1. No category

Why are snowflakes hexagonal?
E. A. Engel, B. Monserrat, R. J. Needs
TCM Group, Department of Physics, University of Cambridge
July 30, 2014
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Overview
1
Ice Ih and Ic and proton disorder.
Introduction.
2
Results.
Results for protonated ice.
3
Sanity checks.
Vibrational pressure.
Beyond principal axes approximation effects.
4
Origin of differences in anharmonicity.
Dominant contributions to anharmonicity.
5
Conclusions and next steps.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Introduction.
Differences in stacking of hexagonal layers.
❛ ✁
❜ ✁
❛ ✁✁
Figure : Hexagonal ice, Ih (blue).
ABAB stacking of bilayers.
a-I and a-II show chair and boat
form hexamers, respectively, the two
basic building blocks of Ih.
E. A. Engel, B. Monserrat, R. J. Needs
Figure : Cubic ice, Ic (red).
ABC stacking of bilayers.
b-I shows a chair form hexamer.
Cubic ice does not contain boat
form hexamers.
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Introduction.
Proton disorder in a nutshell.
Bernal-Fowler ice rules.
1
2
Each oxygen is covalently bonded to two hydrogen atoms.
Each oxygen accepts and donates two hydrogen bonds
from/two other oxygens.
Defect-free ice consists of tetrahedrally coordinated water
molecules bound in a hydrogen bond network.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Introduction.
Proton disorder in a nutshell.
Bernal-Fowler ice rules.
Pauling’s residual configurational entropy.
Paulings residual configurational entropy [1] has been
confirmed experimentally by measuring the entropy differences
between pure and KOH-doped ice [2, 3]. The configurational
free energies of bulk Ih and Ic are almost identical, since they
are effectively determined by the tetrahedral coordination of
the molecules [4].
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Introduction.
Proton disorder in a nutshell.
Bernal-Fowler ice rules.
Pauling’s residual configurational entropy.
Choice of polytypes for this study.
16 symmetry-unique eight-molecule Ih configurations [5] and
11 symmetry-unique proton-ordered Ic configurations [6].
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Energetics.
Ganh
Ghar
Ghar
15
∆Ih→Ic
min Ganh
10
I41md
0
0
Ic
5
Cmc21
5
Pc
Energy [meV/H2O]
Ganh
Ganh(T) - GXIc
anh(0K) [meV/H2O]
10
20
10
15
Ih
20
25
# of proton-ordering
Figure : Harmonic free energies,
Ghar , (empty squares) and total free
energies, Ganh , (filled squares),
measured with respect to
XIh
Ghar
(Cmc21 ).
30
Ih configurations
Ic configurations
Ih P63cm
Ic P43
0
-10
-20
0
-1
-2
-30
0 10 20 30 40 50
-40
0
50
100
150
T [K]
200
250
Figure : Anharmonic vibrations
stabilise Ih with respect to Ic across
a wide temperature range.
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Vibrational pressure.
Beyond principal axes approximation effects.
Vibrational pressure - Results.
Vibrational pressure of ∼ 0.45 ± 0.05 GPa.
Zero temperature expansion of ∼ 4% which agrees well with
other ab initio DFT and path-integral MD studies [7].
Expanded volumes including vibrations agree with experiment
to within ∼ 1%.
Vibrational frequencies and anharmonicity do not change
significantly upon evaluation at the expanded volume.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Vibrational pressure.
Beyond principal axes approximation effects.
Coupling of vibrational modes.
Next level of approximation: coupling of vibrational modes.
Pairwise coupling of vibrational modes already requires
mapping of 2D Born-Oppenheimer surfaces and scales as N 5 .
Calculations for the primitive unit cells of ice Ih and Ic
indicate that including pairwise coupling of vibrational modes
leads to a small increase in the differences in anharmonicity
between ice Ih and Ic.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Dominant contributions to anharmonicity.
Cumulative ∆Ganh(0K) [meV/H2O]
Origin of differences in anharmonicity - I.
16
Results averaged over
proton-orderings.
Ih configurations
Ic configurations
Ih P63cm
Ic P43
14
12
10
Phase
8
6
Ih
Ic
4
2
uanh
[˚
A]
0.225
0.220
uhar
[˚
A]
0.227
0.225
∆u
[˚
A]
-0.002
-0.005
0
0
2
4
6
13
Phonon frequency [meV]
15
Figure : Cumulative anharmonic
energies as a function of frequency.
E. A. Engel, B. Monserrat, R. J. Needs
Table : RMS displacements of the
protons at the harmonic, uhar , and
anharmonic level, uanh , and the
difference due to anharmonicity, ∆u.
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Dominant contributions to anharmonicity.
2.0
0.20
1.5
0.15
1.0
0.10
0.5
0.05
0.0
0.00
-0.5
-1.0
gIh
anh
gIc
anh
1
-0.05
∆ganh
∆ghar
2
-0.10
3
4
r [Ang]
5
Figure : H-H RDFs for Ih (blue) and
Ic (red).
E. A. Engel, B. Monserrat, R. J. Needs
∆gH-H(r)
gH-H(r)
Origin of differences in anharmonicity - II.
For r < 3 ˚
A
At the harmonic level (green),
the difference between Ih and Ic,
Ih − g Ic , is minimal.
∆ghar ≡ ghar
har
At the anharmonic level (black),
the difference between Ih and Ic,
Ih − g Ic , is
∆ganh ≡ ganh
anh
non-negligible.
˚:
For r > 3 A
The differences in the static
structures of Ih and Ic become
dominant.
Why are snowflakes hexagonal?
Structural origin of differences in anharmonicity.
☞☛✌
■✕
❈✓✝✔✠ ✟✆✠✡
■✖
☞☛✵
✙
✚✚
✙✙✙
✎✎
✵☛✘
✎❱
✵☛✗
✙✛
✎
✛
✒
✑
✭
❣
❍✏ ✵☛✹
✎✎✎
❍
✵☛✌
❱
✵☛✵
✵☛✍
❱✎✎✎
❱✎
✵☛✌
❇✆✝✞ ✟✆✠✡
✛✙
✛✙✙
❱✎✎
✵☛☞
✵☛✵
✍☛✵
✍☛✸
✹☛✵
r
✹☛✸
✸☛✵
✁✂✄☎
Figure : Anharmonic H-H RDF decomposed into contributions from
different bonding configurations of fourth-nearest neighbour pairs of
protons.
✛✙✙✙
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Conclusions and next steps.
Relevance of Ih and Ic:
climate modelling and the simulation of ice nucleation and
formation.
potential relevance in biological sciences in the context of
cryopreservation.
Importance of anharmonic vibrations:
in hydrogen bonded molecular crystals: likely to be crucial in
correctly describing the energy differences between very similar
such polymorphs, e.g., in pharmaceutical science.
in various other examples as in B. Monserrat’s talk.
at ice surfaces (basal and prism surfaces in Ih and basal in Ic).
around impurities or other defects.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Acknowledgements.
HECToR
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Bibliography I
[1] L. Pauling.
The structure and entropy of ice and of other crystals with some randomness of
atomic arrangement.
Journal of the American Chemical Society, 57:2680–2684, 1935.
[2] Y. Tajima, T. Matsuo, and H. Suga.
Phase transition in KOH-doped hexagonal ice.
Nature, 299:810–812, 1982.
[3] S. M. Jackson and R. W. Whitworth.
Thermally-stimulated depolarization studies of the Ice XI-Ice Ih phase transition.
Journal of Physical Chemistry B, 101:6177–6179, 1997.
[4] J. F. Nagle.
Lattice Statistics of Hydrogen Bonded Crystals. The Residual Entropy of Ice.
Journal of Mathematical Physics, 7:1484–1491, 1966.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Bibliography II
[5] K. Hirsch and L. Ojam¨
ae.
Quantum-chemical and force-field investigations of ice Ih: computation of
proton-ordered structures and prediction of their lattice energies.
Journal of Physical Chemistry B, 108:15856–15864, 2004.
[6] Z. Raza, D. Alf`
e, C. G. Salzmann, J. Klimeˇs, A. Michaelides, and B. Slater.
Proton ordering in cubic ice and hexagonal ice; a potential new ice phase – XIc.
Physical Chemistry Chemical Physics, 13:19788–19795, 2011.
[7] R. Ram´ırez, N. Neuerburg, M.-V. Fern´
andez-Serra, and C. P. Herrero.
Quasi-harmonic approximation of thermodynamic properties of ice Ih, II, and III.
Journal of Chemical Physics, 137:044502, 2012.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Convergence behaviour.
1
∆Gh
694
Ic
Ih
692
690
0
[meV/H2O]
688
1
-1
Ih
∆Ghar
-
4
16
64
16
64
Ic
∆Ghar
Ih
Ic
-3 ∆Ganh - ∆Ganh
∆Ganh
11
9
7
5
1
-4
4
# molecules
1
2
4
8
# multiples of unit cell
16
32
Figure : Convergence of the harmonic (top) and anharmonic
contributions (bottom) to the vibrational energy with supercell size.
E. A. Engel, B. Monserrat, R. J. Needs
Why are snowflakes hexagonal?
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
0.01
0.01
Ic LDA
Ic BLYP
Ic PBEsol
Ic PBE
Ih LDA
Ih BLYP
Ih PBEsol
Ih PBE
0.03
(∆Eanh 1
∆Eanh [meV/H2O]
0.04
0.01
LDA
BLYP
PBEsol
PBE
-10
-5
0
5
10
Mode amplitude [Ang]
Figure : Anharmonic components of
the BO energies,
∆Eanh ≡ Eanh − Ehar , for the
highest energy
vibrational
modes
in
E. A. Engel,
B. Monserrat,
R. J. Needs
2
Ic
2
0.16
|Ψanh|
0.12
0.08
0.04
0.00
0.00
-10
0.00
Ih
|Ψanh|
-5
0
5
10
Mode amplitude [Ang]
Figure : The differences in ∆Eanh
between the highest energy
vibrational modes in Ih and Ic for
different density functionals are
Why are snowflakes hexagonal?
Vibrational mode density
) [meV/H2O]
0.02
- ∆Eanh
0.01
0.05
Ih Cmc21
0.02
Ic I4 md
.
Ice Ih and Ic and proton disorder.
Results.
Sanity checks.
Origin of differences in anharmonicity.
Conclusions and next steps.
Energetics.
Ghar
20
Energy [meV/D2O]
Ghar
15
∆Ih→Ic
min Ganh
10
0
Ic
5
15
0
Figure : Protonated ice.
25
30
2O
GD
anh
2O
GD
har
2O
GD
har
5
Ih
10
15
20
# of proton-ordering
2O
GD
anh
10
I41md
I41md
0
Cmc21
5
Pc
Energy [meV/H2O]
Ganh
0
Cmc21
Ganh
20
Ic
5
10
15
Ih
20
25
# of proton-ordering
Figure : Deuterated/heavy ice.
Ghar (empty squares) and Ganh (filled squares) measured with
XIh B.
E. A.G
Engel,
Monserrat,).R. J. Needs
Why are snowflakes hexagonal?
respect to
(Cmc2
30
Vibrational pressure - Self-consistency problem.
Initial static equilibrium structure.
Structure relaxed at ambient pressure.
Harmonic approximation to the BO
surface.
Equilibrium structure at expanded
volume due to lattice vibrations.
Evaluate phonon modes and
frequencies.
Rerelax equilibrium structure taking
into account phonon pressure.
Anharmonic calculation.
Check for self-consistency.
Evaluate anharmonicity and resultant
phonon pressure.
If phonon pressure matches ambient
pressure, then exit self-consistency loop.
Final structure at expanded volume due
to lattice vibrations.
The phonon pressure matches the
external ambient pressure.