Computational Investigation of Surface Freezing in a Molecular Model Of

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University the C.D., wrote P.G.D. 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APPLIED PHYSICAL SCIENCES difficult to determine whether this criterion is satisfied for a given might not be representative of how real water behaves close to material due to the challenges of measuring surface tensions in free surfaces, both due to this model’s coarse-grained nature and the supercooled regime (16). Not surprisingly, this criterion is its lack of electrostatic interactions, which can potentially play not quantitatively verified for water in ref. 14, and only a quali- a pivotal role in interfacial phenomena. For the more realistic tative argument is given as to why this substance is expected to molecular models, however, a systematic investigation of sur- satisfy this criterion and undergo surface freezing. face freezing was not possible until recently because direct cal- Because of their insufficient spatiotemporal resolution, the culations of nucleation rates were out of reach. Recently, we existing experimental techniques have so far been incapable of used a coarse-grained variant of the FFS algorithm to perform directly confirming or disproving surface freezing in water. The the first direct calculation of the bulk homogeneous ice nucle- most direct anecdotal evidence obtained so far is the observa- ation rate (35) in the TIP4P/ice system (36), which is one of the tion that contact freezing (i.e., heterogeneous nucleation caused most accurate molecular models of water and the best model for by the presence of a solid particle close to a vapor–liquid inter- studying ice polymorphs. This development enables us to con- face) occurs at rates considerably higher than immersion freez- sider the question of surface freezing in the TIP4P/ice system by ing (i.e., heterogeneous nucleation caused by the presence of a computing volumetric nucleation rates in freestanding nanofilms solid particle in the bulk) (17). All efforts in this regard, includ- and comparing those with the already-computed nucleation rates ing the analysis carried out in ref. 14, have therefore been indi- in the bulk. The rationale for preferring nanofilms over nan- rect and have focused on examining the dependence of volumet- odroplets is the net zero curvature of free interfaces in nanofilms. ric nucleation rates on the droplet size distribution (18, 19), with This allows us to explore exclusively the effect of a free interface the studied droplets usually between 1 and 100 µm in diameter. on the kinetics and mechanism of nucleation. For nanodroplets, Such studies, however, have been inconclusive with respect to however, changes in nucleation kinetics will be due to the com- the core question of surface freezing in water due to the large bined effect of the free interface and the large Laplace pressures uncertainties in droplet size distributions and nucleation rates that it induces inside the droplet, and disentangling the two is and the fact that surface freezing becomes potentially dominant nontrivial. Furthermore, the curvature at free interfaces in atmo- at droplet sizes of a few micrometers, which lies at the lower spheric microdroplets is orders of magnitude smaller than in the end of sizes considered in such studies (20). Probing nucleation case of nanodroplets. Accordingly, the former are accurately rep- kinetics at the nanoscale (e.g., in nanofilms and nanodroplets) resented by zero-curvature freestanding nanofilms. poses further complications. For instance, nanodroplets typically have large Laplace pressures, and their freezing occurs at lower Results and Discussion temperatures and culminates in the formation of stacking disor- Rate Calculations. We compute nucleation rates using the coarse- dered ice, a polymorph different from hexagonal ice that is the grained FFS algorithm that we introduced and discussed in detail final product of freezing in microdroplets (21, 22). This makes in an earlier publication (35). In every FFS calculation, it is nec- the proper comparison of rate measurements in nanodroplets essary to first define an order parameter, λ(xN ), that quantifies and microdroplets nontrivial. The question of surface freezing the progress of the kinetic process of interest—nucleation in our in water is therefore still unresolved and is considered one of the case—with the liquid, L, and crystalline, X , basins characterized 10 major open questions concerning ice and snow (23). N N N by λ(x ) ≤ λL and λ(x ) ≥ λX , respectively, and x denoting The lack of convincing direct experimental evidence for (or the N -body system’s configuration. The nucleation rate is then against) surface freezing in water has made molecular simulation given by the cumulative flux of trajectories that leave λL and an attractive alternative. Depending on the force fields and sys- end up in λX , and is estimated as the cumulative product of Φ0, tem sizes considered, however, such computational studies have λ λ( N ) = λ > λ reached conflicting conclusions. For example, molecular dynam- the flux of trajectories that leave L and cross x 1 L p λ ics (MD) simulations of freestanding nanofilms simulated using a and k values, the transition probabilities of going from k to λ . Here, λL < λ1 < λ2 < ··· < λn = λX are milestones that six-site force field (24) reveal that freezing occurs predominantly k+1 lie between the two basins. We compute the nucleation rate in at the surface (25, 26), an observation attributed to the lack of a 4-nm-thick freestanding film of supercooled water with 3,072 electrostatic neutrality in the subsurface region. Because of the TIP4P/ice molecules. To be consistent with our earlier rate cal- small system sizes (N = 384–576) considered in refs. 25 and 26, culation in the bulk, performed at 230 K and 1 bar (35), we con- however, these findings were likely affected by considerable finite ducted this new calculation at 230 K. We also used the same size effects. The model of choice for the overwhelming majority order parameter (i.e., the size of the largest crystallite in the of computational studies of surface freezing has been the coarse- system). grained monoatomic water (mW) potential (27). Li et al. (28) and Fig. 1 depicts the cumulative growth probability P(λk |λ1) = Haji-Akbari et al. (29) use forward-flux sampling (FFS) (30) to Qk−1 compute nucleation rates in droplets (28) and freestanding thin j =1 pj as a function of λ for both the bulk and confined geome- films (29) of supercooled mW. By comparing the nucleation rates tries. It can be seen, in the first place, that the volumetric ice in the bulk (29, 31) and in confined (28, 29) geometries, they both conclude that surface freezing is suppressed in the mW system. These findings are interesting in the sense that the mW system satisfies (32) the partial wettability condition articulated in refs. 14 and 15, and yet fails to undergo surface freezing. Another fea- ture that correlates with surface freezing propensity in silicon, another tetrahedral liquid, is the negative slope of dP/dT , the melting curve in the phase diagram (33). Materials with negative dP/dT need to undergo expansion upon freezing, which is presumably facilitated at free interfaces. However, it has been demonstrated that the mW system (29) and some of its variants (34) fail to undergo surface freezing despite satisfying this criterion. The fact that surface freezing propensity fails to correlate with simple thermodynamic features, such as partial wettability of the crystal or the negative slope of the melting curve, underscores the complexity of the surface freezing process and the possibility Fig. 1. Cumulative transition probabilities vs. the size of the largest that it can be affected by system-specific structural features. crystallite in the system for the bulk and film geometries at 230 K. −3 −1 Despite providing valuable insight into the underlying physics The volumetric nucleation rates are log10J[m ·s ] = 12.5819 ± 0.4961 and of surface freezing, the findings obtained with the mW model 5.9299 ± 0.5863 for the film and bulk geometries, respectively.

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Fig. trans- tensor the breaks stress confinement the S that of inter- free established isotropy by well affected lational directly is are iden- It that to film faces. need the of first of regions we values the however, different tify distributions, at such of film sense the make within distribution spatial crystallites the largest inspect of we geometries, confined and bulk bulk. the inflec- in the observed of otherwise disappearance region virtual the tion geometry. to growth leads the film on and HCs crystallites the of of effect in adverse potential lower the suppresses significantly This is however, ticipation, 2 (Fig. pathway contribute surviving nucleation not the the do to that because configurations vanishing less geometry the correspondingly film than and HC-rich, the DDC-rich, more in still true are configurations is 2 same Fig. The in depicted D. 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F F aigisnormal, its making ), emtis h surviv- The geometries. ) –F aeparticipation Cage ) λ 1ad66 and 41 = and C λ E and ) To . and E h CDCrto infiatyeceigisbl au of value bulk its exceeding significantly ratio, with HC/DDC DDCs depleted, However, the strongly values. bulk more respective are their than both lower cantly region, order ice-like subsurface 5 Fig. of blue in extent demonstrated com- As local molecule. also the each around We quantifies 3. that local (39) Fig. of parameter in profile shown spatial the profiles pute the computing for used to than scale, length larger decays is geomet- characteristic icantly that that a trivial observable over geometry value a structural film bulk a the to its of exists up fraction there the and bulk, for be “active”), the account should to for rate factor bulk-like overall calculated ric truly the that that not to suggest is similar would region 4 interior Fig. the (otherwise, that at suggests progeny This with 4A). ones the (i.e., λ configurations (Fig. surviving film the for the across that distributed 4B), ther- equally fact bulk-like are the crystallites exhibits largest Despite the presumably 4). that in (Fig. but film behavior regions, subsur- modynamic the blue of shaded the interior the intuitively in in the the is not and peaked observed emerge it is crystallites be bulk), behavior largest opposite would the the Strikingly, to histogram region. face respect the (with that film expected of the enhancement the in Considering freezing crystallites. con- such by the to FFS belong of from that obtained histogram crystallites a largest structing the slightly of over spread tial albeit value, bulk its scales. to length shorter converges sub- be density the to criterion, stress, in a determined as outlined is anisotropy region approach tensor surface stress the the using Using stress 38. films ref. and TIP4P/ice density 4-nm-thick of compute simulations and MD regular out carry the of components lateral and normal simulated tensor. the K stress in 230 anisotropy at from system determined TIP4P/ice the 5 of for film freestanding 4-nm-thick a 3. Fig. ρ nm 0.0319(8) obtain and bar compare deter- 1 we and and bulk, from densities the film DDC for and the As HC values. mine across bulk respective densities, their with in number those crystallites DDC largest compute and difference the we HC stark of geometries, the content film and DDC and bulk and process, HC nucleation the the between in DDCs and HC 6 r lasecuieylctdi h neirrgo (Fig. region interior the in located exclusively always are 66) = pnietfigtesbufc ein eisettespa- the inspect we region, subsurface the identifying Upon osdrn h motneo h nepa ewe HCs between interplay the of importance the Considering (z .Tesae lergoscrepn otesbufc ein as regions subsurface the to correspond regions blue shaded The µs. ) in (B) stress lateral and normal and (A) density of distribution Spatial and ρ −3 DDC epciey o h l emty ecompute we geometry, film the For respectively. , B A (z l s ) h hcns ftesbufc region. subsurface the of thickness the , rfie yaayigteM trajectories MD the analyzing by profiles ρ HC,bulk .627 nm 0.0602(7) = z ρ oriae fwtrmolecules water of coordinates HC q 6 NpT noinainlbn order bond orientational an , (z . mtik(i.3.Like 3). (Fig. thick nm ∼1.5 ψ ) (z and ρ Dsmltosa 3 K 230 at simulations MD := ) HC NSEryEdition Early PNAS (z ρ DDC ) −3 l and ρ c nteshaded the in B, hc ssignif- is which , HC and (z (z ) ρ DDC r signifi- are ρ )/ρ DDC,bulk λ (z DDC 1 | the ), 12, = f6 of 3 (z = ),

APPLIED PHYSICAL SCIENCES Similar to TIP4P/ice, the decay length scale for the deviations A of cage density profiles and ψ(z) from the bulk is significantly larger than the thickness of subsurface region in the mW system (Fig. S3), even though the magnitude of such deviations are considerably smaller. It might therefore be interesting to explore the possible generality of the inequality, lc ls . The stark difference between decay characteristics of simple thermodynamic proper- B ties, such as stress, and complex structural features, such as cage density, underscores the difficulty of identifying regions with bulk- like behavior in confined systems, especially when it comes to phenomena as complex as nucleation. Determining what consti- tutes “ultrathin” is therefore a subtle question and requires care- ful examination of all thermodynamic and structural properties that could be relevant to the phenomenon under consideration. The development of the coarse-grained mW potential has Fig. 4. Spatial distribution of molecules that belong to the largest crystal- been a major breakthrough in computational studies of water lites for different values of λ for surviving configurations that have progeny and has considerably expanded the range of accessible time and at λ = 66 (A) and all configurations at a given λ (B). The shaded blue regions length scales (40). The popularity of mW is not only a con- correspond to the subsurface regions determined from anisotropy of the sequence of its efficiency, but also of its remarkable success stress tensor. in reproducing thermodynamic and structural features of water (27). This work suggests that this very successful model might not ψbulk = 1.88(5) (Fig. 5C). The paucity of both cage types and be able to accurately capture subtle interfacial phenomena of the the stronger depletion of DDCs make the subsurface region an type reported in this work. The mW model has also been shown unfavorable environment for the formation of nascent small crys- to differ qualitatively from atomistic models in other aspects, tallites. This explains why nucleation does not start at the subsur- such as the presence of a liquid–liquid critical point (41). face region. Unlike the subsurface region, ψ(z) is significantly lower than Comparison with Earlier Experimental Work. As outlined above, ψbulk in the interior region, which thus provides a more favor- directly addressing the surface freezing problem in experi- able environment for the formation of DDCs. This, in turn, facil- ments is extremely challenging. Comparing our computed rate itates nucleation and leads to elevated rates by suppressing the with rate measurements in ultraconfined geometries such as HC-induced inflection observed in the bulk. Consistent with this nanodroplets (21, 22) is not straightforward, because the latter hypothesis, the majority of water molecules belonging to the are typically done at lower temperatures and higher (Laplace) largest crystallites reside in this low-ψ region (Fig. 4). Interest- pressures. Extrapolating those rates to 230 K and 1 bar not only ingly, such deviations persist even in films as thick as 9 nm (Fig. involves predicting the thermodynamic and transport properties S1 C and D), pointing to lc values that are much larger than ls ∼ of supercooled water at experimentally inaccessible conditions 1.5 nm. We observe a similar lack of convergence to the bulk in q6(z) profiles that are always larger than and never converge to hq6ibulk = 0.254 in the interior region (Fig. 5A and Fig. S1B). Higher q6 values in the interior region are indeed consistent A with higher cubicity because the value corresponding to cubic ice, hq6icubic = 0.800 is larger than the corresponding hexagonal value, hq6ihex = 0.690 (31, 35).

Surface Freezing in the mW System. Unlike in TIP4P/ice, surface freezing is suppressed in the coarse-grained mW system (28, 29). To explore the origin of this different behavior, we perform the same type of cage statistics analysis in the mW system, at the B same temperature (and pressure) as TIP4P/ice. Note that this corresponds to the same degree of supercooling because both models have realistic melting temperatures (27, 36). In the bulk, at 230 K and 1 bar, the HC and DDC number densities and mW −3 the HD-to-DDC ratio are given by ρHC,bulk = 0.02099(3) nm , mW −3 mW ρDDC,bulk = 0.00465(2) nm and ψbulk = 4.51(2), respectively. (See SI Text for a comparison of bulk cage statistics in the TIP4P/ice and mW systems.) We also compute the spatial pro- C files of density, stress, ρHC(z), ρDDC(z), and ψ(z) for a 20-nm- thick mW film at 230 K (Fig. 6). The subsurface regions, as determined from the anisotropy of the stress tensor, have similar thicknesses in the TIP4P/ice and mW films (Fig. 6A). Fur- thermore, they both have fewer cages (Fig. 6B) and larger ψ(z) values (Fig. 6C) than the bulk. Beyond the subsurface region, mW however, ψ(z) is almost never smaller than ψbulk. This is in sharp contrast to TIP4P/ice films that have extended interior regions with lower-than-bulk ψ(z) values. Similarly, q6(z) behaves like the other thermodynamic properties in the mW system and is mW never higher than hq6ibulk = 0.251 in the interior region (Fig. S2). Fig. 5. Spatial distributions of local q6 (A), cage number densities (B), and In other words, confinement does not lead to the formation of HC/DDC ratio (C) across the 4-nm-thick TIP4P/ice film. The subsurface region, low-ψ regions in mW films, and all it does is to amplify ψ in the as determined from the anisotropy in the stress tensor, is shaded in blue. Bulk subsurface region. It is therefore no surprise that surface freezing values and the corresponding error bars are depicted with straight lines and is suppressed in the mW system. dashed lines, respectively. In B, the symbols are larger than the error bars.

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J hc s71 resof orders 7–10 is which 1.5, r a c r osntdpn ncurva- on depend not does h rtcldoltradius droplet critical the , c a o endetermined been not has d iha fetv volu- effective an with ρ HC (z ) asvl aallsmltr(AMS 4)uigtevlct Verlet velocity the using (46) atomic/molecular (LAMMPS) large-scale using simulator by parallel performed are massively simulations MD All major a Methods remains still thick) and thin, survey challenge. (ultrathin, computational open The regimes characteristics. all possess bulk that of regions all interior not have but they simple some that sense are than work the scales this in in length ultrathin structural considered larger films by The much driven properties. over thermodynamic be decay might that freezing interior features surface find- the Our that proceed. in suggest to uni- easier ings emergence process rather nucleation preferential their the to makes their region due and con- process growth, a nucleation play form the DDCs in earlier, role demonstrated structive favor- As within region. more DDCs interior of film. the formation the the the to for of emerges behavior that interior environment counterintuitive the able emerging within this but crystallites attribute region, the We subsurface with the nature, in in is not and the geometry, “homogeneous” nucleation film of the yet that in one observe is magnitude of is We orders seven water. which by of enhanced field, com- nanofilms, models force water directly molecular TIP4P/ice freestanding to best in the technique nucleation using FFS ice simulated of the rate use the surface We pute of water. question longstanding in the freezing investigate we work, this In Conclusions computational other (45). well in as nucleation demonstrated ice of recently studies been of latter has This effect facilitate—nucleation. point the than order- suppress—rather interpreting “bad” can because in ing kinetics, nucleation caution on ordering of interfacial under- importance This text. nucleation the the facilitate water in to scores discussed the of reasons tend structural ordering of not subtle the orientational does regions for interface words, deeper the other neutral at In molecules electrically 4A). freezing the (Fig. S4), at in (Fig. film neutrality start films electrostatic TIP4P/ice to in of tends observed lack is similar subsurface vapor a the the face though atoms Even hydrogen the phase. which tend in molecules orientations water electro- immediate favor where of authors interface, to the lack free the the the in to at this with neutrality initiate attribute static They different, events interface. of is the freezing of mechanism 26 vicinity most the and that work, 25 observing this refs. reveal as Despite in also 26). conclusion enhancement (25, model same interfaces freestanding the free water at reaching molecular freezing of of six-site enhancement studies the a using computational nanofilms earlier Models. Molecular introduction, Other with would Comparison and nucleation bulk region. by bulk well-developed dominated a be possess Finally, would region. films bulk well-developed and thick a dominant, possess be to to enough stress nucleation thick surface yet and for enough (density thin by be characteristics (q would connected bulk others are lacks some but first, regions has isotropy), the subsurface that In the region films. here, a thick study and we thin, which ultrathin, regimes: three least a Such compute nucleation. properly see to bulk needed not and be do would surface we separation between nucleation, separation ice of clear mechanism a and computationally rate be influ- both to profoundly ences surfaces find free we of presence that the although thicknesses tractable, film of range the agrta h expected the obtain than we larger 35, ref. in −1.5699 reported rate nucleation of the estimate an us rate nucleation metric pdbl ein nthat in region, in nm bulk 9 oped 5; Fig. in nm (4 studied have aus(otatwt i.6C Fig. with (contrast values u acltos nohrwrs ugs h xsec fat of existence the suggest words, other in calculations, Our ± .60 hc sruhyfu reso magnitude of orders four roughly is which 0.7680, log 10 J J v a r , q c [m J 6 ∼ (z a 6 and −2 O sgvnby given is ) n aedsrbto) hnfilms Thin distribution). cage and · and (10 s i.S2 Fig. −1 akapoel devel- properly a lack S1) Fig. −6 ψ ] 3.8829 = h hnfim htwe that films thin The m). (z ) o W.Hne within Hence, mW). for J NSEryEdition Early PNAS ee ec hi bulk their reach never smnindi the in mentioned As a = J ± v d r .91 Using 0.4961. c hsgives This /2. log . 10 r c | = [m] f6 of 5

APPLIED PHYSICAL SCIENCES algorithm with a time steps of 2 and 10 fs for the TIP4P/ice and mW sys- the number of water molecules, and the local q6 of molecule i, respectively. tems, respectively (47). In the TIP4P/ice system, rigidity constraints are main- For ρHC(z) and ρDDC(z) calculations, thicker 0.2-nm bins are used, and each tained by using SHAKE (48), and long-range electrostatic interactions are cage’s contribution is equally distributed among the bins that it intersects computed by using the particle–particle particle–mesh algorithm (49) with (see SI Text for details). For all film calculations, initial configurations are a short-range cutoff of 0.85 nm. Bulk and film simulations are carried out prepared by melting a slab of cubic ice at T = 350 K, saving independent in the NpT and NVT ensembles, respectively, with temperature and pressure configurations separated by a minimum of 0.4 ns, and gradually quenching controlled by using the Nose–Hoover´ thermostat (50, 51) and the Parrinello– each configuration to the final temperature of 230 K. Cage statistics in the Rahman barostat (52), respectively. Nucleation rates are computed using the bulk are computed in NpT MD simulations at 230 K and 1 bar, with the start- coarse-grained variant of the FFS algorithm described in ref. 35 with a sam- ing configurations prepared using the procedure described in refs. 29 and pling time of τ = 1 ps. The order parameter is chosen as the number of s 35. Further details of the conducted MD simulations are given in Table S2. water molecules in the largest crystallite, determined using the Steinhardt q6 order parameter (39) and the chain exclusion algorithm of Reinhardt (53). ACKNOWLEDGMENTS. We thank I. Cosden for his assistance in secur- The FFS calculation amounts to a total of 376 µs of MD trajectories, which ing computational resources for this calculation. P.G.D. was supported by is equivalent to 7.5 million CPU-hours on the Stampede supercomputer. Fur- National Oceanic and Atmospheric Administration Cooperative Institute for ther details of FFS are given in SI Text. HCs and DDCs are detected by using Climate Science Award AWD 1004131; and the Princeton Center for Complex the topological criteria outlined in ref. 35. For both q6 calculations and ring Materials, a Materials Research Science and Engineering Center (MRSEC) identification, a distance cutoff of 0.32 nm between oxygen atoms is used. supported by National Science Foundation (NSF) Grant DMR-1420541. These Spatial profiles of thermodynamic quantities are determined from con- calculations were performed on the Terascale Infrastructure for Ground- breaking Research in Engineering and Science at Princeton University and ventional isochoric, isothermal (NVT) MD simulations, using a simple bin- the Yale Center for Research Computing. This work also used the Extreme ning approach (38) with a bin thickness of 0.05 nm. The q6 spatial profile Science and Engineering Discovery Environment, which is supported by NSF PN PN is defined as q6(z) := h i=1 q6,iδ(zi − z)i/ i=1 δ(zi − z)i, with N and q6,i Grant ACI-1053575.

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