Surface Energy-Mediated Multistep Pathways for Heterogeneous Ice

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Surface Energy-Mediated Multi-Step Pathways for Heterogeneous Ice Nucleation Chu Li, Xiang Gao, and Zhigang Li J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b01358 • Publication Date (Web): 11 Apr 2018 Downloaded from http://pubs.acs.org on April 11, 2018

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The Journal of Physical Chemistry

Surface Energy-Mediated Multi-Step Pathways for Heterogeneous Ice Nucleation Chu Li, Xiang Gao, and Zhigang Li∗ Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

Abstract Heterogeneous ice nucleation (HIN) is the dominant mode of ice formation, while its kinetic pathways remain poorly understood.

The classical nucleation theory suggests a one-step

pathway, i.e. direct change from liquid water to ice (e.g. hexagonal ice), which has been widely accepted.

In this work, however, through molecular dynamics simulations, we observe

intermediate states, square ice, at the early stage of ice nucleation at certain surface energies. The intermediate square ice gives rise to a new, nonclassical pathway for HIN: from liquid water to hexagonal ice via square ice.

This multi-step pathway may coexist with and can be more

probable than the classical, one-step pathway though it may delay the ice nucleation process. The new multi-step pathway offers insights in controlling the kinetics of ice crystallization and understating the mechanisms of HIN.

April 2018 ∗

Email: [email protected]

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Introduction Heterogeneous ice nucleation (HIN), i.e., ice crystallization on a foreign material, is the major mode of ice formation.

A deep understanding of the fundamental kinetics of HIN would help

design materials to either inhibit1-3 or promote ice crystallization.

This will have significant

impacts in a variety of areas, ranging from atmospheric sciences,4,5 food industry,6 aviation,7 to energy systems.8,9

To better understand the kinetics of ice nucleation, it is essential to probe the

nucleation pathways, which provide detailed and insightful information about how water experiences phase changes and transforms into ice. The classical nucleation theory (CNT)10-12 suggests that nucleation emerges directly from liquid to small solid nuclei, which trigger further crystal growth.

For ice nucleation, it is known

that hexagonal ice (Ih) is the expected stable ice structure in nature under atmospheric pressure in a moderate supercooled temperature range, roughly 220 K < T < 273.15 K .13-15

Therefore, the

CNT indicates a classical, one-step pathway for HIN, which is the direct transformation from supercooled liquid water into stable ice, such as Ih. Some previous studies on HIN did find that the classical, one-step pathway is valid for certain surfaces, such as graphene16 and kaolinite.17 However, recent studies show that the CNT may not describe ice nucleation processes well18,19 due to the polymorphs of ice crystals.13,20,21

Ostwald’s rule of stages22,23 also suggests that a

nucleation process may take pathways involving transient states, which are associated with smaller free energy loss, before finally reaching a stable state.

Nucleation processes following

Ostwald’s rule of stages have been found in systems involving minerals,24 colloids,25 and gas hydrate.26

Therefore, theoretically a HIN process may also undergo nonclassical, multi-step

pathways under proper conditions.

Although extensive studies about the effects of surface


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properties,27-29 such as defects,30,31 crystal structures,32-34 surface charges,35 and wettability,36-38 on HIN have been conducted, multi-step pathways for HIN have not been reported. In this study, we investigate the kinetic pathways of HIN through molecular dynamics (MD) simulations.

It is found that intermediate states containing square ice (Is) occur at certain surface

conditions, which lead to a new ice nucleation pathway, i.e. supercooled liquid water → Is → Ih. Depending on the surface energy, this multi-step pathway can be the dominant pathway.

It may

also coexist with the classical, one-step pathway. Results and Discussion The MD simulations are performed using the LAMMPS packages.39 The simulation system is similar to that described in previous work.38

It consists of a rigid substrate with wurtzite

structure and a slab of water containing 1560 molecules, as shown in Figure 1a. described by the TIP4P/Ice water model.40

Water/ice is

Practically, many materials take the wurtzite

structure, such as silver iodide,32 which is a common material known for its ability in promoting ice nucleation. The surface energy is regulated by tuning the water-surface interaction binding energy ε ws .41

Details of the simulations can be found in Ref. 38 and the Supporting

Information. To characterize the structure of water molecules, the average bond-order parameter for water/ice molecules is computed, which is defined as42-45 1/ 2

2  4π  m =l 1 Ni  , ql (i ) =  q j ( ) ∑ ∑ , l m  2l + 1 m =−l N i + 1 j =0   

(1)

where N i is the number of neighbors of molecule i (j = 0 corresponds to molecule i) and ql ,m ( j ) =

1 Nj

∑

Nj

Y

k =1 l , m

(r ) jk

with Yl ,m ( rjk ) being the spherical harmonics and m ∈ [ −l , l ] . Here,


3


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only the positions of oxygen atoms are considered and vector r jk = r j − rk with rj and rk being the position vectors of molecule j and its neighbor, k.

Practically, q4 and q6 have been

widely used to distinguish structures with four- and six-fold symmetries.44,45

For bulk water,

17 N i = 4 is usually used and q6 = 0.45 is applied to distinguish Ih from liquid water.

interfacial molecules, N i = 3 is employed. 4.3-Å-thick layer above the surface.38

For

Here, interfacial molecules are those in a

Figure 2a shows the number distribution of water

molecules at T=300 K and that of Ih at T=230 K at the interface in the q4 ─ q6 coordinates for -1 ε ws = 4.989 KJ mol .

structures.

It is clear that there are two regions, which correspond to two different

For liquid water, usually q4 < 0.6 and q6 < 0.5 , as shown in Figure 2a, where a

snapshot of liquid water is depicted in the lower right inset.

The region with q4 < 0.6 and

q6 > 0.5 , however, corresponds to Ih, as illustrated by the six-fold symmetry structure in the upper left inset of Figure 2a.

The two distinct structures are also confirmed by the radial distribution

function g(r) and Voronoi diagram in Figures S1 and S2 (Supporting Information). For ε ws = 4.989 KJ mol-1 and T=230 K, where HIN occurs and Ih forms eventually,38 another structure different from liquid water and Ih is observed at the interface during nucleation. As shown in Figure 2b, it lies in the region of q4 > 0.6 in the q4 ─ q6 coordinates.

The inset in

Figure 2b shows a snapshot of this structure, which has four-fold symmetry with oxygen atoms on square lattice sites.

The radial distribution function g(r) and Voronoi diagram for this structure

are also given in Figures S1 and S2, which confirm that it is different from liquid water and Ih. This four-fold symmetry structure is similar to that of square ice (Is), which was only found in nanoconfinements46-48 or on certain surfaces as a stable state.34


Figure 2c shows the total

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numbers of Is and Ih molecules in the system, ns and nh , as a function of time t for the case in Figure 2b.

It is found that ns increases with time and nh remains almost constant for t < 150

ns, after which nh increases, while ns decreases and approaches zero when t > 250 ns. Figure 2c indicates that Is is just an intermediate state during nucleation.

It eventually disappears

or transforms into Ih, followed by continuous Ih growth, as shown in Figure 1b. It is known that Is is not as stable as Ih in a bulk system.

Although Is-like structure has been

observed on certain surfaces as a stable structure that acts as a template for triggering the growth of Ih,34,37 Is has not been found as an intermediate state during ice nucleation. Thermodynamically, the occurrence of intermediate Is might be associated with a locally low free energy as an ice nucleation process prefers to take a pathway of relatively low energy.

In this

sense, the occurrence of Is might offer a new, nonclassical ice nucleation pathway. To explore the pathways, the free energy ∆G ( ns , nh ) during ice nucleation is calculated, which is obtained through49-51

β∆G ( ns , nh ) = − log P ( ns , nh ) ,

(2)

where β = 1/ ( kBT ) with kB being the Boltzmann constant and P ( ns , nh ) is the probability of the system assuming a state characterized by

( ns , nh ) .

P ( ns , nh ) is calculated based on the

Markov chain model.51,52 During nucleation, the state of the system, which corresponds to a pair of ns and nh values, usually changes with time.

In simulations, the state of the system is

collected many times at a time interval ∆t . Based on these states, the probability for the system transiting from a state, say, Si at a time t to state S j at t + ∆t , p ( S j | Si ) , is calculated, which


5


satisfies the normalization condition

∑ p (S

j

| Si ) = 1 .

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Then the probability of the system

Si

assuming state Si is obtained by iterating the following equation51

p ( S j ) = ∑ p ( S j | Si ) p ( Si ) .

(3)

Si

Under the condition of

∑ p(S

j

) = 1 , the convergent result p ( S i ) is considered as P ( ns , nh ) .

Sj

(

)

To obtain good statistics for the transition probability p S j | Si , 18 simulations of total 8 µs and about 2.0 ×107 state transition events are used with ∆t = 0.4 ps. Figure 3a presents the free energy as a function of ns and nh for the case in Figure 2b.

It is

seen that there are local minima for the free energy, which accommodate multiple pathways. Initially, the system quickly assumes a state of a small number of ice molecules at point A, where the free energy is a local minimum. overcoming free energy barriers.

Then, the nucleation may take two paths through

One (path I) is from points A to D via saddle point O1, i.e.

A→O1→D. Along this path, only one free energy barrier, ∆GI ≈ 2.98 kBT , which is the free energy difference between points A and O1 (Figure 3b), needs to be overcome.

This pathway can

be viewed as the classical, one-step pathway (liquid water → Ih). The other path (path II), A→O2→B→O3→C→O4→D, as shown in Figure 3a, goes through two local free energy minima, B and C.

States B and C are intermediate states containing notable square ice [( ns , nh )=(85,110)

at B and ( ns , nh )=(32,164) at C] compared with the stable state of Ih at point D, where ( ns , nh )=(0,258).

2 1 Along this path, the system overcomes ∆GII ≈ 1.96 kBT , ∆GII ≈ 1.83 kBT , and

∆GII3 ≈ 1.10 kBT to reach states B, C, and D, respectively. Therefore, it is a nonclassical, multi-step pathway.

Herein, it is referred to as a liquid water→ Is → Ih path.


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Figure 4 shows the typical configurations of the system at points A, B, C, and D along path II in Figure 3a.

At point A, there are some ice-like molecules at the interface, which are mainly

induced by water-surface interactions.

Further studies show that the formation of ice at point A

is intermittent (see the snapshots in Figure S3). As the nucleation evolves along path II, ns increases first (from A to B) and then decreases (from B to C and D) due to the conversion of Is to Ih, as shown in Figure 4. Although both pathways in Figure 3a are accessible during nucleation, the probabilities of taking the pathways, PI and PII, are different.

According to Ostwald’s rule of stages, the

probability of taking a path is mainly governed by the free energy barrier between point A and the 1 next saddle point along the path, i.e. ∆GI and ∆GII for paths I and II, respectively. As the

probability is estimated as P ~ exp ( −∆G kBT ) , the probability ratio for the two paths PII / PI ≈ 2.77 , which indicates that path II is more probable than path I.

This is consistent with

MD simulations, where 5 and 13 out of 18 independent simulations follow paths I and II, respectively (information about the determination of path statistics is provided in the Supporting Information). If the CNT for one-step pathways is employed, the nucleation time for an i-step pathway can

(

)

−1

be approximated as t ≈ ∑  Ki exp −∆G i / k BT  , where Ki is a prefactor, depending on different quantities such as the nucleation site density, the Zeldovich factor, and the molecular mobility of water molecules at the water-ice interface.11,53

The average nucleation times up to

point D in Figure 3b are 196 ± 69 and 370 ± 113 ns for paths I and II, respectively, indicating that the values of K along the two paths are different, given the values of ∆G for the two paths.


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This is reasonable as the mobility of water molecules at the water-ice interface is affected by the structure of ice. The free energy barriers for the formation of Is and Ih can be mediated by changing the surface energy, which is characterized by the water-surface binding energy, ε ws . Hence, whether there are multi-step pathways or not for a HIN case and the probability of taking a multi-step pathway can be controlled by tuning the surface energy or ε ws . Figure 5 shows the free energy of three cases at relatively low surface energies, ε ws = 2.910 , 3.326, and 4.148 KJ mol-1. It is seen that the classical, one-step pathway (liquid water → Ih) dominates in these cases and the nonclassical, multi-step pathway (liquid water→ Is → Ih) is unlikely to occur because ∆ G for the formation of Is is much higher than that for Ih. Nevertheless, Figure 5 also shows that ∆ G for the formation of Is decreases as ε ws is increased, indicating that nonclassical, multi-step pathways may become more possible at a high surface energy. As ε ws is increased, at ε ws = 4.573 KJ mol-1, the free energy barrier for the formation of Is decreases and becomes comparable to that for Ih, leading to the coexistence of the classical, one-step and nonclassical, multi-step pathways, as shown in Figure 6a, where a multi-step pathway is denoted by A→B→O3→C→D→O2→E.

In this case, the nonclassical pathway is less

preferred compared with the classical pathway due to the higher free energy barrier from A to O3, as shown in the inset of Figure 6a.

Figure 6b shows another case of even higher surface energy

( ε ws = 5.404 KJ mol-1), where the nonclassical pathway (Is → Ih) appears to be the only pathway. It is worth mentioning that in certain situations the free energy barrier for the formation of Ih could be sufficiently high such that HIN may not be observed. cases is shown in Figure S4.

The free energy of two such

In Figure S4a, ε ws = 2.494 KJ mol-1, where the water-surface

interaction is too weak to assist interfacial water molecules to reconstruct and form ice.38


In 8

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Figure S4b, water-surface interaction is strong ( ε ws = 5.821 KJ mol-1) and some interfacial water molecules transform into a mixture of Is and Ih, but the further growth of ice is inhibited due to the high energy barrier from Is to Ih (Figure S4b). Conclusion In conclusion, we have found an intermediate state, square ice, during HIN at certain surface properties, which leads to nonclassical, multi-step pathways and offers more opportunities for controlling HIN. Supporting Information Simulation details; Figure S1, radial distribution function g(r) for liquid water, hexagonal ice, and square ice; Figure S2, Voronoi diagrams for different structures; Figure S3, Snapshots of ice-like structures induced by the surface after initialization; Figure S4, free energy for

ε ws = 2.494 and 5.821 KJ mol-1; Figure S5, Determination of path statistics. Acknowledgements This work was supported by the Research Grants Council of the Hong Kong Special Administrative Region under Grant No. 16228216.


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Figure Captions: Figure 1. Snapshots of the MD simulation system. hexagonal ice.

(a) Simulation system.

(b) A snapshot of

Surface, oxygen, and hydrogen atoms are represented by grey, purple,

and green spheres, respectively. Figure 2. Structure symmetry analysis and time variation of ice molecules at ε ws = 4.989 KJ mol-1.

(a) Number distribution of interfacial liquid water (T=300 K) and hexagonal

ice (Ih) (T=230 K) molecules in q4 ─ q6 plane.

Liquid water is located in the lower

group with q6 < 0.5 and Ih is in the upper group with q6 > 0.5 . The lower right and upper left insets are snapshots of liquid water and Ih, respectively.

(b) Number

distribution of molecules at the interface for different structures in q4 ─ q6 plane at T=230 K.

An extra group with q4 > 0.6 shows up, which corresponds to square ice

and a snapshot is shown in the inset.

(c) Number variation of ice molecules of the

system during nucleation for the case in (b). Figure 3. Nucleation pathways at ε ws = 4.989 KJ mol-1 and T = 230 K. function of ns and nh .

(a) Free energy as a

Two pathways are indicated by lines going through points

A(54,54), B(110,85), C(164,32), D(258,0), O1(117,16), O2(82,78), O3(138,47), and O4(170,12).

(b) Free energy along the two paths (shaded areas represent numerical

errors). Figure 4. Snapshots of the system at points A, B, C, and D in Figure 3. Top panel: Side view. Bottom panel: Top view. The surface atoms are presented by grey spheres; the oxygen atoms for liquid water, Is, and Ih molecules are denoted by green, purple, and pink spheres, respectively. Figure 5. Free energy at ε ws = 2.910 (top panel), 3.326 (middle panel), and 4.158 (bottom panel) KJ mol-1, respectively.

The classical, one-step pathway dominates in these three

cases. Figure 6. Free energy at (a) ε ws = 4.573 KJ mol-1 and (b) ε ws = 5.404 KJ mol-1.


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Figure 1


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Figure 2


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Figure 3


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Figure 4


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Figure 5


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Figure 6


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TOC Graphic


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Surface Energy-Mediated Multistep Pathways for Heterogeneous Ice

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