A Review of Condensation Frosting

A Review of Condensation FrostingSaurabh Nath, S. Farzad Ahmadi, and Jonathan B. Boreyko

Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, Virginia, USA

ABSTRACTThe accretion of ice and frost on various infrastructure is ubiquitous in cold andhumid environments, causing economic losses amounting to billions of dollarsevery year worldwide. The past couple of decades have seen unprecedentedadvances in the fields of surface chemistry and micro/nanofabrication,enabling the development of hydrophobic and superhydrophobic surfacesthat promote facile deicing and/or passive anti-icing. However, in the light ofnew discoveries regarding the incipient stages of frost formation, it is becom-ing increasingly clear that the problems of icing and frosting are not one andthe same. Thus, passive anti-icing strategies do not exhibit anti-frosting beha-vior, and the development of passive anti-frosting surfaces remains anunsolved problem. In this review, we provide a critical discussion of condensa-tion frosting and show how the emerging new phenomena of frost halos,interdroplet ice bridges, and dry zones that comprise the incipient stages offrosting set it apart from the conventional problem of icing. Subsequently, wediscuss possible strategies to break the sequential chain of events leading topervasive frost growth.

ARTICLE HISTORYReceived 31 August 2016Accepted 29 October 2016

KEYWORDSFrost; condensation; anti-frosting; icephobicity; icebridges

Introduction

In the fourth century BC, Aristotle wrote [1]:

Let us now deal with the most remarkable conditions which are produced in and around the earth, summariz-ing them in the barest outline. . . . Dew is moisture of fine composition falling from a clear sky; ice is watercongealed in a condensed form from a clear sky; hoar-frost is congealed dew. . . .

It is astonishing to think that over 2,000 years ago, Greek philosophers had such a remarkableunderstanding of condensation and phase change. Indeed, in this quote, Aristotle correctly identifiesthe two possible modes of frosting—desublimation, which is a direct transformation of water vaporto ice—and condensation frosting, where the vapor first condenses into supercooled dew dropletsthat subsequently freeze [2, 3]. However it was not until 1657, 15 years after Galileo’s death, that thefirst systematic experiments on the freezing of water in an enclosed jar were performed to investigateGalileo’s anti-Aristotelian claims that water when frozen becomes lighter than water itself [4].

We have come a long way since and yet we have not. Despite significant advances in the under-standing of ice physics [3, 5–20], we are far from solving the icing problem. From an economicstandpoint, ice accretion today is a multibillion dollar problem in the United States alone [21].Frosting and icing adversely affect multiple industries, including aviation, telecommunication, electricaltransmission, hydropower, wind power, oil rigs, and almost all modes of transportation [22].Accumulation of ice on airplane wings can significantly alter the dynamic characteristics of aircraftflight, causing severe damage and even plane crashes [23–27]. Icing and frosting have also been shown to

CONTACT Jonathan B. Boreyko [email protected] Department of Biomedical Engineering and Mechanics, Virginia Tech,495 Old Turner Street, MC 0219 Norris Hall, Blacksburg, VA 24061, USA.Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/umte.© 2017 Taylor & Francis

NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERINGhttp://dx.doi.org/10.1080/15567265.2016.1256007

http://www.tandfonline.com/umte

cause mechanical damage to helicopter blades and fuselage [28], pose severe safety hazards to offshore oilexploration platforms [29–30], damage locks and dams [22, 31], and account for up to 40% of roadaccidents in winter [27, 32, 33]. Frost accumulating on refrigerators and heat exchangers can reduce theirheat transfer efficiency by as much as 50–75% [27, 34, 35]. Frosting can also cause mechanical damage topower transmission line systems as well as induce electric faults, such as flashovers, due to insufficientclearances [27,36–38]. Onwind turbines, it has been shown that ice accretion can substantially reduce theaerodynamic efficiency and torque, causing power losses as high as 50% [39, 40].

As recently as a few years ago, the underlying mechanism of condensation frosting was consideredto be equivalent to that governing icing, namely, that supercooled droplets freeze in isolation byheterogeneous nucleation at the solid–liquid interface [41–44]. From this false perspective, thedifference between icing and frosting seems merely contextual: the supercooled water is depositedfor icing and nucleated for condensation frosting. However, it has recently been discovered thatheterogeneous ice nucleation is the dominant mechanism only in the case of icing but not forcondensation frosting. The true dominant mechanism of condensation frosting on hydrophobic andmildly hydrophilic surfaces is that of interdroplet ice bridging, where frozen droplets grow ice bridgestoward their neighboring liquid droplets to form an interconnected ice network [45–47].

The discovery of ice bridging, along with other intriguing associated phenomena such as frosthalos [48, 49] and dry zones [50–52], constitute the incipient stages of condensation frosting andfundamentally differentiate the physics of frost growth from ice growth. Consequently, strategiesdeveloped in circumventing the icing problem do not necessarily translate to the frosting problem.There are many reviews [27, 40, 53–55] and research articles [56–79] today that characterize theicing problem and possible anti-icing and deicing strategies. There are also many studies character-izing the densification and growth of macroscopic frost sheets [2, 80–99]. However, the physics offrost incipience at a microscopic level has been largely overlooked until the past several years [45–47,100–103].

The present work reviews recent advances in our understanding of condensation frosting andsummarizes the stages of incipient frost formation. Subsequently we discuss possible means ofdelaying or even halting in-plane frost growth by inhibiting one or more of these stages of incipientfrost formation.

Stages of condensation frosting

There are five prominent stages of condensation frosting: (I) supercooled condensation, (II) onset offreezing, (III) frost halos, (IV) interdroplet ice bridging and dry zones, and (V) percolation clustersand frost densification. Figure 1 shows the chronological order of the stages with illustrativeschematics of the various phenomena. We will now elaborate on each stage in order.

Stage I—Supercooled condensation

The first stage of condensation frosting is the formation of supercooled condensation on a substrate.Broadly, condensation is a two-step process including (a) heterogeneous nucleation on a substrateand (b) subsequent growth (which itself is a multistep process that will be detailed at the end of thissection).

A necessary (but not sufficient) condition to have heterogeneous nucleation on a substrate is thatthe temperature of the substrate must be beneath the dew point. The dew point temperature isdefined as the saturation temperature corresponding to the partial pressure of water vapor in theambient. As such, at the dew point temperature, ambient water vapor and liquid water have the samechemical potential and are in thermodynamic equilibrium. However, nucleation involves creation ofnew interfaces, which necessitates a certain degree of supersaturation in the ambient atmosphere.The extent of supersaturation necessary for heterogeneous nucleation can be determined by equatingthe Gibbs free energy change ΔG of a nucleating embryo to the change in free energy associated with

2 S. NATH ET AL.

supersaturation (Figure 2a) [2, 104]. For a nucleating embryo of radius re, ΔG can be expressed as asummation of the negative change in energy inherent to supersaturated vapor becoming liquid/ice(ΔGv) and the positive energy barrier associated with the creation of the interfaces (ΔGs). If the

Figure 1. The different stages of condensation frosting. The word “local” implies that the given phenomenon is microscopic; thatis, specific to either a single droplet or constitutes a droplet pair interaction. The word “global” implies that the phenomenon ismacroscopic and requires the participation of the entire condensate population. The relevant vapor pressures are the pressurerequired to nucleate a water/ice embryo on a surface with a given wall temperature (pn;w), the saturation vapor pressure withrespect to ice at 0°C (pi;0), and the nucleation pressures corresponding to the condensation mode (pn;w ¼ pn;l) or desublimationmode (pn;w ¼ pn;i). The timescale of thermal conduction through ice is labeled τf and τD is the timescale of vapor diffusion fromthe ice droplet. The non-dimensional length scale S* is the bridging parameter (defined in Stage IV) that dictates the success orfailure of an ice bridge connection.

NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 3

ambient vapor pressure is supersaturated beyond a certain limit, it is possible that the vapor directlytransforms to ice on the substrate, bypassing the liquid phase (desublimation). Figure 2b shows aphase map of the preferred mode of nucleation on a substrate depending on its temperature andwettability [105].

The critical degree of supersaturation SSD required for a particular mode of nucleation isexpressed as

SSD ¼ pn;w � ps;wps;w

; (1)

where pn;w is the critical supersaturation pressure required for nucleation in that mode and ps;w is thesaturation vapor pressure with respect to water (for condensation) or ice (desublimation) corre-sponding to the wall temperature of the substrate. SSD is the abbreviation for supersaturation degree,which describes the metastable supersaturation that occurs locally at the surface to overcome thenucleation energy barrier. Note that a thermodynamic description of nucleation analogous tosupersaturation can be formulated in terms of subcooling: ΔT ¼ Tn;w � Ts;w; where Tn;w, and Ts;w

are the saturation temperatures corresponding to pn;w and ps;w. Using classical nucleation theory andBecker-Döring embryo formation kinetics [106, 107], pn;w can be obtained as [105]

pn;w ¼ ps;w expv

RgTw

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4π3

σ3ij

kTw lnðI0IcÞð2þmÞð1�mÞ2

vuut0@

1A; (2)

Figure 2. (a) Left: p-T diagram showing the extent of supersaturation required to nucleate an embryo of water or ice. Right:schematic of Gibbs free enthalpy variation (ΔG) with the radius of the nucleating embryo (re). The green curve corresponds tohomogeneous nucleation, and the red curve corresponds to heterogeneous nucleation. Note the enthalpic requirement associatedwith supersaturation (blue bracket) equals the critical Gibbs free enthalpy change (red bracket). (b) Phase diagram for thepreferred mode of nucleation for any surface temperature and wettability for embryo formation rates I� ¼ 1024 and 1027.Supercooled condensation is thermodynamically favorable in the phase space above the critical line and desublimation isfavorable below [105]. (c) Spatial control of condensation on a chemically patterned surface chilled to Tw ¼ �10 °C, wherenucleation occurs only on the hydrophilic circles (inset) that were fabricated on a hydrophobic background [50]. (d)Supersaturation degree (SSD) required for condensation mode of nucleation as a function of surface temperature for differentsurface wettabilities (θ ¼ 30, 60, 90, and 120�). (e) Condensation SSD as a function of wettability, θ for different surfacetemperatures (Tw ¼ 0, –10, –20, and –30°C). Reprinted with permission from [105]; copyright 2016, American Chemical Societyand from [50]; copyright 2016, Nature Publishing Group.

4 S. NATH ET AL.

where σij denotes the surface energy per unit area of water vapor with respect to liquid water forcondensation and with respect to ice for desublimation, m ¼ cos θ where θ is the contact angle ofwater with respect to the substrate for condensation and that of ice with respect to the substrate fordesublimation, Tw is the substrate temperature, v is the molar volume of water for condensation andthat of ice for desublimation, Rg is the universal gas constant, k ¼ 1:38� 10�23 J/K is the Boltzmannconstant, and I0 and Ic are the kinetic constant of embryo formation and the critical embryoformation rate for nucleation to successfully occur (both with units of m-2s-1).

Figures 2d and 2e show the dependence of the condensation SSD on substrate temperature (Tw)and wettability (θ), while varying I� ¼ I0=Ic over three orders of magnitude for all cases. As can beseen, the degree of supersaturation required for embryo formation is much larger for hydrophobicsurfaces than for hydrophilic surfaces (SSD ! 0 as θ ! 0°). Thus, a chemically heterogeneoussurface exhibiting both hydrophobic and hydrophilic features would have preferential nucleationevents only within the hydrophilic patterns as shown in Figure 2c. In fact, in the last decade,wettability patterning has been successfully utilized to promote preferential condensation andincrease condensation heat transfer [108–112], enhance water harvesting [113–115], or to controlicing/frosting behavior [50, 116, 117].

After nucleating, the supercooled water droplets keep growing from the ambient vapor. Thegrowth of condensate shows several remarkable features that have been studied in depth by Beysensand others [118–124]. There are three significant stages in droplet growth on a 2D substrate. Thefirst is the isolated growth regime, where the droplets have just nucleated and are sufficiently farfrom each other, such that the pressure profiles about each droplet do not overlap. The vaporpressure distribution about each droplet follows a hyperbolic profile p ¼ p1 � ðp1 � plÞR=r, wherep is the vapor pressure at a distance r from the center of the droplet, p1 is the vapor pressure in theambient atmosphere, pl is the vapor pressure at the liquid–vapor interface, and R is the radius of theliquid droplet. In this regime, under a steady state of condensation, each droplet grows in time asR � t1=2. As the droplets grow larger, the dynamics of droplet growth transition to the secondregime. The pressure profiles about each droplet now overlap, resulting in a pressure gradient that iseffectively linear and out-of-plane such that the droplet pattern can be treated as a homogeneousfilm. Analytically, the pressure field can be expressed as p ¼ pl þ ðp1 � plÞðz � hÞ=ζ, where z is thedistance perpendicular to the substrate, h is the average thickness of the homogeneous film thatapproximates the droplet distribution, and ζ is the concentration boundary layer thickness. In thisregime, the droplet radius evolves as R � t1=3. Over time, a plethora of coalescence events lead to athird, accelerated growth regime that is self-similar and characterized by a constant surface coverage.The growth law follows R � t as long as the film approximation discussed in the second regimeremains valid. Note that the above expressions of pressure profiles are valid only under isothermalconditions; that is, as long as the ambient temperature is not significantly different from that of thesubstrate. Otherwise, the pressure terms should be replaced by their corresponding vapor concen-tration values.

Stage II—Onset of freezing

The second stage of condensation frosting is the onset of freezing in the supercooled condensate.Even if the substrate is at a subfreezing temperature, freezing does not initiate immediately aftercondensation ensues. This is because freezing, just like condensation, requires a certain degree ofsubcooling to overcome the free energy barrier associated with the formation of new interfaces.Supercooled liquid water can in fact remain metastable at temperatures as low as Tw � � 40�Cwithout freezing right away [8].

Freezing of condensate droplets on a substrate starts with a probabilistic nucleation event. Thisnucleation can be either homogeneous if it initiates within the droplet away from the solid substrateor heterogeneous if it starts at the solid–liquid interface (see Figure 3a). The preferred mode of


nucleation at any given condition corresponds to the one that has a lower enthalpic requirement.The Gibbs free energy barrier for heterogeneous nucleation ΔGhet is usually lower than that forhomogeneous nucleation ΔGhom. As such, heterogeneous nucleation at the solid–liquid interface isusually the preferred mode of nucleation. However, it has recently been shown that unsaturated gasflow conditions can induce homogeneous nucleation at the free surface of a supercooled droplet dueto evaporative cooling [125].

The delay in heterogenous nucleation at the solid–liquid interface can be obtained from theembryo formation kinetics, following classical nucleation theory. Assuming that the freezing eventsoccurring in a condensate population are random and uncorrelated for any given nucleation rate, therandom freezing events can be considered to constitute an inhomogeneous Poisson process [127].Therefore, if a substrate is cooled from a temperature T0 to Tw and α ¼ dT=dt is the rate of cooling,then the probability of freezing Nf number of droplets in an ensemble of N0 droplets after time t isgiven by

PðNÞ;Nf

N0¼ 1� exp �α�1 �Tw

ToJðTÞdT

� �; (3)

where JðTÞ is the ice nucleation rate per unit time (s�1). Therefore, JðTÞ ¼ IðTÞA, where IðTÞ is theembryo formation rate per unit area per unit time (m�2s�1) and A is the solid–liquid surface area.From Becker-Döring embryo formation kinetics, I can be expressed as [106, 107]

I ¼ I0 exp �ΔGkT

� �; (4)

where I0 is the kinetic prefactor accounting for the diffusive flux of water molecules across the iceinterface, ΔG is the total change in free energy corresponding to nucleation, and k ¼ 1.38� 10�23J=Kis the Boltzmann constant. The change in free energy is expressed as ΔG ¼ 16πσ3f =3Δ~g2, where f is ageometrical factor that takes into account the roughness and wettability of the substrate, σ is the water–ice interfacial energy per unit area, and Δ~g is the Gibbs energy change per unit volume. Thus, we seethat the nucleation delay is a function of substrate temperature, wettability, and surface roughness.This is succinctly condensed in the following equation: if α is a constant, then for a given substratetemperature T, the expected delay in nucleation time hτi is given by [128]

hτi ¼ 1JðTÞ : (5)

By increasing the nucleation energy barrier, one can decrease the nucleation rate and consequentlydelay freezing substantially for deposited or condensed supercooled water. The nucleation energybarrier increases with larger contact angles of the water, which serves to increase IðTÞ while alsodecreasing A for a droplet of fixed volume. In the extreme case of superhydrophobic surfaces where

Figure 3. (a) Schematic depicting the two possible modes of ice nucleation in a water droplet: heterogeneous nucleation andhomogeneous nucleation. [54]. (b)Top row: Homogeneous nucleation originating at the liquid–vapor interface followed by freezefront propagation in a supercooled sessile droplet. Bottom row: heterogeneous nucleation originating at the solid–liquid interfacefollowed by freeze front propagation in a supercooled sessile droplet [54, 125]. Reprinted (adapted) with permission from [54];copyright 2012 American Chemical Society. (c) The freezing of a water droplet ends in a tip singularity [126]. Reprinted (adapted)with permission from [126]; copyright 2012 American Institute of Physics.

6 S. NATH ET AL.

supercooled droplets exhibit a suspended Cassie state [129] the decrease in A is much more dramaticdue to the presence of air pockets replacing much of the solid–liquid interface. Therefore, the freezingof supercooled droplets can be significantly delayed or in some cases even prevented by maximizingthe hydrophobicity of a substrate. This is precisely the idea that has provided the connective tissuebetween the fields of icephobicity and superhydrophobicity over the past decade of research[27, 54–58, 60–71, 100]. For cases where the deposited droplet is initially warmer than the chilledsubstrate, the superhydrophobic Cassie state also serves to minimize conductive heat transfer betweenthe droplet and substrate to prolong freezing onset [42, 59]. The low hysteresis of superhydrophobic(or liquid-infused) surfaces also facilitates the dynamic removal of supercooled droplets by rebound orsliding before ice nucleation can occur [66, 130–134]. On some nanostructured superhydrophobicsurfaces, even supercooled condensation can be removed prior to freezing via coalescence-inducedjumping [46, 135].

In a few special cases, it has been shown that increasing hydrophobicity or promoting the Cassiewetting state is not always optimal. For example, hydrophobic surfaces can actually exhibit an inferiorfreezing delay compared to ultrasmooth hydrophilic surfaces in cases where the benefit of the decreasedwettability is outweighed by the disadvantage of increased surface roughness (which increases A and/ornucleation-promoting defects) [56, 136]. A thermodynamic analysis by Eberle et al. [128] recentlyshowed that for ultrafine nanoroughness, the tendency of the roughness to promote ice nucleation forimpaled supercooled droplets can be counteracted by the confinement of the interfacial quasiliquid layer,delaying freezing by as long as 25 h under proper conditions. On such a surface, impaledWenzel dropletscould remain in the supercooled liquid state longer than suspended Cassie droplets.

Ice nucleation is followed by two partially overlapping stages of freezing: (a) recalescence and (b)freeze front propagation [125]. Recalescence is the very rapid (kinetically controlled) first stage offreezing where the growth of the ice embryo leads to an explosive release of latent heat, raising thetemperature of the supercooled liquid to the equilibrium freezing temperature of 0°C [137–140]. Therecalescent phase transforms the liquid droplet to a slushy matrix of partially solidified liquid, typically inthe order of ~ 10 ms for milimetric droplets [48]. This is followed by a significantly slower isothermalfreeze front propagation where the liquid in the interstitial space of the ice scaffold is completely frozen.The second stage of freezing is governed primarily by the rate at which the latent heat is conducted intothe substrate and/or dissipated in the ambient. As such, the duration of the second stage of freezing mayvary from fractions of a second to tens of seconds depending on the thermal conductivity of theunderlying substrate. Figure 3b shows experimental images of homogeneous and heterogeneous nuclea-tion in deposited supercooled droplets and the subsequent freeze front propagation that follows.

For the case of heterogeneous nucleation, the freezing of a water droplet eventually ends in abeautiful tip formation at the top [141, 142] (Figure 3c). This tip singularity is a consequence of theexpansion of water upon freezing and the fact that the top of a droplet is the last portion to freeze forheterogeneous nucleation. Intriguingly, the cone angle at the tip of a frozen droplet is a constant139� � 8°, independent of the substrate temperature, droplet size, and wettability [141].

Stage III—Frost halos

The phenomenon of frost halos was first reported in 1970 by Roger Cheng as a spontaneous ejection ofmicrodroplets during the freezing event of a droplet [143]. However, in light of a recent report by Junget al. [48], it appears that these microdroplets were not directly ejected at all but rather are nucleatedcondensate fostered by a recalescence-induced local supersaturation in the vicinity of the freezing droplet.

This third stage of condensation frosting (frost halos) partially overlaps with the second stage; that is,the onset of freezing. This is because the phenomenon of frost halos initiates after recalescence. Duringthe recalescent stage of freezing, the temperature of the droplet increases from its supercooled tempera-ture Tw to 0°C [137–140, 143]. Correspondingly, the pressure at the interface of the ice scaffoldsurrounding the droplet becomes equal to the saturation vapor pressure over ice at 0°C; that is, pi;0 ¼


611:2 Pa. In such a situation, vapor can flow out from the ice interface and deposit on the substrate ascondensate if the supersaturated pressure required for nucleation on the substrate (pn;w, Eq. (2)) is lessthan that of pi;0. This manifests itself as an annular ring of microdroplets around the frozen droplet thatcan quickly freeze over; hence the designation of frost halo (Figure 4a).

The extent and survival of the halo are dependent upon how long pi;0 > pn;w is true. This timescale isessentially the entire duration of the isothermal second stage of freezing combined with a portion of thetime required for the fully frozen droplet to cool back down toward the substrate temperature (Tw) bythermal conduction. Therefore, the sufficient condition for the observability of a frost halo is that theheat transfer timescale τf is greater than the vapor diffusion timescale τD. The diffusion timescale isgiven by τD � R2=D, where R is the radius of the droplet andD is the diffusivity of water vapor in air; adetailed estimation of τf has been provided by Jung et al [48]. The frost halo phenomenon cantherefore be best observed on a low - energy substrate (reducing pn;w, see Eq. (2)) that is also thermallyinsulating (increasing τf ).

Recently, it has been proposed that analogous to a condensation halo, a desublimation halo shouldalso be possible when vapor emanating from the ice interface can deposit on the substrate directly as iceembryos. This would be possible when the pressure required for the desublimation mode of nucleation(pn;i) is less than that for condensation nucleation (pn;l). Figure 4b shows a theoretically estimatedphase space of substrate temperature and wettability that has been posited to be the desublimation haloregime. However, desublimation halos are yet to be experimentally validated.

Stage IV—Interdroplet ice bridging and dry zones

The fourth stage of condensation frosting is essentially what differentiates it from the icing problemyet eluded researchers until the past 5 years. It was long believed that each condensate droplet freezesin isolation due to heterogeneous or homogeneous nucleation without interacting with neighboringdroplets [41–44]. This is true for temperatures below −40°C where the delay in nucleation time isnegligible and almost all the droplets freeze simultaneously [8]. However, for temperatures higherthan −40°C, the vast majority of the droplets do not freeze in isolation but rather are frozen byinterdroplet interactions.

In a population of supercooled condensate, which droplet freezes first is a probabilistic event. Thedroplets that freeze first have a lower vapor pressure above them due to the fact that the saturationvapor pressure over ice is lower than that over water at the same subfreezing temperature [7]. Thisleads to localized vapor pressure gradients in the system where the frozen droplets start behaving aslocal humidity sinks. The source–sink interaction between the frozen droplets and their neighboringwater droplets leads to the fascinating phenomenon of interdroplet ice bridging [45–47] and

Figure 4. (a) Condensation halo surrounding a freezing 5 μL droplet on a poly(methylmethacrylate) substrate [48]. Reprintedwith permission from [48]. Copyright 2012 National Academy of Sciences. (b) Phase map of frost halos showing three distinctregimes: desublimation halos (red region), condensation halos (blue), or no halos (white), as a function of the surfacetemperature and wettability [105]. Reprinted with permission from [105]; copyright 2016 American Chemical Society.

8 S. NATH ET AL.

localized dry zones [50, 52, 105], which are the hallmarks of condensation frosting on hydrophobicsurfaces. Note that interdroplet ice bridging and dry zones are specific to surfaces which exhibitdropwise condensation. Sufficiently hydrophilic surfaces exhibit filmwise condensation and the filmfreezes over all at once [44].

Interdroplet ice bridgingOnce a droplet freezes and equilibrates to the temperature of the substrate, the vapor pressureabove it becomes lower than that over the water droplets surrounding it [7]. If there is acondensation halo around the frozen droplet then the microdroplets in the halo that are nearestto the ice droplet start evaporating [48]. In the absence of a condensation halo, the frozendroplet starts harvesting water molecules from its nearest neighboring water droplets [45–47].These water molecules deposit on the frozen droplet and start growing ice bridges directedtoward the water droplets that are being harvested. The neighboring liquid droplets freeze assoon as the ice bridges connect. These newly frozen water droplets now start harvesting waterfrom their adjacent water droplets and grow ice bridges toward them. Thus, frost propagates in achain reaction of interconnected ice bridges that form a network. The phenomenon of inter-droplet ice bridging in the context of condensation frosting was first reported by J. B. Dooley inhis Ph.D. thesis in 2010 [45]. Figure 5a shows the the first observation of interdroplet icebridging across a population of supercooled condensate on a SiAl substrate at Tw = –10°C.Note that in these image sequences, the frost halos are not visible, most likely because theexperimental conditions were not conducive for them.

Figure 5. (a) First reported observation of interdroplet ice bridging [45] driving frost growth across supercooled condensate on asubstrate. The substrate temperature was Tw ¼ �10�C, the air temperature was Tair ¼ 5.1 �C, and the relative humidity wasRH ¼ 65.4%. Each image in the top row is a 50X magnification of the region marked in red in the bottom image. Reprinted withpermission from the Ph.D. thesis of J.B. Dooley [45]. (b) Time taken for ice bridges to connect to their targeted droplets onhydrophobic (HPB) or superhydrophobic (SHPB) surfaces as a function of the bridging parameter, S�[46]. Reprinted withpermission from [46]; copyright 2013, American Chemical Society. (c) Experimental and schematic depiction of successful (top)versus failed (bottom) ice bridging between a droplet pair [105]. Experiments were performed at Tw ¼ �10�C and p1 ¼ 776:3Pa,time stamps are in seconds, and the scale bar represents 20 μm. (d) Experimental micrograph of a stable dry zone around a frozendroplet, where multiple rows of condensate droplets have evaporated. Tw ¼ �12:5�C and T1 ¼ 17:4�C and RH ¼ 21%.Reprinted with permission from [105]; copyright 2016, American Chemical Society.


The growth rate of ice bridges can be determined by conservation of mass flux between theevaporating droplet and the growing ice bridge. The characteristic velocity of ice bridge growth, vb, isgiven by [105]

vb � Dρi�RgTw

pl � piδ

; (6)

where �Rg ¼ 461.5 J/kg K is the gas constant of water vapor, pl and pi are the vapor pressures overwater and ice, ρi is the density of ice, and δ is the edge-to-edge separation between the frozen dropletand its neighboring liquid water droplet. This yields a dynamic evolution of the ice bridge that islinear in time. In a separate study, Petit and Bonaccurso [144] identified a crossover to a late-timebridging regime that is quadratic in time. They have also observed that individual bridge growthrates are independent of the stiffness of the substrate. The lack of viscoelastic braking in the growthof in-plane frost on soft substrates further demonstrates that the propagation of ice bridges is notself-sustained but externally driven by the evaporating neighboring droplets [44].

The phenomenon of interdroplet ice bridging is highly sensitive to vapor pressure differentialsbetween supercooled droplets and frozen droplets and thus can be utilized to estimate the vaporpressure at the ice–vapor interface. Previous models positing that vapor supersaturation is requiredfor frost growth [3, 80, 104] yield a vapor pressure over ice that exceeds the saturation vapor pressureof water at the same temperature. In this context, the ubiquity of interdroplet ice bridging over awide range of temperatures is thus strong experimental evidence that pressure over growing frostmust be less than that above water and therefore cannot be highly supersaturated as estimated bythese models. We recently used experimental measurements of ice bridge growth rates to demon-strate that the vapor pressure at the ice–vapor interface of growing frost is approximately saturated[105]. This, in tandem with another recent work [145] that used laser confocal microscopy anddifferential interference contrast microscopy to observe the equilibrium pressure of ice, resolves thelong-standing debate of whether the interface of growing frost is saturated [2, 83–85, 92, 145] orsupersaturated [3, 80–82, 104, 146].

However, an ice bridge is not guaranteed to connect to the water droplet it is harvesting. If theneighboring liquid droplet is sufficiently far away from the frozen droplet, it is possible that theliquid droplet completely evaporates before the ice bridge formed at its expense can connect to it.Whether an individual ice bridge can connect to its targeted liquid droplet can be predicted by asimple scaling analysis based on conservation of mass. At the limiting condition, the mass of thecompleted ice bridge, mbridge, is equal to the mass of the liquid droplet being harvested, ml. For a pairof identical droplets, one frozen and the other unfrozen, this yields a geometric constraint that canbe simply expressed by a nondimensional number called the bridging parameter defined asS� ¼ Lmax=d, where Lmax is equal to the distance between the edge of the frozen droplet and thecenter of the liquid droplet and d is the initial (projected) diameter of the liquid droplet prior toharvesting. Conservation of mass mandates that for S� < 1, bridging succeeds and for S� > 1,bridging fails, which was verified experimentally for large populations of supercooled condensate[46]. Thus, just based on the initial condition of a droplet pair system, even before freezing hasoccurred, it is possible to predict whether a sudden freezing event in one will engender an ice bridgeconnection versus the complete evaporation of the unfrozen droplet. Figure 5c shows how thebridging parameter dictates whether a droplet pair interaction will lead to a successful ice bridgeconnection or not.

Dry zonesWhen S� > 1, the liquid droplet fueling the ice bridge evaporates completely before the bridge canconnect, halting the in-plane growth of the frost. When all liquid droplets are sufficiently far fromtheir nearest frozen droplet(s), it is possible to evaporate the entire array of neighboring waterdroplets to create an annular dry zone about the ice. The presence of this dry zone is remarkable

10 S. NATH ET AL.

when considering that the surface is beneath the dew point. Figure 5d shows an experimental imageof a stable dry zone that has formed between the frozen droplet and the nearest supercooledcondensation.

The extent of such a dry zone, δCr , can be estimated by a scaling argument based on a mass fluxbalance. Droplets at the periphery of such a dry zone exhibit approximately zero mass flux, becauseany water just within/outside the dry zone must evaporate/condense by definition. Therefore at thiscritical length the out-of-plane mass flux condensing onto the substrate, _mc, must be equal to the in-plane evaporative mass flux in the direction of ice, _me. Because mass flux is proportional to thepressure gradient, equating the two one can obtain the critical dry zone length as

δCr � βζpl � pip1 � pl

; (7)

where ζ is the concentration boundary layer thickness and β is a geometrical prefactor that accountsfor the ratio of the in-plane and out-of-plane projected areas of the evaporating liquid droplet [52,105]. The above formulation has a few caveats. Firstly, Eq. (7) is true as long as the temperature ofthe substrate Tw is close to the ambient temperature T1. Otherwise, the vapor pressures in Eq. (7)should be replaced by corresponding vapor concentrations. In addition, it is not clear whether thevapor pressure pl above the water droplets at the periphery of the dry zone δCr should be taken to besaturated corresponding to micrometric droplets [52] or supersaturated for nanometric droplets[105]. Furthermore, the above formulation has an underlying assumption that even at the peripheryof the dry zone, the concentration profile in the vertical direction is linear, as is established over a seaof condensate droplets far from ice. The validity of such an assumption near ice is indeed question-able, more so if the frozen droplet is significantly larger in size than the microscopic condensate.Previous works on dry zones around hygroscopic droplets [51, 124] have alternately proposed anucleation dry zone model, where the supersaturated pressure required for embryo nucleation is themechanism for the dry zone rather than a flux balance. We are currently investigating all of thesemodeling issues, as well as the competition between nucleation and flux dry zones, in order to moreaccurately predict dry zones in future reports.

In a population of condensate, disparate droplet sizes and interdroplet distances can lead to localregions of S� > 1. When ice bridges propagate across such a population of supercooled condensate,these regions cause local failures in ice bridge connections of the order of δCr that serve to retard thevelocity of the global freeze front. This explains why for superhydrophobic surfaces exhibitingjumping-droplet condensation and low surface coverage [147], ice bridges are able to propagateacross only about one third of the droplets [46], whereas for hydrophobic surfaces, bridging issuccessful for nearly all of the droplets [46, 47].

Stage V—Percolation clusters and frost densification

Our previous discussion on the success and failure of ice bridging deals with the local pairinteraction of a frozen and a liquid droplet. However, after the first freezing event, ice bridgespercolate globally through the entire population of supercooled condensate in a chain reaction [46,52, 103, 144, 148–150]. Such percolation clusters are characterized by local regions of S�>1 where dryregions exist and S� < 1 where ice bridge connections occur. Figure 6a shows the dynamic evolutionof such an interdroplet network of ice bridge connections. In Figure 6b, one can see the chainreactions of ice bridging happen and the local failures in ice bridge connections that are inherent inany percolating cluster of interdroplet ice bridges. Note that the timescale of an individual bridgegrowth [46, 47, 144] τbridge � δavg=vb (typically 1–10 s) is greater than the duration of freezing, τf , ifthe substrate is thermally conducting and is therefore the dominant timescale even in the propaga-tion of a global freeze front.

The velocity of propagation of such an interdroplet global freeze front is typically of the order1–10 μm/s [46, 47, 50]. As discussed before, this growth rate has been shown independent of


substrate stiffness [144]. However, the global propagation of ice bridges can be suppressed byincorporation of three-dimensional microscale structures with inclined edges on a hierarchicalsuperhydrophobic surface. Microscale structures create a structural barrier for ice bridging, dimin-ishing their global propagation rates [103]. The propagation of ice bridges can also potentially besuppressed by having a bilayer architecture functionalized with an outer porous superhydrophobicepidermis and an underlying dermis that is infused with an antifreeze liquid [151]. Condensationfrosting on such a surface is characterized by discrete frozen droplets surrounded by a film of diluteantifreeze liquid that can suppress in-plane ice bridging and frost growth owing to their hygroscopicnature.

The velocity of a global freeze front can in general be tuned by controlling the interdropletdistances between the droplet. This is possible by designing functionalized surfaces that can spatiallycontrol nucleation sites with desired interdroplet distances [108, 110, 113, 116, 152–155]. Onepossible mechanism for doing so is to have chemical micropatterns that are hydrophilic on ahydrophobic background (Figure 6b). However, note that spatial control of nucleation is not enoughto control interdroplet distances, as the nucleated liquid droplets keep growing larger and larger overtime until the first freezing event happens. In fact, in a recent study with precise temporal controlover the first freezing event, it was shown that at a fixed temperature Tw ¼ �10�C, the longer thedelay in ice nucleation, the faster the ice bridge propagation rates [50]. It was also demonstrated thatan intentional triggering of a very early freezing event can cause a global failure of ice bridgeconnections resulting in a global dry zone (Figure 7c). The extent of such a dry zone was accuratelymodeled by balancing the mass fluxes as given by the expression in Eq. (8).

The percolation process inherent to interdroplet ice bridging is different from the growth of anisolated snow crystal at the expense of the ambient vapor. Unlike the morphology of snow crystals,which often exhibit self-similar fractal characteristics in their dendritic growth, interdroplet icebridging shows very specific preferential growth in the direction of the nearest neighboring waterdroplet that is harvested. The percolation dynamics of interdroplet ice growth across a population ofdroplets has yet to be studied in detail; we encourage more in-depth studies of this topic as a futuredirection of research.

Once the global freeze front has propagated though the entire population condensate, a network of inter-connected frozen droplets provides the foundation upon which out-of-plane frost growth can happen. Thethermodynamics of frost densification has been studied in extensive detail for decades, which has producedmany excellent reviews and articles [2, 80–99].

Figure 6. (a) Interdroplet freezing path lines indicating in-plane ice bridge connections over time. The color contour shows thetime elapsed and the arrows show the direction of propagation of individual ice bridges in the percolation cluster. Substratetemperature Tw ¼ �7:1�C, air temperature Tair ¼ 5�C and relative humidity RH ¼ 64.9%. Reprinted with permission from thethesis of J. B. Dooley [45]. (b) Interdroplet frost growth across patterns of supercooled condensate. The bottom of each imageshows a rectangular water pad, and the rest of the surface has a uniformly distributed condensation pattern against a hydrophobicbackground. The T2P array indicates an interdroplet distance equal to that of the droplet diameter, whereas the T4P array impliesthat the distance is thrice the diameter. The first frame shows that the water pad is frozen, initiating freezing (t ¼ 0), whereas thesecond and third frames correspond to the times where the interdroplet frost has grown to the field-of-view in T2P and T4Parrays [50]. Reprinted with permission from [50]; copyright 2016, Nature Publishing Group.

12 S. NATH ET AL.

Anti-frosting strategies

It should now be clear that the physics of incipient frost growth are quite distinct from the accretion of iceon a surface due to deposited water. But do these differences also extend to anti-icing strategies? Here, wesummarize recent findings regarding the fabrication of passive anti-icing surfaces and discuss whichtechniques also apply to anti-frosting surface technology. In the respects where anti-icing and anti-frostingdiffer, we consider novel approaches to anti-frosting that should be the subject of future research efforts.

The most promising strategy for passively preventing ice formation is to fabricate surfaces that (1)delay the freezing of supercooled water and (2) exhibit a very low contact angle hysteresis. Whenboth of these features are in place, supercooled water impacting the surface is able to bounce/slide offthe substrate before the onset of heterogeneous ice nucleation [66, 130–134]. At first glance, thisstrategy may also seem amenable for the promotion of anti-frosting, considering that the dynamicalremoval of condensate can be achieved by gravity at millimetric length scales [66, 130] or by

Figure 7. (a) Frost-phobic dry zone around a salty water droplet. [52]. (I) t ¼ 0: Salt crystal just after deposition, (II) t ¼ 8 s: partialcrystal dissolution, (III) t ¼ 30 s: condensation dry zone forming around the salty water droplet under humid conditions, (IV) t ¼34 s: frost invades the upper-left corner, but a dry zone δI develops between the ice and the salt crystal while a condensation dryzone δW remains between the salt and the remaining liquid condensate, (V) t ¼ 49:6 s: 40 ms before the growing ice bridge canconnect to the salty water droplet (white circle), and (VI) t ¼ 50 s: upon contact, the salty droplet immediately freezes. Reprintedwith permission from [52]; copyright 2015, IOP Publishing. (b) Using hygroscopic antifreeze liquids to prevent frost growth.Inhibition of condensation frosting around four 2 μL propylene glycol (PG) droplets over time.51 Reprinted from [51]. Copyright2015, American Chemical Society. (c) Using ice itself to prevent frost growth. The freezing of a film of water (visible at the bottomof the images) at Ts ¼ �5�C and subsequent cooling to Tw ¼ �12:5�C evaporates the condensate around it to cause a globalfailure of ice bridge connections. The condensate in these experiments has been grown on hydrophilic stripes and freezing causesevaporation along these stripes to halt interdroplet frost growth. Reprinted with permission from [50]; copyright 2016, NaturePublishing Group.


coalescence-induced jumping at micrometric length scales [46, 100]. However, even when super-cooled condensate is continually removed from the substrate by sliding and/or jumping, frostinevitably forms due to the heterogeneous nucleation of ice at edge defects, which subsequentlypropagates frost across the entire surface via interdroplet ice bridging [46, 150]. If the edge defectsare shielded—for example, by using a rubber gasket—ice nucleation will still occur due to surfacedefects caused by dust particles and other unavoidable imperfections [149]. Even on liquid-impreg-nated surfaces [156], where both hysteresis and surface defects are minimal, ice nucleation isfollowed by spontaneous oil migration onto the frozen droplets, causing irreversible damage to theself-healing properties of the surface [131]. Because heterogeneous ice nucleation will eventuallyoccur somewhere on any real-life surface, promoting a delay in freezing onset is actually harmful inthe context of anti-frosting efforts, because the rate of interdroplet ice bridging propagating fromthis point source will significantly increase due to the increased size and surface coverage of thesupercooled condensate [46, 50]. Thus, we see that anti–icing techniques do not typically translate toanti–frosting except for the unlikely case where the delay in ice nucleation achieved for all super-cooled condensation is greater than the time for which the system needs to be kept frost-free.

So, what is an appropriate anti-frosting strategy? Now that it is clear that interdroplet ice bridging isthe primary mechanism for in-plane frost growth, we suggest that the only viable anti-frosting strategiesare to (1) promote the failure of interdroplet ice bridges (i.e., water droplets evaporate before bridgeconnects) or (2) prevent the nucleation and growth of any nearby supercooled droplets so that bridgescannot grow at all. Both of these approaches to creating a dry zone, free from condensation and frost, arebest accomplished by the use of hygroscopic humidity sinks such as salty water [52, 157, 158], nectar[159], or glycols [51]. Historically, the germ of the idea of dry zones was seeded in the works of Lopezet al. [160] in 1993 and Aizenberg et al. [161] in 1999 that established how patterned surface functio-nalization can be utilized for spatial control of nucleation. However, these dry zones were reported asregions where nucleationwas inhibited because the Gibbs free energy requirement had been increased bysurface functionalization. Such nucleation dry zones are different from flux dry zones that emerge fromthe cooperative diffusion mechanism between condensing droplets [162] (see Stage IV), where growth issuppressed by the presence of neighboring humidity sinks that evaporate any condensing embryos. Thefirst demonstration of overlapping flux dry zones was performed by Schäffle and coworkers in 2003 [152]well before the discovery of ice bridging. This clever study distributed an array of diethylene glycoldroplets on a chemically patterned substrate to keep the intermediate surface area dry from condensate.More recent works have extended the concept of overlapping dry zones to supercooled condensate, inorder to help suppress condensation frosting. Guadarrama–Cetina et al. showed that a salt crystal canpromote an annular dry zone for both supercooled condensate and frost (Figure 7a) [52]. Sun et al. [51]scaled this concept up by depositing arrays of droplets (composed of propylene glycol or salty water)across a substrate, confirming that when the dry zones overlap the intermediate area initially remains dryfrom condensate and frost even under supersaturated conditions (Figure 7b).

When using traditional humidity sinks such as salt crystals or glycols, their hygroscopic propertiesare gradually lost as they continually harvest water vapor from the ambient and become diluted. As aresult, the dry zones eventually break down and the frost proceeds to invade across the surface afteronly a few minutes have passed [51, 52]. It follows that the applicability of such an approach isconstrained to the (rather impractical) case where the time required to dilute the humidity sinksexceeds the desired operation time of the system. Therefore we argue here that the best choice ofhygroscopic material is, ironically enough, ice itself. After all, as ice harvests water vapor from theambient (and any neighboring supercooled droplets), it remains pure ice. Therefore, the depressedvapor pressure of ice with respect to water continues unabated, regardless of how much water hasbeen harvested. In Figure 5d, we see a stable dry zone around ice, which is analogous to the dryzones we see around hygroscopic droplets [105]. Figure 7c shows the dynamic evolution of such adry zone around a droplet that has just been frozen [50]. This shows a global failure of all ice bridgeconnections from the frozen droplet. Thus, it seems only logical to have sequential arrays of ice itselfto uniquely promote stable and overlapping dry zones in between. This should in principle severely

14 S. NATH ET AL.

suppress in-plane frosting in favor of out-of-plane ice growth at the sacrificial humidity sinks. Weare currently investigating the validity and robustness of such an idea.

Despite the emerging anti-frosting strategies discussed above, to date no passive anti-frosting surfacetechnology has been successful. For this reason, many works have been dedicated to maximizing theefficiency of the active removal of existing frost sheets by physical or thermal means [66, 101, 102, 163].Interestingly, though anti-icing strategies do not translate to anti-frosting, defrosting and deicing strategiesare in fact conceptual analogues of each other. For both deicing and defrosting, the mechanical removal ofice can be optimized by minimizing the adhesion strength of ice with the substrate, whereas thermaldefrosting is accelerated when the melt water exhibits a low contact angle hysteresis to facilitate shedding atlow tilt angles. Both of these criteria can be met by suspending the ice/frost in a Cassie state on asuperhydrophobic surface. In this sense, the only major difference between defrosting and deicing is thatCassie frost is more difficult to achieve than Cassie ice [101]. This is because a microstructured super-hydrophobic surface is sufficient to suspend deposited water in most cases, whereas a robust nanostructureis required to suspend condensate (and by extension frost) in a Cassie state [147, 164–166].Wenzel frost, onthe other hand, can be dynamically removed by thermal defrosting where the drainage of the frost melt canbe enabled by physical microgrooves on a metal substrate [167–169]. Alongside superhydrophobicsubstrates, liquid-infused surfaces (SLIPS) have also been reported to demonstrate low contact anglehysteresis as well as low ice adhesion strength, facilitating both thermal [66, 131, 170] and mechanicaldefrosting [66, 151]. However, the lack of durability of both superhydrophobic and liquid-infused surfacesmakes it difficult for practical implementations [131]. Very recently, it has been shown that smooth andhydrophobic elastomeric coatings can durably exhibit remarkably low ice adhesion if the physical stiffnessof elastomers is tuned to enable interfacial slippage [79].

Conclusion

In this review, we have detailed the five-stage process of condensation frosting in chronologicalorder and elaborated on the specifics of each stage. Subsequently, we have discussed how anti-icingstrategies do not necessarily translate to anti-frosting strategies. We have reviewed the present anti-frosting strategies and discussed future pathways.

Despite the abundant amount of research articles in anti-icing and anti-frosting, the incipientstages of condensation frosting that lead to frost densification have just been discovered. To this end,the discovery of frost halos, ice bridging, and dry zones marks the departure of the physics offrosting from icing physics. In the wake of these new discoveries relating to condensation frosting,anti-frosting studies have just begun. Ideas are nascent and strategies inconclusive. Experiments havejust started to open up a vast unanswered domain in the physics of bridging and dry zones. We hopethat this review will inspire experimentalists and theoreticians alike to investigate and unravel theintricacies of these phenomena, leading to an exceptional control of in-plane frost growth onsurfaces.

Funding

This work was supported by the National Science Foundation (CBET-1604272) and startup funds from theDepartment of Biomedical Engineering and Mechanics at Virginia Tech.

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A Review of Condensation Frosting

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