Thermodynamic origin of surface melting onice crystalsKen-ichiro Murataa,1, Harutoshi Asakawab, Ken Nagashimaa, Yoshinori Furukawaa, and Gen Sazakia

aInstitute of Low Temperature Science, Hokkaido University, Kita-ku, Sapporo 060-0819, Japan; and bGraduate School of Sciences and Technology forInnovation, Yamaguchi University, Yoshida, Yamaguchi 753-8512, Japan

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved September 8, 2016 (received for review June 2, 2016)

Since the pioneering prediction of surface melting byMichael Faraday,it has been widely accepted that thin water layers, called quasi-liquidlayers (QLLs), homogeneously and completely wet ice surfaces.Contrary to this conventional wisdom, here we both theoreticallyand experimentally demonstrate that QLLs have more than twowetting states and that there is a first-order wetting transitionbetween them. Furthermore, we find that QLLs are born not onlyunder supersaturated conditions, as recently reported, but also atundersaturation, but QLLs are absent at equilibrium. This means thatQLLs are a metastable transient state formed through vapor growthand sublimation of ice, casting a serious doubt on the conventionalunderstanding presupposing the spontaneous formation of QLLs inice–vapor equilibrium. We propose a simple but general physicalmodel that consistently explains these aspects of surface meltingand QLLs. Our model shows that a unique interfacial potential solelycontrols both the wetting and thermodynamic behavior of QLLs.

surface melting | quasi-liquid layer | advanced optical microscopy |pseudo-partial wetting | wetting transition

In general, surfaces and interfaces yield unique phase transi-tions absent in the bulk (1–5). Surface melting (or premelting)

of ice (3, 4) is one typical and classical example that has beenknown since the first prediction by Michael Faraday in 1842 (6).He hypothesized that thin water layers, now called quasi-liquidlayers (QLLs), wet ice crystal surfaces even at a temperaturebelow the melting point. Since then, this phenomenon hasattracted considerable attention not only because of its impor-tance in the fundamental understanding of melting (a solid-to-liquid transition) itself but also as a link to a diverse set of naturalphenomena in subzero environments: making snowballs, slippageon ice surfaces, frost heave, recrystallization and coarsening ofice grains, morphological change of snow crystals, electrificationof thunderclouds, and ozone-depleting reactions (3, 4, 7). Fur-thermore, it is now recognized that surface melting is not specificto ice but rather is universally seen in a wide range of crystallinesurfaces such as metals, semiconductors, ceramics, rare gases, andorganic and colloidal systems (8–12). Its underlying physics istherefore also inseparable from material science and technology.Although the origin of surface melting, including the nature of

QLLs themselves, is still far from completely understood and amatter of active debate (13–18), it is at least phenomenologicallybelieved that surface melting is driven by the reduction of thesurface free energy by the presence of intervening liquid betweenthe solid and gas phases (3, 4, 13, 19). More sophisticated ap-proaches have also been proposed in terms of surface phasetransitions (1, 3, 4, 20). In contrast to such theoretical specula-tions, however, the direct observation and the accurate charac-terization of QLLs by experiments are still highly challengingbecause of their thinness, assumed to be less than tens ofnanometers (21). Experimental efforts in the past have oftenbeen bedeviled by large uncertainties depending on the experi-mental methods and researchers (see table S1 in ref. 22 for de-tails). Even the first convincing evidence for the existence ofsurface melting of ice was not provided until 1987 (13, 14), morethan one century after Faraday’s suggestion. Thus, the conventional

theories, although rigorous themselves, have suffered from the lackof reliably experimental support.Recently, we succeeded in making in situ observations of

QLLs on ice surfaces using an advanced optical microscope(laser confocal microscopy combined with differential interfer-ence contrast microscopy: LCM-DIM), whose resolution in theheight direction reaches the order of an angstrom (22, 23).Surprisingly, this work revealed that, contrary to the commonbelief that QLLs completely and homogeneously wet ice sur-faces, they are spatiotemporally heterogeneous and are absent inthe equilibrium conditions (22, 24–26). Furthermore, we haveobserved that QLLs exhibit more than one wetting morphology:droplet type, thin-layer type, and their coexistence (sunny-side-uptype) at supersaturation (22, 24–26). This finding fundamentallyrequires us to recast the conventional understanding based onspatiotemporally averaged equilibrium theories and experiments(e.g., scattering, spectroscopy, and ellipsometry), because of ig-norance of the counterintuitive nature of QLLs.In this paper, we present a simple physical model bridging the

gap between the conventional interpretation and the above as-pects of surface melting based on in situ observations with ouradvanced optical microscopy combined with a two-beam in-terferometer. Here we revisit the thermodynamics of wetting(27). The general nature of surface melting suggests the rele-vance of the phenomenological approach. Starting from thephenomenological interfacial free energy, we robustly de-termine a full interfacial potential between ice and vapor in themedium of a QLL, governing both the selection and stability ofthe wetting states, and the thermodynamic condition for theexistence of QLLs. As a theoretical consequence, we extendthe concept of surface melting into nonequilibrium regimes,

Significance

Phase transitions of ice are a major source of a diverse set ofnatural phenomena on Earth. In particular, quasi-liquid layers(QLLs) resulting from surface melting are recognized to be keyplayers involved in various natural phenomena spanning frommaking snowballs to electrification of thunderclouds. With theaid of in situ observations with our advanced optical micros-copy combined with two-beam interferometry, we elucidate athermodynamic origin of the formation of QLLs and theirunique wetting behavior (pseudo-partial wetting and wettingtransitions) on ice surfaces. We show that QLLs are a meta-stable transient state formed through vapor growth and sub-limation of ice that are absent at equilibrium.

Author contributions: K.-i.M. and G.S. designed research; K.-i.M., H.A., K.N., Y.F., and G.S.performed research; K.-i.M. analyzed data; and K.-i.M. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Freely available online through the PNAS open access option.

See Commentary on page 12347.1To whom correspondence should be addressed. Email: murata@lowtem.hokudai.ac.jp.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1608888113/-/DCSupplemental.

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more specifically, supersaturation and undersaturation, which has asignificant implication for exploring the possible existence of thisphenomenon in a wider range of crystalline surfaces. Our modelprovides not only a clear-cut answer to the long-standing questionof the origin of surface melting of ice but also offers a general in-sight into the origin of surface melting of other solid–gas interfaces.

Results and DiscussionDescription of the Interfacial Free Energy and Interfacial Tension ofQLLs.A series of our recent studies (22, 24–26) have assumed thepresence of two different liquid phases (α- and β-QLL phases) toexplain a variety of wetting morphologies of the QLLs withoutcharacterizing the order parameter distinguishing these two liquidphases. Hereafter we will show that such a nontrivial assumption isnot necessary in the context of the standard wetting theory (27).We start from a simple layered system (having flat interfaces)

where a liquid (QLL) is sandwiched by solid (ice) and gas (va-por) with a thickness, e. The surface free energy of this systemcan be written as

FðeÞ= γSL + γ +PðeÞ, [1]

where γSL and γ are the solid–liquid (ice–QLL) and the liquid–gas (QLL–vapor) interfacial tension, respectively (27). We setPðeÞ as an interfacial potential that includes both long-rangevan der Waals (vdW) forces (e � a) and short-range forces(e∼ a). Here a is the molecular size (3.7 Å for water). BecauseQLLs are quite thin (9 nm) (21), the PðeÞ term cannot be ig-nored. Although we do not know its specific form, PðeÞ is trivialfor two limiting cases: Pð0Þ= γSV − γSL − γ = S and Pð∞Þ= 0,where γSV is the solid–gas (ice–vapor) interfacial tension andS the spreading coefficient. The former comes from Fð0Þ= γSVbecause e= 0 in this system corresponds to just the solid–gas(ice–vapor) interface. The latter is due to the nature of vdWforces generally showing power-law decay (28). Here note thatγ and γSL are not always constant but are thickness-dependentvariables in some cases. Although our model does not includevariables in γ and γSL explicitly, these can be incorporated intoPðeÞ consistently (Materials and Methods). Thus, our theoreticalframework is applicable regardless of whether γ and γSL areconstant or variable.Given FðeÞ explicitly, one can write the interfacial tension

under the existence of the effective interfacial potential as fol-lows (27):

γðeÞ= dðFðeÞAÞdA

− γSL = γ +PðeÞ+ eΠðeÞ, [2]

where A is the area of the wetting surface and ΠðeÞ=−dPðeÞ=dethe so-called disjoining pressure. The subtraction by γSL meansthat we now consider the interfacial tension not of the wholesystem but just of the film. Here the constant volume, Ae= const,is assumed to derive this equation. Strictly speaking, however, thiscondition weakly breaks down because the formation of QLLs oc-curs at supersaturation and undersaturation, and their amount canvary with time. Even so, the wetting morphology itself is determinedby the force balance in the equilibrium condition (where Ae= constis validated), as with the case of the usual heterogeneous nucleation.

Characterization of the Wetting Morphologies of QLLs. A key mor-phology for understanding the unique wetting behavior of QLLsis the so-called sunny-side-up-type state (Fig. 1A), also known aspseudo-partial wetting or as frustrated-complete wetting in somecases (29). First we consider this wetting state from the forcebalance of interfacial tensions. As shown in Fig. 1A, a dropletwets its own liquid film, whose thickness is em, with a contactangle θE. In this state, two kinds of force balances hold (Fig. 1B),the first of which is simply represented as γe = γ cos θE (at point

A). Here γe is the interfacial tension of the film with em, givenexplicitly by γe = γ +PðemÞ+ emΠðemÞ from Eq. 2. The second isthe force balance acting on the film in the horizontal direction,written as γe = γSV − γSL. In addition to these, coexistence betweentwo different thicknesses (e= 0 and em) requires Pð0Þ= Sc =PðemÞ+emΠðemÞ (Materials and Methods). Substitution of this relation into γestraightforwardly yields γe = γ + Sc. This can be rewritten as followsusing the relation of γe = γ cos θE:

Sc = γðcos θE − 1Þ. [3]

Due to the finite contact angle θE, the spreading coefficient forthis state is of negative sign. From the interference image in Fig. 1Cits value is estimated as 0.6°, consequently giving Sc =−4× 10−6 N/m.Note that the value of bulk water at 0  °C (7.56 ×10−2 N/m) was usedas γ. Furthermore, the combination of the force balance equa-tions at points A and B recovers the Young–Dupré equation:

cos θE = ðγSV − γSLÞ=γ. [4]

This means that, for S= Sc, partial wetting coexists with thepseudo-partial wetting with the same contact angle θE. Actually,the interference image in Fig. 1C, showing coexistence of thesetwo states, demonstrates that the contact angle of the droplet

Ice γS L γS V

Water(QLL)

γγe γeθE

A BIce

Water(QLL)γθA

γS V γS L

λ/2

λ/2

A

Bare ice surface

B

C

D

E

F

Bare ice surface

Thin liquid layer

Bare ice surface

Fig. 1. Interferometric analysis of the pseudo-partial and partial wettingstate. (A) A characteristic form of pseudo-partial wetting. The blue and theblack arrows correspond to the growth of an elementary step on the barebasal faces and that in a QLL. (B) Schematic of the cross-sectional view ofpseudo-partial wetting. The force balances of the interfacial tensions actingat point A and B. (C) An interferogram (top view) of the coexistence of thepseudo-partial (blue arrow) and the partial wetting (red arrow) states atsupersaturation (T = −0.6 °C). The fringes shift in response to the heightprofile of the droplets. Note that the thickness of thin liquid layers is belowthe detection limit of this interferometer (9 nm) (21). (D) A characteristicform of partial wetting. The blue arrow indicates the growth of an ele-mentary step. (E) Schematic of the cross-sectional view of partial wetting.The force balances of the interfacial tensions hold at point A. (F) An in-terferogram (top view) of the partial wetting state at supersaturation (T =−0.6 °C). Here, the spacing between adjacent interference fringes corre-sponds to a height difference of a half-wavelength, (λ=2= 316.5 nm). (Scalebars in A, C, D and F: 20 μm.)

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(the red arrow) is 0.8°, which almost coincides with θE. Here,note that the change in γ induced by the curvature effect issufficiently small to ignore because the curvature of the dropletis negligible due to its extremely small contact angle (also dis-cussed below). This also validates the assumption in Eq. 1 that γis constant.Next we focus on the partial wetting state (Fig. 1D). Consider

again the force balance acting at point A (Fig. 1E). In this state,the thin liquid layer emerging ahead of the nominal contact lineis spontaneously destabilized, which requires γSV − ðγe + γSLÞ< 0.Inserting γe = γ + Sc in this equation yields S< Sc as a criterion forpartial wetting. As well as pseudo-partial wetting, we can experi-mentally assess the value of the contact angle as 2.3° using the in-terferometer (Fig. 1F). We consequently obtain S=−6.1× 10−5 N/mfrom the well-known relation of S= γðcos θ− 1Þ, which actuallysatisfies S< Sc. It is noteworthy that S and θ are varied by thetemperature and the vapor pressure.

Link Between the Interfacial Potential and the Wetting Behavior ofQLLs.Although no specific forms have been defined in PðeÞ so far,the presence of pseudo-partial wetting imposes a strong re-striction on its shape because PðeÞ should have a local minimumcorresponding to the thin-layer state. From Sc < 0 and the co-existence condition (Materials and Methods), each PðeÞ must bedrawn as shown in Fig. 2. Note that the chemical potential dif-ference between ice and vapor, Δμ= kBT lnðp=peÞ (kB and pebeing the Boltzmann constant and equilibrium vapor pressure ofice, respectively), is not explicitly included in PðeÞ (strictly

speaking, eΔμ) because this term is cancelled in PðeÞ+ eΠðeÞ, andthus has no influence on the stability and the coexistent condi-tion of QLLs. We summarize below the tangent construction ineach PðeÞ according to the type of wetting state. First, we showPðeÞ for partial wetting in Fig. 2A (S< Sc). In this case, co-existence between e= 0 and e→∞ (droplet) is energetically fa-vored. Although one can also draw the common tangent to e= 0and em, this coexistence, indicating thin liquid layers, has higherenergy than that between e= 0 and e→∞. In Fig. 2B, we showPðeÞ for pseudo-partial wetting (S> Sc). There exist two types ofcoexistence between e= 0 and em′ (thin layer), and between emand e→∞ (sunny side up), both of which are acceptable as thestable wetting state. Which states are selected basically dependson the total amount of QLLs present. In contrast, coexistencebetween e= 0 and e→∞, the partial wetting state, whose contactangle is less than θE, is energetically disfavored. Finally, we showin Fig. 2C the boundary case of S= Sc. We can see only onecommon tangent to e= 0, em and e→∞. This means that threetypes of wetting morphologies, droplets, thin layers, and/or sunnyside up, are allowed to coexist with each other. In this case, asdiscussed above, the pseudo-partial and the partial wetting statecoexist with the same contact angle θE (Fig. 1C). We concludethat the competition between S and the local minimum of PðeÞsolely controls the wetting transition of QLLs. The tangentconstruction of PðeÞ also allows us to characterize its local min-imum straightforwardly. As shown in Fig. 2C, on noting ΠðemÞ=0 for S= Sc, PðemÞ is simply given by Sc =PðemÞ= −4 ×10−6 N/m .Furthermore, our recent analysis of the spreading dynamics ofQLLs revealed em = 9 nm (21). Here we note that Elbaum andSchick (30) have predicted the presence of the local minimum inthe vdW potential with the Lifthitz theory. Remarkably, ourexperimental characterization of the local minimum in PðeÞagrees well with their theoretical prediction [em = 3.6 nm,PðemÞ=−4.4× 10−6 N/m (30)]. This excellent coincidence indi-cates that the potential minimum purely comes from the long-range vdW potential, governed by the dielectric properties of ice,water, and vapor (28, 30). Because of the absence of the di-electric anomaly at the triple point, the shape of the vdW po-tential is expected to display almost no change with temperatureand vapor pressure in our experimental regime. This implies thatthe thickness of thin liquid layers does not diverge but remains afinite value, even when approaching the triple point.

Instability of Thin Liquid Layers: Rupture by Dewetting. Examiningthe (in)stability of thin liquid layers provides direct insight intothe short-range nature of PðeÞ (a< e � em), which is not acces-sible by the Lifthitz theory (a being the molecular size). Here wefollow the rupturing dynamics of thin liquid layers by dewetting.We found two types of transformation kinetics by quenching athin layer to droplet stable regimes: (i) hole nucleation andgrowth (Fig. 3A and Movie S1) and (ii) spinodal dewetting (Fig.3B and Movie S2). Note that these two are the characteristicpathways observed in dewetting of thin liquid films, which is,more generally, reminiscent of the first-order phase transition(31). For i, holes exposing a bare ice surface are nucleated in thematrix of a thin liquid layer and grow with time. Hole nucleationoccurs stochastically when overcoming the potential barrier byinterfacial fluctuations (discussed in the next section), whichmeans that the thin liquid layer is a metastable wetting mor-phology against the droplet. Along the line of the tangent con-struction discussed above, PðeÞ for i should be illustrated asshown in Fig. 3C. Due to the metastability, P″ðeÞ is of positivesign around em despite S< Sc, and PðeÞ has two common tan-gents, corresponding to the stable and the metastable wettingstate. For ii, however, the thin liquid layer is spontaneouslydestabilized with accompanying growth of zig-zag-like interfacialfluctuations, and finally transforms into droplets as a result ofcoarsening. This means that the thin liquid layer becomes unstable

P(e)e

P(e)e

P(e)e

Sc

S

S

A

B

C

es

em

es

es

e's

e's

e's

em

eme'm

Fig. 2. Schematic representation of interfacial potentials: (A) partial wet-ting, (B) pseudo-partial wetting, and (C) coexistence of the partial andpseudo-partial wetting states. The red solid lines indicate common tangentscorresponding to the stable state and the blue dashed lines show the tan-gents corresponding to the metastable state. Here, em′ and em in B corre-spond to the thickness of the thin layer and that emerging ahead of thenominal contact line, respectively (em′ < em). The upper and lower stabilitylimits are indicated by es and es′ , respectively [P″ðesÞ= P″ðes′ Þ= 0].

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against the droplet; P″ðemÞ≤ 0. Fig. 3D shows PðeÞ for ii; thereexists only one common tangent corresponding to the droplet state.Here, note that the spinodal ring clearly appears at 2.69 s and thewavelength of the fast growing mode is λm = 2π=qm = 3.6 μm,where qm is the peak wavenumber (Fig. 3E). These experimentalconditions indicate that the temperature gap between the bistable(coexistence or binodal; S= Sc) and the unstable (spinodal) statelies within only a 0.5-K window, strongly suggesting that the localminimum of PðeÞ is quite shallow.The vdW potential proposed by Elbaum and Schick (30) is

composed of the nonretarded repulsive part, favoring thickeningof wetting films, and the retarded attractive part, favoring thinning,the competition of which yields the minimum in the potential. Asstated above, the proposed vdW potential qualitatively supports ourobservations. Therefore, we can decompose PðeÞ as sketched in Fig.3F, which uncovers the existence of a short-range attractive po-tential not taken into account in the vdW approach. Although itsorigin is not clear at this stage, the structural coupling between γSLand γ may be a promising candidate for it. It is well known thatwater molecules localized in the vicinity of vapor are highly struc-tured (32, 33), which may reduce γSV due to their structural simi-larity when the two interfaces approach each other. This means thatγSL is a thickness-dependent variable, γSLðeÞ, which can be alsoregarded as a kind of interfacial potential (Materials and Methods).Because the correlation length of the structural ordering induced bythe vapor is about 1 nm at most (32), γSLðeÞ is expected to imme-diately recover the bulk value with increasing e, which can plausiblyexplain the behavior of this short-range potential shown in Fig. 3F.

Thermal Fluctuation Effects: Droplet Nucleation from Thin LiquidLayers. In earlier discussions, we have ignored thermal fluctua-tion effects on the interface between QLL and vapor. Surpris-ingly, our advanced optical microscopy is fully capable of in situvisualizing not only QLLs themselves but also their interfacialfluctuations in real space, which is conventionally accessible onlythrough scattering techniques (34) and colloidal systems (35). InFig. 4, the interfacial fluctuations of a thin liquid layer are foundto be thermally excited, which induces nucleation of droplets fromthe thin layer covering the ice surface (Movie S3). For small fluctu-

ations (∇e � 1), the interfacial Hamiltonian can be described as

H= γeRdx½

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ ð∇eÞ2

q−1�∼ ðγe=2Þ

Rdxð∇eÞ2= ðγe=2Þ

Pkjekj2k2,

where ek is the thickness of the film for wavevector k in Fourierspace. Here we used the constant γe instead of including PðeÞ ex-plicitly in this Hamiltonian. This approximation is permitted at thefirst-order level because the thin layer is the state stably trapped inthe local minimum in PðeÞ. Using the equipartition theorem, westraightforwardly obtain the following relation: hjekj2i=kBT=γek

2.Hence, the correlation length in a direction perpendicular to theliquid film is given by ξ2⊥ =kBT

R ½dk=ðγek2Þ� ¼ ½kBT=ð2πγeÞ�  ln ξ∥a.

Note that a and ξ∥ in the factor are the small and the large wave-length cutoff for the integration, the latter of which is the lateralcorrelation length, corresponding to the so-called healing length(27). Here we roughly estimated ξ∥ as 5 μm (the same order as λm)from Fig. 4C and Movie S3. From the earlier estimation of γe, wecan calculate ξ⊥ as 0.3 nm. This means that the layer thicknessfluctuates with ξ⊥ in the well of the potential minimum (Fig. 2).Droplet (hole) nucleation is triggered when e reaches beyond theupper (lower) stability limit, es∼ em+ ξ⊥ (es′∼ em−ξ⊥). Therefore,for sufficiently large layers where we can ignore the suppression ofthe fluctuation by the finite size effect (� ξ∥), the interfacialfluctuation helps nucleation, a transition from thin layer to sunnyside up (or from thin layer to droplet).

Thermodynamic Origin of the Formation of QLLs. A thermodynamiccriterion for the equilibrium presence of QLLs is given by thebalance of the interfacial free energy between wet and bare icesurfaces: FðeÞ−Fð0Þ=ΔF =PðeÞ− S. This interfacial term is ofcrucial importance in the phase behavior because the chemicalpotentials of ice, water, and vapor are almost all the same nearthe triple point. For ΔF < 0, ice surfaces lower the total freeenergy by ΔF by getting wet, which drives the spontaneous for-mation of QLLs and permits the equilibrium surface melting.However, we previously found the absence of QLLs at and nearthe ice–vapor equilibrium (26). As shown in Movie S4, a droplet-type (partially wet) QLL gradually disappears on an ice surfaceat the ice–vapor equilibrium (T =−1.5  °C and p= 540 Pa) forsufficiently long time to observe the whole kinetics directly. Note

P(e) e

S

em

P(e) e

S

emP''(e )<0m

P''(e )>0m

P(e)

e

S vdW potential

short-range potential

repulsive

attractive

A

B

F

C D

0.00 s 2.69 s 6.14 s

0.00 s 21.4 s 40.4 s E

Fig. 3. Rupturing dynamics of thin liquid layers. (A and B) Morphological evolution of hole nucleation and growth (T = −0.6 °C and p= 609 Pa; see Movie S1)and spinodal dewetting (T = −1.0 °C and p= 609 Pa; see Movie S2), respectively. Here, the differential interference contrast was adjusted as if the ice crystalsurface was illuminated by a light beam slanted from the upper left to the lower right direction. Thus, convex (concave) objects showed brighter (darker) anddarker (brighter) contrast on the upper left sides and the lower right sides, respectively. (Scale bars: 10 μm.) (C and D) The interfacial potential correspondingto A and B, respectively. Note that in A and B we first quenched the thin liquid layers prepared at T = −0.5 °C and p= 609 Pa (S≥ Sc), whose interfacialpotential is shown in Fig. 2C, to the droplet stable regimes (or metastable and unstable regimes for thin layers) and then observed their dynamics at aconstant temperature. (E) A radially averaged scattering function of the image (2.69 s) in B. (Inset) Its 2D power spectrum (2.69 s). The red curve is the fit withGaussian. (F) The decomposition of the full interfacial potential into the short-range attractive and the long-range vdW potential. In the vdW potential, theattractive part originates from retardation whereas the repulsive part comes from the native potential associated with no retardation (30).

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that we can strictly define the ice–vapor equilibrium by observingadvancing and receding elementary steps with our advancedmicroscopy. On noting that ΔF > 0 for partial wetting as shown inFig. 2A, ΔF is found to be positive at and near the ice–vaporequilibrium. For ΔF > 0, ice surfaces pay the penalty by ΔF to thetotal free energy when forming QLLs, which rules out theirequilibrium presence. This fact casts a serious doubt on theconventional theory presupposing S> 0 and ΔF < 0, that is, thespontaneous formation of QLLs in ice–vapor equilibrium (e.g.,see refs. 3, 4, 13, and 19). Movie S4 indicates that even one smalldroplet, whose radius is only 15 μm, takes 19 min to completelydisappear, meaning that QLLs survive persistently on ice sur-faces even in the thermodynamically unstable region once theyform. Such a long timescale of the equilibration (the annihilationkinetics of QLLs) may mimic the equilibrium presence of QLLs.Despite the absence of QLLs in ice–vapor equilibrium, they

can form as an intermediate state on their way to ice from vapor,or vice versa. Actually, we recently obtained supportive evidencethat QLLs form kinetically under supersaturation conditions asthe elementary steps of ice advance (22, 24, 26). A plausible

explanation of this behavior is that the relationship of the freeenergy between vapor and QLL is reversed by crossing thevapor–water equilibrium line extrapolated inside the ice stableregime, which enables a vapor–ice transformation via metastablesupercooled water. Although all of our observations of QLLs sofar have been performed under supersaturated conditions (22,24–26), this scenario further proposes the formation of QLLsat undersaturation.To demonstrate this experimentally, we directly followed the

formation and the annihilation process of QLLs in undersatu-rated conditions (Fig. 5). In Fig. 5A and Movie S5, we found that,starting from the equilibrium point and increasing the temper-ature at a constant vapor pressure, the bare ice surface is firsthollowed by sublimation as the steps of ice recede, and thennucleation and growth of droplet-type QLLs follow from thehollowed surface. With further increasing the temperature thinliquid layers subsequently appear through both nucleation andspreading of the existing droplets. Such a formation via nucle-ation reflects the presence of the energy barrier arising from theinterfacial tension (for droplets), and that of the fringe and theline tension (for thin layers). Note that these tensions surviveeven for thin layers with ΔF =PðemÞ− S< 0. We also confirmedthe same nucleation (and spreading) behavior under supersatu-ration conditions (Movies S1 and S2). Therefore, QLLs can beregarded as a metastable state possessing an intermediate freeenergy between ice and vapor. Moreover, we checked the re-verse process (Fig. 5B and Movie S6); first thin liquid layerstransform into droplets through dewetting, and then the drop-lets shrink and finally disappear on approaching the ice–vaporequilibrium point.Fig. 5C shows the wetting and phase behavior of QLLs in the

p−T plane. The red symbols represent the points at which QLLsdisappear on ice surfaces, whereas the blue symbols represent thepoints at which thin layers are destabilized and transform intodroplets through dewetting. The former process, disappearanceof QLLs, is a normal first-order transition, exhibiting hysteresiscaused by the metastablity; their disappearance occurs at adifferent temperature from their formation (via droplet nu-cleation) at a fixed vapor pressure, as shown in Movies S5 andS6. In contrast, although the latter process, the wetting tran-sition of QLLs, is also first-order, the metastable wetting stateis easily and immediately destabilized due to interfacial fluc-tuations of QLLs. Actually, we have confirmed no hysteresis;dewetting (complete-to-partial wetting) and spreading (partial-to-complete wetting) occur at almost the same temperature andvapor pressure (Movies S5 and S6). Thus, the blue lines cor-respond to thermodynamically defined wetting transition lines,satisfying S= Sc.Here it is also worth noting that no hysteresis in the wetting

transition rules out the possibility that metastable thin liquidlayers remain to cover ice surfaces even in ice–vapor equilibrium.In addition, the absence of interfacial fluctuations on ice surfacesis another direct piece of evidence denying this possibility. Notethat ice basal faces have no roughening transition below themelting point (25), which means that there exists no interfacialfluctuation on bare ice basal faces. Hence, if there exist thinliquid layers in this condition, their interfacial fluctuations mustbe observed. However, our direct observations have never de-tected these in this condition.In Fig. 5C we found that the metastable coexistence lines of

QLLs for both supersaturation (QLL–vapor) and undersaturation(QLL–ice) significantly deviate from those of bulk water, al-though the slope of each line shows a similar relationship tothe corresponding bulk metastable lines. Particularly for thecase of undersaturation, the metastable line of QLL penetratesinside the region where water never exists even as an intermediatestate, which cannot be explained by only kinetic effects: the bal-ance of a growth or evaporation rate (a mass transfer of water

A

B 0.0 s 4.0 s 7.9 s

C

Bare ice surface

Fig. 4. Interfacial fluctuations of thin liquid layers. (A) QLLs spreading onthe ice basal face (T = −0.1 °C and p= 603 Pa). The pink arrowheads indicateelementary steps on the bare ice basal face. Interfacial fluctuations were notobserved at the ice–vapor interface (the bare basal face), in contrast to theQLL–vapor interface (see Movie S3). (B) Droplet nucleation induced by thethermal fluctuation of thin liquid layers covering the ice surface (the areasurrounded by the blue frame in A). The blue arrows indicate the birth of thenew droplets. (C) The differential images of the fluctuation obtained from A(see also Movie S3). (Scale bars: 20 μm.)

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molecules) among three phases. Thus, this deviation suggests thatthe thermodynamic properties of QLLs themselves are differentfrom those of bulk water, which is consistent with our recentstudy assessing the surface tension-to-shear viscosity ratio ofQLLs from their wetting dynamics (21).We summarize in Fig. 6 a thermodynamic route to form QLLs

on ice surfaces. For weak supersaturation (undersaturation) nearthe vapor–ice equilibrium where the free energy of a QLL ishighest among the three phases, vapor (ice) directly transformsinto ice (vapor) through the layer-by-layer process driven by 2Dnucleation and/or spiral growth (surface hollowing mediated bydefects). This is the standard process of crystal growth based onthe terrace-step-kink picture (23, 36–38). However, far from thevapor–ice equilibrium where the free energy of a QLL is in-termediate between that of ice and vapor, vapor (ice) transformsinto ice (vapor) with accompanying nucleation and growth ofmetastable QLLs on growing (sublimating) ice surfaces. It istherefore fair to categorize the formation of QLLs not into justsurface melting in the conventional and ambiguous way, but intotwo distinct types, surface condensation for supersaturation andsurface melting for undersaturation. Here we note that, althoughwe have observed a signature of the advancing elementary stepsin thin liquid layers (black arrow in Fig. 1A), a detailed mecha-nism of crystal growth at QLL–ice interfaces, and more generallyliquid–crystal interfaces, is still unknown and remains an in-teresting topic for future studies.Finally, we briefly remark on a possible link between surface

melting and the thermal roughening transition. In our previousstudy (25) we investigated this link by observing QLLs on theprism face of ice crystals, exhibiting the roughening transition at−4 °C ∼ −2 °C. We have demonstrated that both the wettingbehavior of QLLs and the condition of their existence on theprism face are basically same as those on the basal face, havingno roughening transition below the bulk melting point, whichtells us that the thermal roughening and surface melting areseparate phenomena and do not compete with each other.

ConclusionsWe have shed a light on the nature of the wetting transition ofQLLs and the stability of each wetting state from statical anddynamical viewpoints, including their interfacial fluctuation ef-fects. With support from in situ observations with our advancedoptical microscopy combined with two-beam interferometry, wehave demonstrated that the unique shape of PðeÞ is responsiblefor the QLL wetting behavior, and it also plays a prominent rolein determining the thermodynamic condition for the existence ofQLLs. This implies that the wetting and the phase behavior areseriously affected by the change in PðeÞ, easily achieved by in-troducing defects on ice surfaces (24) and by replacing vaporwith acid gases (39), metals, or other dielectrics (40) in oursystem. From a more general perspective, our model also sug-gests that QLLs can manifest not only at equilibrium, as con-ventionally proposed, but also in nonequilibrium conditions viaan Ostwald-like pathway (Fig. 6), owing to the nature of PðeÞ.Thus, by extending the focus into the nonequilibrium regime,QLLs might potentially be observed on diverse crystalline sur-faces where they are as yet unknown. We also note that ourtheoretical framework is widely applicable regardless of the kindof substance due to its phenomenological description.In addition, our findings may apply to another longstanding

problem: how water–vapor interfaces affect freezing in their vi-cinity, which plays a crucial role in ice crystallization of super-cooled water nanodroplets (32, 41–43). In contrast to thenucleation in the bulk, at the nanoscale not only S, which isconventionally discussed (41), but also PðeÞ becomes an impor-tant factor governing the ice nucleation behavior. Our modelsuggests that ΔF =PðeÞ− S is responsible for whether the icenucleation is induced (ΔF > 0) or suppressed (ΔF < 0) by thewater–vapor interface.Finally, one important question still remains unanswered. Why

are ice surfaces not wet by their own melt, water, even thoughthese two are made of same molecules? Elucidating its micro-scopic origin, which is intrinsically linked to the short-range

A BC

p (P

a)

T( C)

-1.3 C

-1.1 C

-0.6 C

-0.4 C

-0.3 C

-0.5 C

-0.6 C

-0.9 C

Fig. 5. Formation and annihilation of QLLs at undersaturation. (A) A formation process of QLLs on a bare ice surface upon heating from the ice–vaporequilibrium point while keeping the vapor pressure constant, p= 555 Pa (Movie S5). (B) A reverse annihilation process of QLLs upon cooling from the thinlayer stable regime at the same constant vapor pressure as A (Movie S6). For the differential interference contrast see Materials and Methods. (Scale bars:50 μm.) (C) Metastable p− T phase diagram near the triple point. The blue symbols represent the transition points at which thin layers are destabilized andtransform into droplets through dewetting (S= Sc) and the red symbols represent the points at which QLLs disappear on ice surfaces. The solid colored linesare guides for the eye. The solid black lines are the ice–vapor and the ice–water equilibrium lines. The filled black circle indicates the triple point. The blue andthe pink dashed lines correspond to the metastable coexistent lines of QLLs for QLL–vapor (blue) and QLL–ice (pink), respectively. The data plotted in the super-saturation regime, shown by the open symbols, were recently obtained by Asakawa et al. (26). Here we note that, for red symbols, the data for undersaturation (thegreen area) are rather more scattered than those for supersaturation. This is because undersaturated conditions hollow clear ice faces by sublimation, making ourmicroscopy observations difficult, and finally making the ice surfaces themselves completely disappear. This hampers observations over sufficiently long time mea-surements to deal with the slow kinetics of the shrinking and disappearance of QLLs, which leads to the large variations in the data.

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attractive potential in PðeÞ predicted in Fig. 3F, will lead to amore fundamental understanding of this phenomenon. This willalso provide a quantitative basis for our decomposition of PðeÞshown in Fig. 3F. Actually, a similar low wettability behavior ofmelts on their own crystal surfaces has already been observed inalkali halide crystals and some metals (44) and has recently beeninvestigated for the NaCl (100) surface by using extensive mo-lecular dynamics simulations (45). A similar numerical approachbeyond the continuum picture would also be a promising path tofurther studies of our system. Experimentally, the direct andsystematic assessment of the temperature and the vapor pressuredependence of S and θ are also important to understanding thephase behavior of QLLs more deeply.

Materials and MethodsAdvanced Optical Microscopy. Our advanced microscopy system is a combi-nation of laser confocal microscopy and differential interference contrastmicroscopy (LCM-DIM). The laser confocal system is designed to maximize thesensitivity (the height resolution) of differential interference contrast mi-croscopy by avoiding deterioration of a high polarization ratio of the laserbeam and by reducing the noise attributed to scattered light from out-of-focus planes. We used a confocal system (FV300; Olympus Optical Co. Ltd.)attached to an inverted optical microscope (IX70; Olympus Optical Co. Ltd.).A super luminescent diode (ASLD68-050-B-FA, 680 nm; Amonics Ltd.) and aHe-Ne laser (05-LHP-991, 633 nm; Melles Griot) were used for LCM-DIMobservations and interferometric observations, respectively. Further detailedinformation of our microscopy is presented in refs. 22–24. In this study, thedifferential interference contrast was adjusted as if the ice crystal surfacewere illuminated by a light beam slanted from the upper left to the lowerright direction. Thus, convex (concave) objects showed brighter (darker) anddarker (brighter) contrast on the upper left sides and the lower right sides,respectively, compared with a flat crystal surface.

Samples. In this study, we used ultrapure water (>18.2 MΩ · cm) as a source ofice and vapor. The purity of dry nitrogen gas filling the observation chamberis more than 99.99% (oxygen ≤ 50 vol ppm and the dew point ≤−58 °C). Thesurface cleanliness of ice achieved by our experimental setup is generally farfrom the cleanliness achieved by so-called ultrahigh vacuum experiments.

Even so, it is worth noting that the pinning of elementary steps typicallyinduced by surface impurities (46), which may affect the behavior of QLLs,has never been observed in all of our experiments. It therefore seems thatour ice crystal surfaces are clean enough for our purposes and surface im-purities can be ignored.

Control of Temperature and Vapor Pressure of Samples. The observationchamber is composed of upper and lower Cu plates whose temperatures areseparately controlled using Peltier elements. At the center of the upper Cuplate, a cleaved AgI crystal was attached as an ice nucleating agent. We grewIce Ih sample crystals on this AgI crystal at −15 °C from supersaturated vaporin a nitrogen environment. We also prepared other ice crystals on the lowerCu plate, as a source of vapor to the sample ice crystals. The vapor pressurein the chamber is controlled by the temperature of the source ices, that is,their equilibrium vapor pressure at the corresponding temperature becausea larger amount of source ices are attached to the lower plate comparedwith the sample crystals on the upper one. Thus, separate control of thetemperatures of the sample and source ice crystals allows us to adjust thetemperature of the sample (T) and vapor pressure (p) independently. Notethat p is the partial pressure of H2O in a nitrogen environment. The totalpressure is the atmospheric pressure. In this study, the observation of QLLswas only performed on basal faces of ice crystals.

Thickness Dependence of Interfacial Tensions. Here, we shall simply assumethat γSL is a function of e in Eq. 1. For instance, this corresponds to the casethat γSL is affected by the presence of the (highly structured) water–vaporinterface. First we set γSLðeÞ as

γSLðeÞ= γ0SL −ΔγfðeÞ, [5]

where γ0SL is γSLð∞Þ (the bulk value of γSL), Δγ is the deviation of γSL led by thecoupling between the two surfaces, and fðeÞ is a thickness-dependent con-tribution decaying from 1 to 0 as e increases from 0 to ∞. We then re-construct FðeÞ in Eq. 1 as

FðeÞ= γ + γ0SL + PðeÞ−ΔγfðeÞ. [6]

Thus, even when γSL is a variable, our theoretical framework still holds,replacing PðeÞ−ΔγfðeÞ with P′ðeÞ. Conversely, γSLðeÞ can be regarded as akind of the interfacial potential.

Coexistence Between Different Thicknesses. Coexistence between two dif-ferent thicknesses (e′ and e″) requires that the same interfacial tensionacts between e′ and e″: γðe′Þ= γðe″Þ. This condition is explicitly repre-sented as (27)

Pðe′Þ+ e′Πðe′Þ=Pðe″Þ+ e″Πðe″Þ. [7]

This demonstrates that, under the coexistence between e′ and e″, one candraw a common tangent in PðeÞ at these two points. By replacing e and ΠðeÞwith c and μ, respectively, this tangent construction can be regarded as thecoexistence condition for a phase separated system. Note that c and μ are aconcentration of a binary mixture and its chemical potential, respectively.

However, we need to pay attention when dealing with e→∞, where thewetting morphology becomes the droplet due to its escaping from the in-fluence of the interfacial potential. Strictly speaking, our model cannotdescribe a partial wetting state appropriately, assuming a simple layeredsystem such that the thickness of QLLs is only one variable. Note that thepositional dependence of e (or eðxÞ) is not taken into account here. Forexample, according to Eq. 7, coexistence between e=0 and e→∞ (partialwetting) incorrectly yields Pð0Þ= Pð∞Þ, that is, S= 0, meaning completewetting of a macroscopic film. Although this relation, or the force balance inthe horizontal direction, is correct on the top of the droplet, γ should bereplaced by its cosine component in Eq. 2 except on the top. In particular, atthe vicinity of the contact line, however, where e→∞ still holds, the cosinecomponent can be written using the macroscopic contact angle as γ cos θ.This replacement again gives the correct coexistence relation between e= 0and e→∞.

ACKNOWLEDGMENTS. We thank Y. Saito and K. Ishihara (Olympus Engi-neering Co., Ltd.) for their technical support of LCM-DIM and G. Layton(Northern Arizona University) for providing AgI crystals. This work waspartially supported by Grants-in-Aid for Young Scientists (A), ScientificResearch (A), and Challenging Exploratory Research Grants 16H05979,23246001, 15H02016, and 24656001 from the Japan Society for the Pro-motion of Science.

Vapour

Ice

Vapour growth

Sublimation

QLL QLL

Evaporation

Nucleation

Nucleation

Melt growth(layer by layer ?)

Supersaturation Undersaturation

2D nucleation

Spiral growth

Surface hollowing

ice surface

ice surface

ES

Fig. 6. A thermodynamic pathway to form QLLs. For weak supersaturation(undersaturation) near the vapor–ice equilibrium where Gice <Gvapor <Gqll

(Gvapor <Gice <Gqll), vapor (ice) directly transforms into ice (vapor) throughthe layer by layer process following the terrace-step-kink picture (23, 36–38).Here note that Gice, Gqll, and Gvapor are the bulk free energy of ice, QLLs, andvapor, respectively. However, far from the vapor–ice equilibrium where thefree energy of QLLs enters between that of ice and vapor, Gice <Gqll <Gvapor

(Gvapor <Gqll <Gice), vapor (ice) transforms into ice (vapor) with accompany-ing nucleation and growth of metastable QLLs on growing (sublimating) icesurfaces. Based on this picture, the formation of QLLs can be classified intotwo distinct processes, surface condensation for supersaturation and surfacemelting for undersaturation.

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