misidentification of water as ice-Ic. This indicates there is a measurable bias for water to be incorrectly identified as ice-Ih over ice-Ic by a factor of 5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 P (B on d s) Bonds 0 1 2 3 4 5 6 7 8 9 Bonds

Figure 5.7: q3 bond types in liquid (black lines) and solid ice-I (red lines). Solid lines are s-bonds, dotted lines are e-bonds. Left shows the distributions in ice-Ic and water, right shows the distributions in ice-Ih and water.

0 0.01 0.02 0.03 0.04 0.05 -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 5.8: Theq3 bond correlation distributions in ice-0 and water over the first coordination shell (left) and second coordination shell (right).

While q3 can distinguish between ice-Ic and ice-Ih, it is not sufficient to identify between ice-0 and water over either the first or second coordination shells, as shown in figure 5.8. There is a greater propensity to identify ice-0 when using a cutoff of 3.5 ˚A as opposed to 5.2 ˚A, but maintains ice-0/water bond correlation overlaps of 55% and 82% respectively. The distribution in the former case does show different bond peaks similar to the existence of s-bonds and e-bonds in ice-Ih. It was investigated whether q3 could distinguish between different bond-types in ice-0, when using a spatial cutoff of 3.5 ˚A, and use such bond-type distributions to distinguish between ice-0 and water. However, it was not possible to clearly resolve the secondary peak in ice-0, from that of water. This is unsurprising givenq6 could not reliably distinguish between ice-Ih and water with an overlap of 37%, and hence the 55% overlap in this case is discouragingly high. Another order parameter is therefore necessary to clearly resolve ice-0 against water.

A further point to mention regardingq3 concerns the choice of the value of the order parameter in solids and liquids. Unlikeq6 where the intrinsic atom order parameter value is the number of solid-like bonds an atom possesses, q3 instead identifies the type of atom directly as a combination of the number of neighbours an atom has and the proportion of different bond types. In order to obtain an order parameter profile, as discussed in chapter 4, each atom must be assigned a value based on its phase. The value assigned is an arbitrary choice, rather than a calculated quantity. It could be suggested that the choice of order parameter in different phases affects the measurement of the local position of the interface. This is demonstrated not to be the case in figure 5.9.

Figure 5.9 shows that the inflexion point, measuring the position of the interface, for the same arrangement of atoms when using ranges of 10 and 4, is

h0 = 202.588 ˚A and h0 = 202.799 ˚A respectively. Clearly there is no significant

deviation in the position of the interface measured by the two different sets of order parameter values. In the case of “Range 10”, solid-like particles have been assigned a value of 10, liquid-like particles a value of 0 and interfacial particles a value of 5. The “Range 4” choice of values instead uses the number of connected bonds, where the order parameter per atom is the sum of the number of e-bonds and s-bonds; with the exception that liquid-like particles are assigned a value of 0, to keep with the convention that the order parameter is high in the solid region and low in the liquid region. This defined value in the liquid, smooths out the profile liquid region, which would otherwise have a varying number of connections per atom.

Given that there is no difference to the measurement of the inflexion point between the choice of values assigned to atoms when usingq3, the decision was made

0 2 4 6 8 10 12 0 50 100 150 200 250 300 350 400 ω ( x ) x(˚A) Range 10 Range 4

Figure 5.9: Theq3 order parameter profile in ice-Ic (111)[1¯10] simulation using dif- ferent assigned order parameter values to identify solid, interface and liquid particles. The vertical line is the average position of the two inflexion points (x= 202.694 ˚A). to use the “Range 10” set of values to identify all local interface positions in ice-I. This choice was made because a larger range in the difference between atom phases allows for a clearer distinction between the solid and liquid regions when having to set thresholds in the interface identification program.