Modelling crystal nucleation and growth
David Quigley
Overview
• Why study nucleation?
• Strategies for atomistic nucleation
• How quantitative can we be and under what circumstances?
• Polytypes and stacking disorder in ice nuclei
• Nucleation via metastable phases - ice 0
Motivation
Sheets of aragonite tablets
https://www.youtube.com/watch?v=0JtBZGXd5zo
Mollusc shells
Nudelman et al
Faraday Discuss.
136, 9-25 (2007)
Columns of calcite
[Vedre et al Atherosclerosis, March 2009, Abela et al American Journal of Cardiology, April 2009]
Classical nucleation theory (CNT)
Energy benefit (bulk) :
Energy penalty (surface) :
Classical nucleation theory (CNT)
Nucleation rates connect directly to experiment
Diffusivity of
Density of liquid
Zeldovich factor, related to curvature at barrier top
For spherical nuclei
Strategies for computer simulation
Use classical nucleation theory (CNT)
1. Use biased sampling (MD/MC) method to measure
2. Fit
and
3. Simulate at
to resulting curve to calculate
and compute
and
from MSD of
Requires specialist codes, but structure of critical
nucleus extracted from the MD/MC
Useful when barrier is high and dynamics are slow
and diffusive
Reinhardt, A. & Doye, J. P. K. J. Chem. Phys., 2012, 136, 054501
Strategies for computer simulation
Use classical nucleation theory (CNT)
1. Run swarms of standard MD trajectories with seed nuclei of size
2. Compute mean drift velocity
3. Identify
4. Compute
5. Fit
and
as
at each
with zero mean drift
from gradient of
to CNT
at
using
Requires assumption of particular nucleus
structure, but only off-the-shelf MD codes
Useful when barrier is high, and dynamics are slow
and diffusive
Knott et al. J. A. C. S., 2012, 134, 19544
Strategies for computer simulation
Brute force molecular dynamics simulation
Useful when barrier is low and dynamics are fast
B. Vorselaars and DQ, in preparation 2015
Strategies for computer simulation
Use a transition path sampling method
1. Count the rate at which some small
is reached
2. Use specialist methods (FFS/TIS) to sample pathways crossing
3. Determine fraction of these pathways which reach solid phase
Various attempts to implement as wrapper to
standard MD packages, however how best to
compute
is an issue
Useful when barrier is high and dynamics are fast
enough to sample a large number of trajectories
Li et al PCCP., 2011, 13, 19807
Modelling water
Short-ranged two and three-body potential
Nucleation simulations with mW
•
Forward Flux Sampling from 220K to 240K
•
Critical nuclei contain ~ 50% cubic
•
Brute force molecular dynamics at 180K
•
Critical nuclei ~ 66% cubic
•
Umbrella sampling Monte Carlo at 220K
•
Critical nuclei “predominantly cubic”
Ice 1 has two polytypes
Hexagonal ice 1h
Cubic ice 1c
Layers stack as ABABABA…..
Layers stack as ABCABCABC…
Growing ice from supercooled water
Malkin et al. P.N.A.S. (2012)
273.15
200
180
120
Temperature (Kelvin)
Hexagonal Ice
Ice homogeneously nucleated at low temperatures is stacking disordered.
NOT cubic OR hexagonal but a mixture.
Stacking fault statistics in nuclei
• Stacking model
• E.g. L = 5
Δg = free energy difference per
molecule between cubic
and hexagonal ice.
Nl = molecules per layer.
Sequence
Energy
Entropy
ABABA
ABCBC
ABACA
0
Nl Δg
0
kln(2)
ABCAC
ABCBA
ABCAB
2Nl Δg
kln(2)
3Nl Δg
0
(Removing sequences identical under
cyclic permutations of labels or reversal)
of t his paper is con
ΔG and classical nucleation theory ∆ Ghc for t he molecu
employed in ice nuc
els molecules
t o correctly repro
• Expectation of % cubic
polytypes
in critical
FIG. 1: Averagepercentageof moleculesin
cubic
layers
within
FIG.
1:
Averagepercenta
show CNT
estimate
of critical
t o assess
t heir
suita
ice nuclei of size N• atPoints
temperature
T
according
to
the
stack
ice nuclei
of size N at −te1
nucleus
size
using:
st
udies.
verage percent age
of
molecules
in
cubic
layers
wit
hin
ing model described in the text with
∆ Gh c =
50J molin
ing
model
described
of size N at t emperat
ure T denote
according
o t he st nucleus
ackOpen circles
thetcritical
size predicted by clas
− 1 Open circles denote the c
described in t he t ext
wit h ∆ Gh c = 50J mol .
-1
sical
nucleation
theory using γsl = sical
33mJnucleation
m− 2 and ∆
H fus u
=
ΔG = 30 J mol
theory
es denot e t he crit ical nucleus
ed by clasMW
− 1 size predict
−
2
6.01kJ
mol
.
−
1
eat ion t heory using γsl = 33 mJ m and ∆ H
= 6.01kJ mol .
ol
−1
fus
.
The mW model w
t er molecule is repre
Units:
layer therefore increases
toward the
centre
the
nucleus
t eract
sofwit
hincrease
it s neigh
layer
therefore
efore increases t oward t he cent re of the nucleus.
t erms,
lathc
ter, favo
Experimental estimates
for
∆ GExperimental
=pertNhe
δg
the
-1 ≈ 0.0001
hc
AO
10
J
mol
eV
H
estim
2
ment al ΔG
estimat
es
for
∆ Ghc = N A δghc , t he
-1
ronment
s.
In(200K)
t hetwo
ab
= 50
J
mol
Gibbs free energy di↵erence per
mole
between
the
≈ 0.006
k
T
per
H
O
B
2
free
energy ice
di↵e
ee energy di↵erence per mole between t he two Gibbs
and
hexagonal
p
polytypes,
are
obtained
from
the
heat
evolved
∆
H
on
hc
s, are obt ained from t he heat evolved ∆ H hc on polytypes,
arelonsdalit
obtainede
mond and
cubic
iceto hexagonal
ice. Measurementso
ming cubic ice transforming
to hexagonal ice.
Measurements
of transforming
cubic
conduct ors.
Theicet
me
∆ H measurement
are varied.s span
Several
span the range
e varied. Several
themeasurements
range
Lattice switching Monte Carlo
• 200K, 64 molecule unit cells
20
P(m)
bh(m)
30
10
0
-400
-200
0
m
200
• 200-400 times faster than TIP4P
400
DG / J mol
-1
40
38
36
34
cubic
(diamond)
32
30
0
5
10
15
20
MC sweep / 10
25
6
30
hexagonal
(lonsdalite)
Finite size effects at 200K
40
-1
30
DG / J mol
20
10
0
-10
-20
0
0.02
0.04
1/N
ΔG = 0.4 to 2.6 J mol-1 in thermodynamic limit
c.f. published estimate 0 ± 30 J mol-1
(Moore and Molinero PCCP 2011)
0.06
Temperature dependence
• N = 384
P(m)
T = 180 K
T = 240 K
-2000
-1000
cubic
0
m
1000
2000
hexagonal
T (K)
ΔG / J mol-1
180
2.47(3)
200
4.15(2)
220
3.55(3)
240
5.77(4)
Comparison of stacking energetics
• mW has almost exactly zero penalty for
formation of cubic stacking faults
• Will nucleate stacking disordered ice at
all temperatures
ΔG = 30 J mol-1
Li, Donadio & Galli 2011
Reinhardt & Doye 2011
Moore & Molinero 2011
mW
DQ J. Chem. Phys. 2014, 141, 121101
ARTICLES
ICLES
PUBLISHED ONLINE: 18DOI:
MAY10.1038/
2014 | DOI:
10.1038/ NMAT3977
NATUREMATERIALS
NMAT3977
a
New metastable form of ice and its role in the
homogeneous crystallization of water
John Russo1†, Flavio Romano1,2† and Hajime Tanaka1*
Ice Ih
Ice Ic
Clathrate (HS-III)
Ice 0
P (kbar)
The
b homogeneous crystallizationcof water at low temperature is believed to occur through the direct nucleation of cubic (I c)
and hexagonal (I h) ices. Here, we provide evidence from molecular simulations that the nucleation of ice proceeds through
the formation of a new metastable phase,
which we name Ice 0. We find that Ice 0 is structurally similar to the supercooled
10
liquid, and that on growth it gradually converts into
a stacking of IceIce
Ic Icand
Ice 0
/ Ih I h. We suggest
Liquidthat this mechanism provides a
thermodynamic explanation for the location and pressure dependence of the homogeneous nucleation temperature, and that
Ice 0 controls the homogeneous nucleation of low-pressure ices, acting as a precursor to crystallization in accordance with
Ostwald’s step rule of phases. Our findings
show that metastable crystalline phases of water may play roles that have been
5
largely overlooked.
T
b DF + shift
b DF (n)
he crystallization of water into ice 0is the most ubiquitous arrangement of nearest neighbours. Thelocally tetrahedral structure
and important freezing transition on Earth, playing a central of water iscompatiblewith alargevariety of crystal structures, which
role in many scientific and technological fields1–4 such as despiteCS-II
being metastable with respect to the I h and I c crystals can
atmospheric ice cloud formation, geological processes such as frost be reflected in the structure of the supercooled liquid state and
150
heaving, and biological
ones such as freezing-induced cell-damage
have200
a significant effect250
on the nucleation pathway of ice. To assess
Ice 0 unit cell
(K)
and food preservation2,5–8. In the case of ice clouds, for example, it theTimportance
of metastable crystal phases we have undertaken a
isdbelieved that ice crystallites in the atmosphere can form
through systematic investigation of the thermodynamic stability of several
e 40
P = 3,1
40 bar; T = 206.2
K
homogeneous
nucleation
in
pre-existing
liquid
aerosols.
Water
also tetrahedral crystalline structures (F. R., J. R. and H. T., manuscript
50
P = 6,280 bar; T =to
196.5
K crystallization, becausepurewater
showsasurprisingability
resist
in preparation), borrowed from the known crystalline phases of
can be
easily
supercooled
far
below
its
melting
temperature;
water
group 14 elements and four-coordinated compounds. We have
30
40
dropletshaveindeed been found in deep convectivecloudsdown to found that one such structure, a recently proposed allotrope for
◦
− 37.5
C (ref. 6). At even lower temperatures, alimit of supercooling silicon, carbon and germanium called T12 (ref. 24), is also a
30
is reached, called the homogeneous nucleation line, below20which mechanically stable phase for two of the most popular models of
water20undergoes rapid crystallization. Why liquid water has such a water, mW (monoatomic water) and TIP4P/2005. A representation
P = 0 bar
large temperature range of metastability, and what determines the of the crystal structure is shown in Fig. 1b, where the oxygen
P = 3,140 bar
10
P = 0 bar; T = 215.2 K
slopeof
thehomogeneous-nucleation lineasa function of pressure, atoms occupy the positions of the T12 structure and the hydrogens
10
P = 3,140 bar; T = 194.7 K
P = 6,280 bar
arestill unanswered questions.
form a disordered
network of hydrogen bonds according to the
P = 6,280 bar; T = 171.0 K
Crystallization
of
deeply
supercooled
water
has
been
studied
Bernal–Fowler
rules for water 1. The unit cell is tetragonal and
0
0
9,100
60 molecules.
80
100 we report a crystallization
0
20by freezing
40
60
80
100
experimentally
water
inside
nanopores
, by 20
contains4012 water
Here,
decompression or warming ofn high-pressure ices11,12, by solutions scenario for nwater centred around this metastable form of ice
of electrolytes13, or inside water droplets emulsified in an oil that we call Ice 0 (we provide an in-depth characterization of
14–16 ice phase and its role in the mW phase diagram. a, Crystal phases relevant to crystallization in order (from left to right) of
new
metastable
matrix
. Diffraction patterns from these experiments showed its structural and crystallographic properties in Supplementary
chemical
potential
in the supercooled
region at ambient
b, Representation
the tetragonal
cellthat
of Ice
0. 0
Oxygen
atoms (red
peaksthat
deviatefrom
what isexpected
from pressure.
thepurecubic
(I c) or of
Section
I). Weunit
show
Ice
has several
structural features close
ccupy
the positions
the
T12 crystal
hydrogen
atoms areand
placed randomly
to the
Bernal–Fowler
rules.
Hydrogen
hexagonal
(I h )ofice
phases,
andstructure
revealedwhereas
a stacking
of hexagonal
to thoseaccording
of liquid
water,
and argue
that
it is this ‘friendliness’
17
shown
as yellow
lines to
between
a pair of oxygen
atoms within
unit cell boxwith
(greenthe
lines).
c, P–Tthat
phasemakes
diagramthis
of mW
water.
cubic
layersdashed
. Owing
the experimental
difficulties
inthe
accessing
liquid
ice
polymorph important in
s lines
the liquid
and di erent which
crystal structures:
Ice Iinitial
Ice 0 (red) and clathrate
CS-II (green).
h / Ic (blue),
theindicate
short coexistence
time and between
length scales
of phase
ice nucleation,
is an the
liquid-to-crystal
transformation
process. We also show
Is ice 0 real?
Slater, B. & DQ
Crystal nucleation: Zeroing in on ice
Nat. Mater., 2014, 13, 670-671
DQ; Alfè, D. & Slater, B.
J. Chem. Phys., 2014, 141, 161102
Summary
•
Quantitative (if inaccurate) calculation of nucleation rates from molecular simulation is
becoming possible for ‘real’ systems..
•
.. provided (very) cheap models are available.
•
Unlikely to be be predictive for some time, but trends should be accessible.
•
Quantifying sources of error likely to be very challenging.
•
Existing simulations do open up new questions!
Acknowledgements
Ice 0 stability
Ben Slater (UCL)
Dario Alfé (UCL)