the many faces of heterogeneous ice...
TRANSCRIPT
Gabriele C. Sosso Trieste, Oct. 2016
The Many Faces of Heterogeneous Ice Nucleation
Insights from Molecular Dynamics Simulations
• Heterogeneous Ice Nucleation • Why do we care? • Open Issues: Hydrophobicity vs surface morphology
• The tools of the trade • Coarse grained water on ideal crystalline surfaces • Brute force molecular dynamics simulations:
getting nucleation rates
• Results • Simple models, complex behaviours • Boosting nucleation rates: three microscopic motivations
• What’s next • Current limitations & future perspectives
• People & acknowledgments
Outline
!Atmospheric science: Clouds formation and dynamics (climate change)
Heterogeneous Ice Nucleation
determined by DSC (results not shown). This result was inaccordance with previous studies [5,25]. Our study thusindicates that in addition to the detrimental e!ect of theextraliposomal ice, the abrupt increase in leakage foundin Fig. 1a could also be due to the phase transition ofEPC. In contrast to EPC, DPPC undergoes the phase tran-sition at 41 !C [6,11,12,32]. Therefore, extraliposomal iceformation was believed to be solely responsible for theleakage of DPPC LUV that started at !10 !C and the sub-sequent rapid increase of leakage between !10 !C and!25 !C (Fig. 1b). The detrimental e!ect of the extracellularice has been previously reported [16,19,50,54]. Such delete-rious e!ect of the extraliposomal ice was apparent for theDPPC LUV but not for the EPC LUV. This suggestionwas made as the temperature of the extraliposomal ice for-mation and the EPC phase transition both fall within thetemperature range in which the abrupt increase in the leak-age of EPC LUV was observed.
As water forms ice, the phase volume of the unfrozenmatrix decreases. The reduction of the phase volume ofthe unfrozen matrix freeze-concentrated the EPC LUVand DPPC LUV to a closer proximity that eventually ledto the aggregation of vesicles (Fig. 3). The e!ect offreeze-concentration and aggregation of the EPC LUV,however, did not lead to a continuous leakage of theEPC LUV as temperatures decreased at 10 !C/min(Fig. 1a).
On the other hand, the freeze-concentrated, compressedand aggregated DPPC LUV in the unfrozen matrix (Fig. 3cand d) attributed to the gradual increase in leakage ofDPPC LUV as temperatures were lowered to !40 !C at10 !C/min (Fig. 1b). It should be noted that the reductionof the phase volume of the unfrozen matrix also caused anincrease of salt concentration in the unfrozen matrix[29,34,40]. The increase of salt concentration and the sub-sequent increase in osmotic pressure in the unfrozen matrixhad been ascribed to promote cellular dehydration, whichwas accounted for the slow freeze-injury involving freezingrates of below 1 !/min [34,41].
In the current study, extraliposomal ice formation of theDPPC LUV was observed at !9 !C and the intraliposomalfreezing was observed at !43 !C when cooled at a relatively‘fast’ cooling rate of 10 !C/min (Fig. 9a). Consequently, theDPPC LUV could have been exposed to osmotic stressacross the bilayer upon the extraliposomal ice formationand before the intraliposomal ice formation. The exposureof DPPC LUV to osmotic stress, however, was short sincethe cooling rate was relatively ‘fast’. The innocuous e!ectof the short exposure of DPPC to osmotic stress was evi-dent in Fig. 3c and d in which the DPPC LUV appearedto be solid spherical at !15 !C and !40 !C. Thus, lesionand leakage of the DPPC LUV was mainly caused by thefreeze-concentration and compression e!ects, which werea result of the phase volume reduction of the unfrozen
Fig. 3. Scanning electron micrographs of EPC LUV and DPPC LUV that were cooled to !15 !C, !40 !C and !55 !C at 10 !C/min. (a) EPC LUV at!15 !C; arrows show the squashed and flattened EPC LUV, (b) EPC LUV at !55 !C, (c) DPPC LUV at !15 !C, (d) DPPC LUV at !40 !C; arrows showthe freeze-concentrated DPPC LUV at the unfrozen channels. The inset in the micrographs shows a magnified unfrozen channel.
L.F. Siow et al. / Cryobiology 55 (2007) 210–221 215
Why do we care?
Homogeneous vs Heterogeneous nucleation
Ice formation on top of lipid bilayers
Cryobiology, 55, 210 (2007)
Figure 1: Illustration of ice formation in clouds (left), together with the typical simulation setup (right)we will employ to unravel the mechanism of heterogeneous ice nucleation on the clay mineral kaolinite.The simulation box on the right side depicts an ice nucleus (blue/light blue balls and sticks) within afilm of liquid water (red and white balls stands for oxygen and hydrogen atoms respectively) on top ofkaolinite layer (light blue and yellow polyhedra represent octahedral hydroxide and tetrahedral silicatesheets respectively, see text.)
Figure 2: a) Side view of a single kaolinite layer, constituted by a tetrahedral silicate sheet (yellowpolyhedra) and an octahedral hydroxide sheet (light blue polyehdra). Oxygen and hydrogen atoms aredepicted in red and white respectively. b) Top view of the (001) hydroxylated surface of kaolinite. Siliconand aluminium atoms are depicted in yellow and light-blue respectively. Only oxygen and hydrogen atomsbelonging to the surface hydroxile groups are shown. c) The (001) hydroxylated surface of kaolinite isamphoteric. Pink (dotted) lines highlight two hydrogen bonds where two hydroxile groups on the surfaceact as donor or as acceptor.
Scientific goals and objectives
Being able to investigate the heterogeneous ice nucleation mechanism on kaolinite represents a break-through in the simulation of rare events. While several works have succeeded in describing heterogeneousice nucleation on model systems, in here we aim to obtain an unprecedented insight into a realistic sys-tem of utmost relevance within the atmospheric science community as well as within the field of crystal
3
Cryobiology: Intracellular freezing
(cryotherapy and cryopreservation)
Heterogeneous Ice Nucleation
6546 Chem. Soc. Rev., 2012, 41, 6519–6554 This journal is c The Royal Society of Chemistry 2012
sensitive to smaller active site densities than either of theaerosol based studies. Similarly, in their single particle aerosolsystem Hoyle et al.181 worked with a much smaller number ofdroplets than used in the cloud chamber experiments,136 andcorrespondingly Hoyle et al. observed the lowest freezingtemperatures. When the nucleation events are normalised toimmersed surface area (i.e. active site density) the data from allthree studies falls on a curve (r2 = 0.96) spanning more thanseven orders of magnitude. This consistency is despite thevariability in experimental technique, implying that there is acharacteristic ice nucleating ability of volcanic ash. However,we have only been able to characterise ash from two sources interms of active site density. Quantitative measurements frommore sources and compositions are needed in order to assessthe hypothesis of Durant et al.320 that all volcanic ashes havesimilar ice nucleating ability.
7 Summary and discussion of ice nucleatione!ciency of heterogeneous ice nuclei
In order to make a meaningful comparison of the ice nucleatinge!ciency of di"erent materials we have estimated the cumula-tive ice active site density (ns) for mineral dust, volcanic ash,soot, fungal spores, pollen grains and bacteria. Our estimates,based on the literature data discussed in the preceding sections,are presented in Fig. 18. In order to estimate ns values we havehad to make assumptions about surface areas of materials suchas pollen and bacteria, due to which our estimates are prone toerrors on the order of a factor of 10. However, the ns valuespresented here extend over nine orders of magnitude and henceeven with these large uncertainties a comparison is still valid.Although there are caveats in the interpretation (see below),Fig. 18 provides a benchmark with which to compare various
materials and also serves to highlight potential future researchdirections.As discussed in Section 4.2, the singular approximation used
here treats the time dependence of nucleation as a secondorder e"ect. We justify this approach on the basis that itprovides a convenient first order approximation of the e!ciencywith which a material nucleates ice. However, it should beborne in mind that time dependence of nucleation may beimportant in some cloud types,29,105,120,128,327 and future studiesexamining the IN activities of substances should aim to quanti-fy the importance of the stochastic nature of ice nucleation.A further important point regarding the calculation of ns
values can be made in relation to the normalization by surfacearea. Surface area is quantified in di"erent ways in di"erentexperiments. For example, some experiments use gas adsorp-tion surface areas (which are quoted as specific surface areas,surface area per unit mass) and provide a total surface area ofall the grains and other small scale features.105,120 Gas adsorp-tion measurements for kaolinite samples are in excellentagreement with surface areas determined from atomic forcemicroscopy measurements,105,183 which suggests that this is anaccurate way of determining surface area. This approach iswell suited to experiments in which a bulk suspension of solidin water is generated and subsequently finely divided. Anotherapproach is to determine the surface area using the size ofaerosolised particulates given by aerosol instrumentationsuch as the mobility diameter (see for example ref. 131, 132and 184). Basing surface area on mobility size measurements isclearly a sensible approximation, but it should be borne inmind that dust particles tend to be agglomerates of manysmaller particles.120,184 Hence, this assumption may lead to asubstantial under-estimate of particle surface area. Broadleyet al.120,184 estimated that a 500 nm diameter particle of
Fig. 18 Summary plot of ns values based on literature data. The surface area of a bacteria is assumed to be 5 mm2.239 For birch pollen, a surface
area of 1520 mm2 is assumed (d = 22 mm). Note that the data of Murray et al.105 and Broadley et al.120 were determined using a gas adsorption
surface area which results in a shift to smaller ns values compared to the other mineral dust results where a spherical approximation was made
(see discussion in Section 7).
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What makes a material a good Ice Nucleating Agent (INA)?
1. Long-standing and 2. Complex question
But we lack insights at the molecular level
Experiments can now quantify the activity of many different INA:
Soot Clay Minerals
!!!!!!
Bacteria !
PollenChem. Soc. Rev. 41, 6519 (2012)
Heterogeneous Ice NucleationAtomistic simulations could help!
Simulating heterogenous ice nucleation is still a formidable task: • It is difficult to describe water at interfaces properly • Nucleation is a rare event (enhanced sampling techniques needed) • Even the simplest scenarios are debated
Would a generic crystalline surface promote nucleation? Most importantly: why and how?
Hydrophobicity Surface morphologyJ. Phys. Chem. A 118, 7330 (2014)
J. Am. Chem. Soc. 136, 3156 (2014)
J. Chem. Phys. 142, 184705 (2015)
J. Appl. Phys. 18, 593 (1947)
Surface Science 601, 5378 (2007)
J. Chem. Phys. 142, 184705 (2015)
In here: Systematic investigation of the interplay between the two
The tools of the trade
• (111), (100), (110) and (211) surfaces (surface morphology) • Different lattice parameters afcc [3.52 - 4.66 Å] (surface morphology) • Different water-surface interaction (LJ potential) strength Eads [0.2-12 kcal/mol]
(hydrophobicity)
Coarse grained (mW) waterJ. Phys. Chem. B 113, 4008 (2009)
on top ofIdeal FCC crystal (LJ particles, frozen)
• Computationally fast • Fast water dynamics even at strong supercooling
FIG. 1. a) Example of a simulation box used in a heterogeneous ice nucleation run. The coarse-
grained water molecules are depicted as blue spheres while surface atoms are gray. The average box
dimensions were 60× 60× 70 A. b) Top and side view of the four crystalline surfaces considered.
Atoms are colored according to their z-coordinate. Red boxes highlight the symmetry of the surface
unit cells.
that the density is converged to the bulk homogeneous value at ∼ 12 A above the interface.
We note that in general in this study we do not aim to mimic a specific system but to
extract instead generic insight and trends from idealized model substrates. To this end we
have taken into account four different crystallographic planes of a generic fcc crystal, namely
the (111), (100), (110) and the (211) surfaces, which exhibit significant differences in terms
of atomic roughness and the symmetry of the outer crystalline layer (see Figure 1b). For
each of the above mentioned surfaces, we have built a dataset of ten different slabs varying
the fcc lattice parameter afcc from 3.52 to 4.66 A61. This range encompasses the majority
of fcc metals. The interaction of the water with the substrate is given by a truncated
Lennard-Jones potential:
U(r) =
4�
��σr
�12−�
σr
�6�
r < rc
0 r ≥ rc
(2)
where r is the distance between a water oxygen and a surface atom. The cutoff distance
was set to rc = 7.53 A.
To measure the interaction strength of water with the surface the adsorption energy Eads
of a single water molecule was computed. In order to vary this quantity � and σ were
changed accordingly. Eads was computed by minimizing the potential energy of a single
water molecule on top of the surface. In this manner well defined adsorption energies can
5
Back to the basics!
Simple system General trends
The tools of the tradeThe method:
Brute force molecular dynamics simulations at 205 K
Code: LAMMPS. NVT ensemble (Nose-Hoover thermostat). Timestep = 2 fs Geometry: Slab
For each combination of afcc and Eads: 15 nucleation events (1 to 500 ns needed)
! "#$%&'
()*#+,
- .+/
$%&0/,1 21 311 321 411
211
3111
3211
4111
421151678
51679
5167:
!"#$
-.+/
$0
FIG. 2. An illustration of how the nucleation induction time tn is established by monitoring the
change in the potential energy Epot in blue. The green data shows the number of water molecules
within the biggest ice-like cluster Ncls63
and that the jump in Ncls coincides with nucleation. The
data refers to the (111) surface for Eads=1.04 kcal/mol and afcc = 4.16 A.
be determined for the (111), (100) and (110) surfaces since only one adsorption site is found
by the minimization algorithm. However, for the (211) geometry multiple adsorption sites
with considerable energy differences were found62. For this reason we have chosen to assign
every (afcc, Eads) combination for the (211) orientation the same (�, σ) pair as for the (111)
surface. This is also motivated by the (111) terrace exhibited by the (211) surface. The final
adsorption energy for the (211) geometry as reported in Figure 5 is the arithmetic average
of the different adsorption energies found on this particular surface. The averaged results
deviate by ca. 5 % from those for the (111) surface, e.g. the highest Eads on (111) is around
12.76 kcal/mol while the average value for the (211) surface with the same (�, σ) parameters
is 13.18 kcal/mol.
B. Obtaining Nucleation Rates
Heterogeneous ice nucleation events have been simulated by means of brute force molecu-
lar dynamics (MD) simulations, employing the LAMMPS simulation package64. We follow a
similar protocol to the one of Cox et al.37. A time step of 10 fs has been used with periodic
boundary conditions in the xy-plane while sampling the NVT canonical ensemble with a
chain of 10 Nose-Hoover thermostats65,66 with a relaxation time of 0.5 ps. The positions
of the surface atoms were fixed throughout the simulations. Every point of the (afcc, Eads)
6
Induction time
The tools of the tradeNucleation rates J from survival probability
grid corresponds to a specific configuration which has been equilibrated at 290 K for 170 ns.
Then 15 uncorrelated (separated by at least 10 ns) snapshots have been selected from the
resulting trajectories as starting points for production runs, after having instantaneously
quenched the system from 290 to 205 K. Nucleation simulations were terminated 10 ns after
a significant drop of the potential energy (> 0.53 kcal/mol per water) was registered or if
the simulation time exceeded 500 ns. In total, we report results from 6000 nucleation and
400 equilibration simulations.
The induction time tn of a nucleation event has been detected by monitoring the drop
in the potential energy Epot of the system associated with the formation of a critical ice
nucleus, as shown in Figure 2. We have calculated tn by fitting the potential energy to:
Epot(t) = a+b
1 + exp[c(t− tn)](3)
where tn, a, b and c are fitting parameters. Due to the smoothness of the potential energy
surface characterizing the mW model, crystal growth at the supercooling considered here
(∼70 K) is extremely fast, resulting in a very sharp potential energy drop that takes place
within - at most - 1 ns for all values of Eads and afcc considered. Thus, the resulting value of
tn does not depend on a specific functional form. We thereby estimate the error associated
with the calculation of tn as ± 1 ns. We also verified that no substantial discrepancy with
respect to tn can be observed by using other order parameters like e.g. the number Ncls63 of
mW molecules in the biggest ice-like cluster, as reported in Figure 2.
From the tn dataset, a survival probability Pliq(t) with respect to the metastable liquid
can be built, which was then fit by a stretched/compressed exponential function:
Pliq(t) = exp[−(J · t)γ] (4)
where J is the nucleation rate and γ is a parameter accounting for possible non-exponential
kinetics. In fact, having quenched each starting configuration instantaneously from 290 to
205 K, we have to take into account that the relaxation of the system, when nucleation
is comparably fast, could lead to a time dependent nucleation rate characterized by a non
exponential behavior67. Examples of Pliq(t) for two very different nucleation events can be
found in the supporting information (SI, Figure S2).
It is difficult to quantify the error in the nucleation rates from the fitting previously
described. Instead, we have employed the Jackknife resampling technique68 to quantify
7
B. Compressed Exponential Fit
The simulation protocol involves an instantaneous quench from the equilibration tem-
perature to the one at which we study nucleation. Because the system has to relax into
quasi-equilibrium first, the nucleation rate increases with time, resulting in a deviation from
perfect exponential characteristics. The effect of this non-exponential behavior can be ap-
preciated in Figure S2, where we show the tn datasets and the resulting Pliq(t) for two
dissimilar nucleation scenarios observed on the (110) surface as a function of the strength
of the water-surface interaction. In this case of a) the nucleation typically proceeds on a
timescale ranging from 1 to 100 ns, resulting in well behaved exponential decay (γ ∼1 in
equation 4 for the survival probability). On the other hand, the fitting of the data shown in
Figure S2b gave γ � 1, which in turn implies a nucleation rate that increases with time, as
the timescale for tn (0.1-1 ns) is indeed comparable with the relaxation time of the system.
This occurrence takes place mainly for those (Eads,afcc) values for which we observe the
basically instantaneous (10-1000 ps) formation of almost perfect ice-like overlayers on top
of the surface.
!"#$%
& '()*!+
,- .- /- 0--
-1,
-1.
-1/
-10
21-
-
&*!+
3+ 4+
565212,5
FIG. S2. Stretched exponential fitting results for two dissimilar nucleation events. Pliq(t) (red
circles) and fit after equation 4 (blue lines) for the (110) surface and afcc = 3.9 A. a) Eads =
11.63 kcal/mol and b) Eads = 5.3 kcal/mol.
3
Pliq(t) = 1− 1
Nsim
Nsim�
i=1
Θ(t− t(i)n )
ResultsSimple models, complex behaviours
FIG. 5. a) Heat maps representing the values of ice nucleation rates on top of the four different
surfaces considered, plotted as a function of the adsorption energy Eads and the lattice parameter
afcc. The lattice mismatch δ on (111) is indicated below the graph. The values of the nucleation
rate J are reported as log10(J/J0), where J0 refers to the homogeneous nucleation rate at the same
temperature. b) Sketches of the different regions (white areas) in the (Eads,afcc) space in which
we observe a significant enhancement of the nucleation rate. We label each region according to
the face of Ih nucleating and growing on top of the surface (basal, prismatic or (1120)), together
with an indication of what it is that enhances the nucleation. “temp”, “buck”, and “highE” refer
to the in-plane template of the first overlayer, the ice-like buckling of the contact layer, and the
nucleation for high adsorption energies on compact surfaces, as explained in section III B.
III. RESULTS
A. No Simple Trend for Nucleation Rates
The nucleation rates on the four surfaces considered are shown as bi-dimensional heat
maps as a function of the lattice constant and adsorption energy in Figure 5a. Regions in the
2D plots72 for which a strong enhancement of the nucleation rates is observed are sketched
in Figure 5b and snapshots of representative trajectories for all the classified regions can be
found in the SI (Figures S4 to S7). Before even considering any microscopic details of the
water structure or nucleation processes, several general observations about the data shown
10
(211) face
FIG. 5. a) Heat maps representing the values of ice nucleation rates on top of the four different
surfaces considered, plotted as a function of the adsorption energy Eads and the lattice parameter
afcc. The lattice mismatch δ on (111) is indicated below the graph. The values of the nucleation
rate J are reported as log10(J/J0), where J0 refers to the homogeneous nucleation rate at the same
temperature. b) Sketches of the different regions (white areas) in the (Eads,afcc) space in which
we observe a significant enhancement of the nucleation rate. We label each region according to
the face of Ih nucleating and growing on top of the surface (basal, prismatic or (1120)), together
with an indication of what it is that enhances the nucleation. “temp”, “buck”, and “highE” refer
to the in-plane template of the first overlayer, the ice-like buckling of the contact layer, and the
nucleation for high adsorption energies on compact surfaces, as explained in section III B.
III. RESULTS
A. No Simple Trend for Nucleation Rates
The nucleation rates on the four surfaces considered are shown as bi-dimensional heat
maps as a function of the lattice constant and adsorption energy in Figure 5a. Regions in the
2D plots72 for which a strong enhancement of the nucleation rates is observed are sketched
in Figure 5b and snapshots of representative trajectories for all the classified regions can be
found in the SI (Figures S4 to S7). Before even considering any microscopic details of the
water structure or nucleation processes, several general observations about the data shown
10
ResultsSimple models, complex behaviours
FIG. 5. a) Heat maps representing the values of ice nucleation rates on top of the four different
surfaces considered, plotted as a function of the adsorption energy Eads and the lattice parameter
afcc. The lattice mismatch δ on (111) is indicated below the graph. The values of the nucleation
rate J are reported as log10(J/J0), where J0 refers to the homogeneous nucleation rate at the same
temperature. b) Sketches of the different regions (white areas) in the (Eads,afcc) space in which
we observe a significant enhancement of the nucleation rate. We label each region according to
the face of Ih nucleating and growing on top of the surface (basal, prismatic or (1120)), together
with an indication of what it is that enhances the nucleation. “temp”, “buck”, and “highE” refer
to the in-plane template of the first overlayer, the ice-like buckling of the contact layer, and the
nucleation for high adsorption energies on compact surfaces, as explained in section III B.
III. RESULTS
A. No Simple Trend for Nucleation Rates
The nucleation rates on the four surfaces considered are shown as bi-dimensional heat
maps as a function of the lattice constant and adsorption energy in Figure 5a. Regions in the
2D plots72 for which a strong enhancement of the nucleation rates is observed are sketched
in Figure 5b and snapshots of representative trajectories for all the classified regions can be
found in the SI (Figures S4 to S7). Before even considering any microscopic details of the
water structure or nucleation processes, several general observations about the data shown
10
FIG. 5. a) Heat maps representing the values of ice nucleation rates on top of the four different
surfaces considered, plotted as a function of the adsorption energy Eads and the lattice parameter
afcc. The lattice mismatch δ on (111) is indicated below the graph. The values of the nucleation
rate J are reported as log10(J/J0), where J0 refers to the homogeneous nucleation rate at the same
temperature. b) Sketches of the different regions (white areas) in the (Eads,afcc) space in which
we observe a significant enhancement of the nucleation rate. We label each region according to
the face of Ih nucleating and growing on top of the surface (basal, prismatic or (1120)), together
with an indication of what it is that enhances the nucleation. “temp”, “buck”, and “highE” refer
to the in-plane template of the first overlayer, the ice-like buckling of the contact layer, and the
nucleation for high adsorption energies on compact surfaces, as explained in section III B.
III. RESULTS
A. No Simple Trend for Nucleation Rates
The nucleation rates on the four surfaces considered are shown as bi-dimensional heat
maps as a function of the lattice constant and adsorption energy in Figure 5a. Regions in the
2D plots72 for which a strong enhancement of the nucleation rates is observed are sketched
in Figure 5b and snapshots of representative trajectories for all the classified regions can be
found in the SI (Figures S4 to S7). Before even considering any microscopic details of the
water structure or nucleation processes, several general observations about the data shown
10
• The crystalline surfaces mostly do promote nucleation compared to homogeneous nucleation.
• Both afcc and Eads do not influence nucleation on top of each surface in the same manner.
• There is no optimal value for Eads, but nucleation rate is generally inhibited for very low adsorption energies
• The lattice mismatch " cannot be regarded as the only requirement for an INA
Results
FIG. 6. Analysis of certain factors important to nucleation. Each row represents data obtained
from a representative trajectory for events classified as temp, buck and both combined mechanisms
(see section III B). The first column depicts the density of water molecules above the surface after
freezing (filled curves) and during equilibration before freezing (dashed black line). The second
column shows side views and the third column snapshots viewed from above. In all cases the
contact layer is colored red while higher layers are colored blue. For ease of visualization in the
top view only part of the second layer is shown.
B. Microscopic Factors for Nucleation
It is somehow unexpected that a simplistic model like the one used here can foster such
diverse behavior. However, when we examine the water structures and nucleation processes
in detail, general trends do emerge. We now discuss the key features important to nucleation.
1. In-Plane Template of the First Overlayer
The in-plane structure of the first water overlayer plays an important role in nucleation,
because it can act as a template to higher layers. This is particularly evident on the (111)
surface, which possesses an hexagonal symmetry compatible with the in-plane symmetry of
the basal face of ice (honeycomb). Where nucleation is significantly enhanced, we find that
an hexagonal overlayer (HOL) of water molecules forms on top of the surface (Figure 6,
13
Boosting nucleation rates: three microscopic motivations
In-plane templating of the first water overlayer
Buckling of the first water overlayer
In-plane templating + Buckling of the first water overlayer
ResultsInhibition and nucleation away from the surface
!
!"!#
!"!$
!"!%
!"!&
#! '! ! '! #!
( )*+,-
.*/*012
( ,32456
7
89:;1,2<4=)*/4>?@64ABC
.*/*
FIG. S8. Probability density distribution Pnuc(z) of the z-coordinate of center of mass (COM) of
pre-critical ice-like clusters. The x-axis refers to the distance from the COM of the mW water slab.
The gray shaded region highlights the extent of the 1st and 2nd water overlayer on top of the LJ
surface. The legend refers to a bulk model of 4000 mW molecules (Homo), the same as a free-
standing slab (HomoVAC) and scenarios (Inh and Pro) in which we observe inhibition/promotion
of J on the (100) surface (afcc = 3.90 A for both, Eads = 3.21 and 5.30 kcal/mol respectively). All
data was collected at 205 K.
8
In other cases Jhet > Jhomo even when nothing happens
within the contact layer
In some cases (typically weak Eads ) Jhet < Jhomo Ice-like nuclei tend not to form on the crystalline surface
(~water-air interface) Phys. Chem. Chem. Phys. 16, 25916 (2014)
Water dynamics at interface and [ice] nucleation
mW water1st-2nd
ResultsDifferent ice faces on top of the very same surface
e.g. : the (211) face
FIG. 7. (a) Representative snapshots of the three different faces (basal, prismatic and (1120) face)
of hexagonal ice growing on top of the (211) surface (side view). Surface atoms are depicted as balls
(grey), while the bonding network of water molecules is represented by sticks (blue). The θ angle
in the top left panel illustrates that the basal face and the normal of the (111) terrace deviate. b)
Nucleation rates (circles) and spline interpolation (line) on the (211) surface as a function of the
step distance d. The red lines indicate the measured characteristic distances d1 and d2 as well as
their standard deviation (red shaded area). The meaning of d, d1 and d2 is illustrated in the top
panels.
growth of three different faces of ice are observed. The three regimes roughly correspond to
different values of afcc (Figure 7a). The (211) substrate has a rectangular in-plane symmetry,
but it features (111) micro-facets (see Figure 1a). For small values of afcc (Figure 7a), the
spacing between the steps allows for rows of hexagons to form on top of these terraces.
This template has a symmetry consistent with the basal face of Ih which in fact exclusively
nucleates in this first regime. As an aside we note that the growth direction of the basal
face is not exactly parallel to the surface normal of the (111) terraces, leading to the small
angle mismatch shown in Figure 7a. As we move on to larger lattice constants, the spacing
between the steps becomes too large to accommodate an hexagonal overlayer. Rather a
rectangular overlayer appears on top of the surface, wiping out the templating effect of the
hexagons. These overlayers are buckled in a manner that follows the corrugation of the
15
Different afcc Different templating
Hexagonal overlayer Rectangular overlayer
Different ice faces
Eads not that important for (211), but each surface has its own story to tell
ResultsSummary
• Even on top of the simplest substrates (frozen LJ particles), the interplay between hydrophobicity and surface morphology can affect heterogeneous ice nucleation in many different ways !• Lattice mismatch can play a role, but there is so much more going on
in the contact layer and even quite far away from the interface !• Simple trends are nice and useful, but:
• It looks like every surface has its story to tell • Can we get closer to reality?
What’s next
• Realistic surfaces: Reliable force fields needed for complex interfaces !
• Fully atomistic models of water: Correct dynamical properties at strong supercooling !
• Methods: Brute force molecular dynamics won’t work Enhanced sampling simulations are still challenging
free-energy barrier associated with the formation of a criticalnucleus, given by
!Gc =16!"3
3#2s j!$j2. [1]
Here " is the solid!liquid surface tension, #s is the number den-sity of the solid, and !$ is the free-energy difference between thecrystalline and liquid phases. The exponential dependence of thenucleation rate on !Gc, and the sensitivity of !Gc to " and !$implies that only a slight deviation of any of these quantities fromthe experimental value can shift the nucleation rate by severalorders of magnitude. Both these quantities are difficult to mea-sure at large supercoolings, mostly because of the difficulty ofstabilizing supercooled water at such low temperatures.
Free-Energy Difference. If !Hf , the latent heat of fusion, is not astrong function of temperature, !$ can be approximated as!$"!Hf !Tf !T"=Tf . This approach, which yields a value of!$" 0.2215 kcal ·mol!1 at a supercooling of 42 K, is, however,
not very accurate for water due to its heat capacity anomaly. Toobtain a more accurate estimate, we take the heat capacitymeasurements for Ih (38) and supercooled water (39) (Table 1),and use thermodynamic integration to obtain a more accurateestimate of !$exp = 0.1855 kcal ·mol!1. Similarly, we use MDsimulations in the isothermal isobaric (NpT) ensemble to com-pute enthalpies of Ih and supercooled water at 230 K#T # 272 Kand use those enthalpies to compute !$ using thermodynamicintegration. We obtain a value of !$TIP4P=Ice = 0.147 kcal ·mol!1for the TIP4P/Ice system, which is around 20% smaller than!$exp. This discrepancy alone can lead to an overestimation ofthe nucleation barrier by as much as 60% if everything else isidentical. To be more quantitative, the classical nucleation the-ory predicts a nucleation barrier of !Gc = !1=2"j!$jNc " 51! kBTfor the TIP4P/Ice system at T = 230 K. However, if we use !$expinstead of !$TIP4P=Ice, and #s,exp = 0.922g · cm!3 (40) instead of#s,TIP4P=Ice = 0.908g · cm!3 (obtained from NpT MD simulation ofIh at 230 K and 1 bar) in Eq. 1, we obtain a barrier of $31 kBT,which corresponds to an increase in the nucleation rate by eightto nine orders of magnitude. This is very close to the discrepancybetween our calculation and the experimental estimates of rate.
Surface Tension.At temperatures below Tf , the supercooled waterthat is in contact with ice is not stable and will immediately freeze.This makes experimental measurements of " in the supercooledregime extremely challenging. Therefore, " is typically estimatedindirectly from the nucleation data assuming the validity of theclassical nucleation theory. Consequently, there is a large variationin the reported estimates of " for supercooled water that spanbetween 25 mJ ·m!2 and 35 mJ ·m!2 (41). Similarly, it is verychallenging to compute " directly from molecular simulations atT <Tf , and all of the existing estimates are obtained from nucle-ation calculations (13, 17, 42). The existing direct calculations haveall been performed at coexistence conditions (43, 44). The com-puted numbers cover even a wider range, from 20.4 mJ ·m!2 in ref.17 to 35 mJ ·m!2 in ref. 42. This large variability underscores thesensitivity of the computed value to the particulars of the watermodel, and to the thermodynamic conditions at which the calcu-lation has been made. In light of the mechanism that is proposedfor freezing in this work, the problem of determining " is furthercompounded by the stacking disorder nature of the critical nucleus.Considering the cubic dependence of the nucleation barrier on", even the slightest deviation from the experimental value canshift the nucleation rate by several orders of magnitude. Forinstance, a 7% deviation can change the nucleation barrier by asmuch as 22%, which can shift the nucleation rate by severalorders of magnitude.Overall, the existing classical models of water inevitably predict
certain thermodynamic properties of water to be at variance withexperiments by a significant margin, and a model that predicts allthermodynamic properties accurately has yet to be developed (45).Therefore, the agreement between the orders of magnitude of thecomputational estimate of the nucleation rate in a classical modelof water, like TIP4P/Ice, and the corresponding experimentalvalue is difficult to achieve with the existing models due to strongsensitivity of the nucleation rate to the particular thermodynamicfeatures of the water model used (e.g., the free energies of theliquid and the solid, and the liquid!solid surface tension).
Earlier Computational Studies of Nucleation Rate. It is inherentlyproblematic to compare computational estimates of nucleationrate obtained for different force fields using different method-ologies. This is not only because of the large uncertainties as-sociated with the methods used (e.g., the validity of classicalnucleation theory in the seeding technique) but also due to theempirical and approximate nature of the force fields used, which,as shown above, can shift the computed rates by several orders of
Table 1. Numerical correlations used for fitting theexperimental heat capacity measurements of refs. 38 and 39
Phase Correlation
Supercooled water (39) Cp!T"=4! 1015T!5.824 +0.7131T ! 202Ih (38) Cp!T"=0.032T +0.3252
Units are in calories per mole per Kelvin. For Ih, we use a linear fit,whereas, for supercooled water, we use a combination of a power lawand a linear fit. The actual experimental data for supercooled water arefor T % 236 K. We thus use the numerical fit provided above to extrapolateCp at 231 K # T # 236 K.
0 50 100 150 200
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Fig. 6. Nucleation beyond the inflection region. (A) Average cage participa-tion of the molecules in the largest crystallite. The solid black line has a slopeof unity. The molecules that participate in a DDC (or HC) are included in thecorresponding count even if they also participate in a neighboring cage of theother type. The overwhelming majority of molecules are at least part of a DDC,whereas very few molecules are only a part of an HC. (B!G) Several repre-sentative configurations obtained at different milestones after the inflectionregion. B!E are precritical, F is critical, and G is postcritical. Molecules that are apart of a DDC, an HC, or both are depicted in dark blue, dark red, and lightyellow, respectively. Here, we use the same size convention used in Fig. 5.
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Proc. Nat. Acad. Sci. 112, 10582 (2015)
MB-pol (Paesani), NN (Dellago, Morawietz )