A Review of Condensation Frosting

Transcription

1 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING A Review of Condensation Frosting Saurabh Nath, S. Farzad Ahmadi, and Jonathan B. Boreyko Department of Biomedical Engineering and Mechanics, Virginia Tech, Blacksburg, Virginia, USA ABSTRACT The accretion of ice and frost on various infrastructure is ubiquitous in cold and humid environments, causing economic losses amounting to billions of dollars every year worldwide. The past couple of decades have seen unprecedented advances in the fields of surface chemistry and micro/nanofabrication, enabling the development of hydrophobic and superhydrophobic surfaces that promote facile deicing and/or passive anti-icing. However, in the light of new discoveries regarding the incipient stages of frost formation, it is becoming increasingly clear that the problems of icing and frosting are not one and the same. Thus, passive anti-icing strategies do not exhibit anti-frosting behavior, and the development of passive anti-frosting surfaces remains an unsolved problem. In this review, we provide a critical discussion of condensation frosting and show how the emerging new phenomena of frost halos, interdroplet ice bridges, and dry zones that comprise the incipient stages of frosting set it apart from the conventional problem of icing. Subsequently, we discuss possible strategies to break the sequential chain of events leading to pervasive frost growth. ARTICLE HISTORY Received 31 August 2016 Accepted 29 October 2016 KEYWORDS Frost; condensation; antifrosting; icephobicity; ice bridges 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, summarizing them in the barest outline.... Dew is moisture of fine composition falling from a clear sky; ice is water congealed 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 remarkable understanding of condensation and phase change. Indeed, in this quote, Aristotle correctly identifies the two possible modes of frosting desublimation, which is a direct transformation of water vapor to ice and condensation frosting, where the vapor first condenses into supercooled dew droplets that subsequently freeze [2, 3]. However it was not until 1657, 15 years after Galileo s death, that the first systematic experiments on the freezing of water in an enclosed jar were performed to investigate Galileo 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 understanding of ice physics [3, 5 20], we are far from solving the icing problem. From an economic standpoint, ice accretion today is a multibillion dollar problem in the United States alone [21]. Frosting and icing adversely affect multiple industries, including aviation, telecommunication, electrical transmission, 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 aircraft flight, causing severe damage and even plane crashes [23 27]. Icing and frosting have also been shown to CONTACT Jonathan B. Boreyko boreyko@vt.edu 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 Taylor & Francis

2 2 S. NATH ET AL. cause mechanical damage to helicopter blades and fuselage [28], pose severe safety hazards to offshore oil exploration platforms [29 30], damage locks and dams [22, 31], and account for up to 40% of road accidents in winter [27, 32, 33]. Frost accumulating on refrigerators and heat exchangers can reduce their heat transfer efficiency by as much as 50 75% [27, 34, 35]. Frosting can also cause mechanical damage to power transmission line systems as well as induce electric faults, such as flashovers, due to insufficient clearances [27,36 38]. On wind turbines, it has been shown that ice accretion can substantially reduce the aerodynamic 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 considered to be equivalent to that governing icing, namely, that supercooled droplets freeze in isolation by heterogeneous nucleation at the solid liquid interface [41 44]. From this false perspective, the difference between icing and frosting seems merely contextual: the supercooled water is deposited for icing and nucleated for condensation frosting. However, it has recently been discovered that heterogeneous ice nucleation is the dominant mechanism only in the case of icing but not for condensation frosting. The true dominant mechanism of condensation frosting on hydrophobic and mildly hydrophilic surfaces is that of interdroplet ice bridging, where frozen droplets grow ice bridges toward 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 frost halos [48, 49] and dry zones [50 52], constitute the incipient stages of condensation frosting and fundamentally differentiate the physics of frost growth from ice growth. Consequently, strategies developed 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 the icing problem and possible anti-icing and deicing strategies. There are also many studies characterizing the densification and growth of macroscopic frost sheets [2, 80 99]. However, the physics of frost incipience at a microscopic level has been largely overlooked until the past several years [45 47, ]. The present work reviews recent advances in our understanding of condensation frosting and summarizes the stages of incipient frost formation. Subsequently we discuss possible means of delaying or even halting in-plane frost growth by inhibiting one or more of these stages of incipient frost formation. Stages of condensation frosting There are five prominent stages of condensation frosting: (I) supercooled condensation, (II) onset of freezing, (III) frost halos, (IV) interdroplet ice bridging and dry zones, and (V) percolation clusters and frost densification. Figure 1 shows the chronological order of the stages with illustrative schematics 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 substrate and (b) subsequent growth (which itself is a multistep process that will be detailed at the end of this section). A necessary (but not sufficient) condition to have heterogeneous nucleation on a substrate is that the temperature of the substrate must be beneath the dew point. The dew point temperature is defined as the saturation temperature corresponding to the partial pressure of water vapor in the ambient. As such, at the dew point temperature, ambient water vapor and liquid water have the same chemical potential and are in thermodynamic equilibrium. However, nucleation involves creation of new interfaces, which necessitates a certain degree of supersaturation in the ambient atmosphere. The extent of supersaturation necessary for heterogeneous nucleation can be determined by equating the Gibbs free energy change ΔG of a nucleating embryo to the change in free energy associated with

3 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 3 Figure 1. The different stages of condensation frosting. The word local implies that the given phenomenon is microscopic; that is, specific to either a single droplet or constitutes a droplet pair interaction. The word global implies that the phenomenon is macroscopic and requires the participation of the entire condensate population. The relevant vapor pressures are the pressure required to nucleate a water/ice embryo on a surface with a given wall temperature (p n;w ), the saturation vapor pressure with respect to ice at 0 C (p i;0 ), and the nucleation pressures corresponding to the condensation mode (p n;w ¼ p n;l ) or desublimation mode (p n;w ¼ p n;i ). The timescale of thermal conduction through ice is labeled τ f and τ D is the timescale of vapor diffusion from the ice droplet. The non-dimensional length scale S* is the bridging parameter (defined in Stage IV) that dictates the success or failure of an ice bridge connection. supersaturation (Figure 2a) [2, 104]. For a nucleating embryo of radius r e, ΔG can be expressed as a summation of the negative change in energy inherent to supersaturated vapor becoming liquid/ice (ΔG v ) and the positive energy barrier associated with the creation of the interfaces (ΔG s ). If the

4 4 S. NATH ET AL. 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 (r e ). The green curve corresponds to homogeneous nucleation, and the red curve corresponds to heterogeneous nucleation. Note the enthalpic requirement associated with supersaturation (blue bracket) equals the critical Gibbs free enthalpy change (red bracket). (b) Phase diagram for the preferred mode of nucleation for any surface temperature and wettability for embryo formation rates I ¼ and Supercooled condensation is thermodynamically favorable in the phase space above the critical line and desublimation is favorable below [105]. (c) Spatial control of condensation on a chemically patterned surface chilled to T w ¼ 10 C, where nucleation 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 different surface wettabilities (θ ¼ 30, 60, 90, and 120 ). (e) Condensation SSD as a function of wettability, θ for different surface temperatures (T w ¼ 0, 10, 20, and 30 C). Reprinted with permission from [105]; copyright 2016, American Chemical Society and from [50]; copyright 2016, Nature Publishing Group. ambient vapor pressure is supersaturated beyond a certain limit, it is possible that the vapor directly transforms to ice on the substrate, bypassing the liquid phase (desublimation). Figure 2b shows a phase map of the preferred mode of nucleation on a substrate depending on its temperature and wettability [105]. The critical degree of supersaturation SSD required for a particular mode of nucleation is expressed as SSD ¼ p n;w p s;w p s;w ; (1) where p n;w is the critical supersaturation pressure required for nucleation in that mode and p s;w is the saturation vapor pressure with respect to water (for condensation) or ice (desublimation) corresponding 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 the nucleation energy barrier. Note that a thermodynamic description of nucleation analogous to supersaturation can be formulated in terms of subcooling: ΔT ¼ T n;w T s;w; where T n;w, and T s;w are the saturation temperatures corresponding to p n;w and p s;w. Using classical nucleation theory and Becker-Döring embryo formation kinetics [106, 107], p n;w can be obtained as [105] 0 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 v u4π σ 3 ij p n;w ¼ p s;w exp@ t R g T w 3 kt w lnð I ð2 þ mþð1 mþ2 A; (2) 0 Ic Þ

5 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 5 where σ ij denotes the surface energy per unit area of water vapor with respect to liquid water for condensation and with respect to ice for desublimation, m ¼ cos θ where θ is the contact angle of water with respect to the substrate for condensation and that of ice with respect to the substrate for desublimation, T w is the substrate temperature, v is the molar volume of water for condensation and that of ice for desublimation, R g is the universal gas constant, k ¼ 1: J/K is the Boltzmann constant, and I 0 and I c are the kinetic constant of embryo formation and the critical embryo formation rate for nucleation to successfully occur (both with units of m -2 s -1 ). Figures 2d and 2e show the dependence of the condensation SSD on substrate temperature (T w ) and wettability (θ), while varying I ¼ I 0 =I c over three orders of magnitude for all cases. As can be seen, the degree of supersaturation required for embryo formation is much larger for hydrophobic surfaces than for hydrophilic surfaces (SSD! 0 as θ! 0 ). Thus, a chemically heterogeneous surface exhibiting both hydrophobic and hydrophilic features would have preferential nucleation events 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 and increase condensation heat transfer [ ], enhance water harvesting [ ], or to control icing/frosting behavior [50, 116, 117]. After nucleating, the supercooled water droplets keep growing from the ambient vapor. The growth of condensate shows several remarkable features that have been studied in depth by Beysens and others [ ]. There are three significant stages in droplet growth on a 2D substrate. The first is the isolated growth regime, where the droplets have just nucleated and are sufficiently far from each other, such that the pressure profiles about each droplet do not overlap. The vapor pressure distribution about each droplet follows a hyperbolic profile p ¼ p 1 ðp 1 p l ÞR=r, where p is the vapor pressure at a distance r from the center of the droplet, p 1 is the vapor pressure in the ambient atmosphere, p l is the vapor pressure at the liquid vapor interface, and R is the radius of the liquid droplet. In this regime, under a steady state of condensation, each droplet grows in time as R t 1=2. As the droplets grow larger, the dynamics of droplet growth transition to the second regime. The pressure profiles about each droplet now overlap, resulting in a pressure gradient that is effectively linear and out-of-plane such that the droplet pattern can be treated as a homogeneous film. Analytically, the pressure field can be expressed as p ¼ p l þðp 1 p l Þðz hþ=ζ, where z is the distance perpendicular to the substrate, h is the average thickness of the homogeneous film that approximates the droplet distribution, and ζ is the concentration boundary layer thickness. In this regime, the droplet radius evolves as R t 1=3. Over time, a plethora of coalescence events lead to a third, 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 regime remains valid. Note that the above expressions of pressure profiles are valid only under isothermal conditions; that is, as long as the ambient temperature is not significantly different from that of the substrate. Otherwise, the pressure terms should be replaced by their corresponding vapor concentration 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 after condensation ensues. This is because freezing, just like condensation, requires a certain degree of subcooling 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 T w 40 C without freezing right away [8]. Freezing of condensate droplets on a substrate starts with a probabilistic nucleation event. This nucleation can be either homogeneous if it initiates within the droplet away from the solid substrate or heterogeneous if it starts at the solid liquid interface (see Figure 3a). The preferred mode of

6 6 S. NATH ET AL. Figure 3. (a) Schematic depicting the two possible modes of ice nucleation in a water droplet: heterogeneous nucleation and homogeneous nucleation. [54]. (b)top row: Homogeneous nucleation originating at the liquid vapor interface followed by freeze front propagation in a supercooled sessile droplet. Bottom row: heterogeneous nucleation originating at the solid liquid interface followed 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. nucleation at any given condition corresponds to the one that has a lower enthalpic requirement. The Gibbs free energy barrier for heterogeneous nucleation ΔG het is usually lower than that for homogeneous nucleation ΔG hom. As such, heterogeneous nucleation at the solid liquid interface is usually the preferred mode of nucleation. However, it has recently been shown that unsaturated gas flow conditions can induce homogeneous nucleation at the free surface of a supercooled droplet due to evaporative cooling [125]. The delay in heterogenous nucleation at the solid liquid interface can be obtained from the embryo formation kinetics, following classical nucleation theory. Assuming that the freezing events occurring in a condensate population are random and uncorrelated for any given nucleation rate, the random freezing events can be considered to constitute an inhomogeneous Poisson process [127]. Therefore, if a substrate is cooled from a temperature T 0 to T w and α ¼ dt=dt is the rate of cooling, then the probability of freezing N f number of droplets in an ensemble of N 0 droplets after time t is given by PðNÞ; N f ¼ 1 exp α 1 ò T w T N o JðTÞdT ; (3) 0 where JðTÞ is the ice nucleation rate per unit time (s 1 ). Therefore, JðTÞ ¼IðTÞA, where IðTÞ is the embryo formation rate per unit area per unit time (m 2 s 1 ) and A is the solid liquid surface area. From Becker-Döring embryo formation kinetics, I can be expressed as [106, 107] I ¼ I 0 exp ΔG ; (4) kt where I 0 is the kinetic prefactor accounting for the diffusive flux of water molecules across the ice interface, ΔG is the total change in free energy corresponding to nucleation, and k ¼ J=K is the Boltzmann constant. The change in free energy is expressed as ΔG ¼ 16πσ 3 f =3Δ~g 2,wheref is a geometrical 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 see that 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 substrate temperature T, the expected delay in nucleation time hτi is given by [128] hτi ¼ 1 JðTÞ : (5) By increasing the nucleation energy barrier, one can decrease the nucleation rate and consequently delay freezing substantially for deposited or condensed supercooled water. The nucleation energy barrier increases with larger contact angles of the water, which serves to increase IðTÞ while also decreasing A for a droplet of fixed volume. In the extreme case of superhydrophobic surfaces where

7 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 7 supercooled droplets exhibit a suspended Cassie state [129] the decrease in A is much more dramatic due to the presence of air pockets replacing much of the solid liquid interface. Therefore, the freezing of supercooled droplets can be significantly delayed or in some cases even prevented by maximizing the hydrophobicity of a substrate. This is precisely the idea that has provided the connective tissue between 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 chilled substrate, the superhydrophobic Cassie state also serves to minimize conductive heat transfer between the 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 or sliding before ice nucleation can occur [66, ]. On some nanostructured superhydrophobic surfaces, even supercooled condensation can be removed prior to freezing via coalescence-induced jumping [46, 135]. In a few special cases, it has been shown that increasing hydrophobicity or promoting the Cassie wetting state is not always optimal. For example, hydrophobic surfaces can actually exhibit an inferior freezing delay compared to ultrasmooth hydrophilic surfaces in cases where the benefit of the decreased wettability is outweighed by the disadvantage of increased surface roughness (which increases A and/or nucleation-promoting defects) [56, 136]. A thermodynamic analysis by Eberle et al. [128] recently showed that for ultrafine nanoroughness, the tendency of the roughness to promote ice nucleation for impaled 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, impaled Wenzel droplets could 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 of freezing where the growth of the ice embryo leads to an explosive release of latent heat, raising the temperature of the supercooled liquid to the equilibrium freezing temperature of 0 C [ ]. The recalescent phase transforms the liquid droplet to a slushy matrix of partially solidified liquid, typically in the order of ~ 10 ms for milimetric droplets [48]. This is followed by a significantly slower isothermal freeze 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 into the substrate and/or dissipated in the ambient. As such, the duration of the second stage of freezing may vary from fractions of a second to tens of seconds depending on the thermal conductivity of the underlying substrate. Figure 3b shows experimental images of homogeneous and heterogeneous nucleation 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 a beautiful tip formation at the top [141, 142] (Figure 3c). This tip singularity is a consequence of the expansion of water upon freezing and the fact that the top of a droplet is the last portion to freeze for heterogeneous nucleation. Intriguingly, the cone angle at the tip of a frozen droplet is a constant 139 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 of microdroplets during the freezing event of a droplet [143]. However, in light of a recent report by Jung et al. [48], it appears that these microdroplets were not directly ejected at all but rather are nucleated condensate 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. During the recalescent stage of freezing, the temperature of the droplet increases from its supercooled temperature T w to 0 C [ , 143]. Correspondingly, the pressure at the interface of the ice scaffold surrounding the droplet becomes equal to the saturation vapor pressure over ice at 0 C; that is, p i;0 ¼

8 8 S. NATH ET AL. 611:2 Pa. In such a situation, vapor can flow out from the ice interface and deposit on the substrate as condensate if the supersaturated pressure required for nucleation on the substrate (p n;w,eq.(2))isless than that of p i;0. This manifests itself as an annular ring of microdroplets around the frozen droplet that can quickly freeze over; hence the designation of frost halo (Figure 4a). The extent and survival of the halo are dependent upon how long p i;0 > p n;w is true. This timescale is essentially the entire duration of the isothermal second stage of freezing combined with a portion of the time required for the fully frozen droplet to cool back down toward the substrate temperature (T w )by thermal conduction. Therefore, the sufficient condition for the observability of a frost halo is that the heat transfer timescale τ f is greater than the vapor diffusion timescale τ D. The diffusion timescale is given by τ D R 2 =D, where R is the radius of the droplet and D is the diffusivity of water vapor in air; a detailed estimation of τ f has been provided by Jung et al [48]. The frost halo phenomenon can therefore be best observed on a low - energy substrate (reducing p n;w, see Eq. (2)) that is also thermally insulating (increasing τ f ). Recently, it has been proposed that analogous to a condensation halo, a desublimation halo should also be possible when vapor emanating from the ice interface can deposit on the substrate directly as ice embryos. This would be possible when the pressure required for the desublimation mode of nucleation (p n;i ) is less than that for condensation nucleation (p n;l ). Figure 4b shows a theoretically estimated phase space of substrate temperature and wettability that has been posited to be the desublimation halo regime. 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 problem yet eluded researchers until the past 5 years. It was long believed that each condensate droplet freezes in isolation due to heterogeneous or homogeneous nucleation without interacting with neighboring droplets [41 44]. This is true for temperatures below 40 C where the delay in nucleation time is negligible and almost all the droplets freeze simultaneously [8]. However, for temperatures higher than 40 C, the vast majority of the droplets do not freeze in isolation but rather are frozen by interdroplet interactions. In a population of supercooled condensate, which droplet freezes first is a probabilistic event. The droplets that freeze first have a lower vapor pressure above them due to the fact that the saturation vapor pressure over ice is lower than that over water at the same subfreezing temperature [7]. This leads to localized vapor pressure gradients in the system where the frozen droplets start behaving as local humidity sinks. The source sink interaction between the frozen droplets and their neighboring water 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]. Reprinted with permission from [48]. Copyright 2012 National Academy of Sciences. (b) Phase map of frost halos showing three distinct regimes: desublimation halos (red region), condensation halos (blue), or no halos (white), as a function of the surface temperature and wettability [105]. Reprinted with permission from [105]; copyright 2016 American Chemical Society.

9 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 9 localized dry zones [50, 52, 105], which are the hallmarks of condensation frosting on hydrophobic surfaces. Note that interdroplet ice bridging and dry zones are specific to surfaces which exhibit dropwise condensation. Sufficiently hydrophilic surfaces exhibit filmwise condensation and the film freezes over all at once [44]. Interdroplet ice bridging Once a droplet freezes and equilibrates to the temperature of the substrate, the vapor pressure above it becomes lower than that over the water droplets surrounding it [7]. If there is a condensation halo around the frozen droplet then the microdroplets in the halo that are nearest to the ice droplet start evaporating [48]. In the absence of a condensation halo, the frozen droplet 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 directed toward the water droplets that are being harvested. The neighboring liquid droplets freeze as soon as the ice bridges connect. These newly frozen water droplets now start harvesting water from their adjacent water droplets and grow ice bridges toward them. Thus, frost propagates in a chain reaction of interconnected ice bridges that form a network. The phenomenon of interdroplet ice bridging in the context of condensation frosting was first reported by J. B. Dooley in his Ph.D. thesis in 2010 [45]. Figure 5a shows the the first observation of interdroplet ice bridging across a population of supercooled condensate on a SiAl substrate at T w = 10 C. Note that in these image sequences, the frost halos are not visible, most likely because the experimental conditions were not conducive for them. Figure 5. (a) First reported observation of interdroplet ice bridging [45] driving frost growth across supercooled condensate on a substrate. The substrate temperature was T w ¼ 10 C, the air temperature was T air ¼ 5.1 C, and the relative humidity was RH ¼ 65.4%. Each image in the top row is a 50X magnification of the region marked in red in the bottom image. Reprinted with permission from the Ph.D. thesis of J.B. Dooley [45]. (b) Time taken for ice bridges to connect to their targeted droplets on hydrophobic (HPB) or superhydrophobic (SHPB) surfaces as a function of the bridging parameter, S [46]. Reprinted with permission 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 T w ¼ 10 C and p 1 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 frozen droplet, where multiple rows of condensate droplets have evaporated. T w ¼ 12:5 C and T 1 ¼ 17:4 C and RH ¼ 21%. Reprinted with permission from [105]; copyright 2016, American Chemical Society.

10 10 S. NATH ET AL. The growth rate of ice bridges can be determined by conservation of mass flux between the evaporating droplet and the growing ice bridge. The characteristic velocity of ice bridge growth, v b,is given by [105] v b D p l p i ; (6) ρ i R g T w δ where R g ¼ J/kg K is the gas constant of water vapor, p l and p i are the vapor pressures over water and ice, ρ i is the density of ice, and δ is the edge-to-edge separation between the frozen droplet and its neighboring liquid water droplet. This yields a dynamic evolution of the ice bridge that is linear in time. In a separate study, Petit and Bonaccurso [144] identified a crossover to a late-time bridging regime that is quadratic in time. They have also observed that individual bridge growth rates are independent of the stiffness of the substrate. The lack of viscoelastic braking in the growth of in-plane frost on soft substrates further demonstrates that the propagation of ice bridges is not self-sustained but externally driven by the evaporating neighboring droplets [44]. The phenomenon of interdroplet ice bridging is highly sensitive to vapor pressure differentials between supercooled droplets and frozen droplets and thus can be utilized to estimate the vapor pressure at the ice vapor interface. Previous models positing that vapor supersaturation is required for frost growth [3, 80, 104] yield a vapor pressure over ice that exceeds the saturation vapor pressure of water at the same temperature. In this context, the ubiquity of interdroplet ice bridging over a wide range of temperatures is thus strong experimental evidence that pressure over growing frost must be less than that above water and therefore cannot be highly supersaturated as estimated by these models. We recently used experimental measurements of ice bridge growth rates to demonstrate 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 and differential interference contrast microscopy to observe the equilibrium pressure of ice, resolves the long-standing debate of whether the interface of growing frost is saturated [2, 83 85, 92, 145] or supersaturated [3, 80 82, 104, 146]. However, an ice bridge is not guaranteed to connect to the water droplet it is harvesting. If the neighboring liquid droplet is sufficiently far away from the frozen droplet, it is possible that the liquid 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 a simple scaling analysis based on conservation of mass. At the limiting condition, the mass of the completed ice bridge, m bridge, is equal to the mass of the liquid droplet being harvested, m l. For a pair of identical droplets, one frozen and the other unfrozen, this yields a geometric constraint that can be simply expressed by a nondimensional number called the bridging parameter defined as S ¼ L max =d, where L max is equal to the distance between the edge of the frozen droplet and the center of the liquid droplet and d is the initial (projected) diameter of the liquid droplet prior to harvesting. 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 has occurred, it is possible to predict whether a sudden freezing event in one will engender an ice bridge connection versus the complete evaporation of the unfrozen droplet. Figure 5c shows how the bridging parameter dictates whether a droplet pair interaction will lead to a successful ice bridge connection or not. Dry zones When S > 1, the liquid droplet fueling the ice bridge evaporates completely before the bridge can connect, halting the in-plane growth of the frost. When all liquid droplets are sufficiently far from their nearest frozen droplet(s), it is possible to evaporate the entire array of neighboring water droplets to create an annular dry zone about the ice. The presence of this dry zone is remarkable

11 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 11 when considering that the surface is beneath the dew point. Figure 5d shows an experimental image of a stable dry zone that has formed between the frozen droplet and the nearest supercooled condensation. The extent of such a dry zone, δ Cr, can be estimated by a scaling argument based on a mass flux balance. Droplets at the periphery of such a dry zone exhibit approximately zero mass flux, because any water just within/outside the dry zone must evaporate/condense by definition. Therefore at this critical length the out-of-plane mass flux condensing onto the substrate, _m c, must be equal to the inplane evaporative mass flux in the direction of ice, _m e. Because mass flux is proportional to the pressure gradient, equating the two one can obtain the critical dry zone length as δ Cr βζ p l p i p 1 p l ; (7) where ζ is the concentration boundary layer thickness and β is a geometrical prefactor that accounts for 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 of the substrate T w is close to the ambient temperature T 1. Otherwise, the vapor pressures in Eq. (7) should be replaced by corresponding vapor concentrations. In addition, it is not clear whether the vapor pressure p l above the water droplets at the periphery of the dry zone δ Cr should be taken to be saturated corresponding to micrometric droplets [52] or supersaturated for nanometric droplets [105]. Furthermore, the above formulation has an underlying assumption that even at the periphery of the dry zone, the concentration profile in the vertical direction is linear, as is established over a sea of condensate droplets far from ice. The validity of such an assumption near ice is indeed questionable, 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 a nucleation dry zone model, where the supersaturated pressure required for embryo nucleation is the mechanism for the dry zone rather than a flux balance. We are currently investigating all of these modeling issues, as well as the competition between nucleation and flux dry zones, in order to more accurately predict dry zones in future reports. In a population of condensate, disparate droplet sizes and interdroplet distances can lead to local regions 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 δ Cr that serve to retard the velocity of the global freeze front. This explains why for superhydrophobic surfaces exhibiting jumping-droplet condensation and low surface coverage [147], ice bridges are able to propagate across only about one third of the droplets [46], whereas for hydrophobic surfaces, bridging is successful 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 pair interaction of a frozen and a liquid droplet. However, after the first freezing event, ice bridges percolate globally through the entire population of supercooled condensate in a chain reaction [46, 52, 103, 144, ]. Such percolation clusters are characterized by local regions of S >1 where dry regions exist and S < 1 where ice bridge connections occur. Figure 6a shows the dynamic evolution of such an interdroplet network of ice bridge connections. In Figure 6b, one can see the chain reactions of ice bridging happen and the local failures in ice bridge connections that are inherent in any percolating cluster of interdroplet ice bridges. Note that the timescale of an individual bridge growth [46, 47, 144] τ bridge δ avg =v b (typically 1 10 s) is greater than the duration of freezing, τ f,if the substrate is thermally conducting and is therefore the dominant timescale even in the propagation of a global freeze front. The velocity of propagation of such an interdroplet global freeze front is typically of the order 1 10 μm/s [46, 47, 50]. As discussed before, this growth rate has been shown independent of

12 12 S. NATH ET AL. Figure 6. (a) Interdroplet freezing path lines indicating in-plane ice bridge connections over time. The color contour shows the time elapsed and the arrows show the direction of propagation of individual ice bridges in the percolation cluster. Substrate temperature T w ¼ 7:1 C, air temperature T air ¼ 5 C and relative humidity RH ¼ 64.9%. Reprinted with permission from the thesis of J. B. Dooley [45]. (b) Interdroplet frost growth across patterns of supercooled condensate. The bottom of each image shows a rectangular water pad, and the rest of the surface has a uniformly distributed condensation pattern against a hydrophobic background. The T2P array indicates an interdroplet distance equal to that of the droplet diameter, whereas the T4P array implies that the distance is thrice the diameter. The first frame shows that the water pad is frozen, initiating freezing (t ¼ 0), whereas the second and third frames correspond to the times where the interdroplet frost has grown to the field-of-view in T2P and T4P arrays [50]. Reprinted with permission from [50]; copyright 2016, Nature Publishing Group. substrate stiffness [144]. However, the global propagation of ice bridges can be suppressed by incorporation of three-dimensional microscale structures with inclined edges on a hierarchical superhydrophobic surface. Microscale structures create a structural barrier for ice bridging, diminishing their global propagation rates [103]. The propagation of ice bridges can also potentially be suppressed by having a bilayer architecture functionalized with an outer porous superhydrophobic epidermis and an underlying dermis that is infused with an antifreeze liquid [151]. Condensation frosting on such a surface is characterized by discrete frozen droplets surrounded by a film of dilute antifreeze liquid that can suppress in-plane ice bridging and frost growth owing to their hygroscopic nature. The velocity of a global freeze front can in general be tuned by controlling the interdroplet distances between the droplet. This is possible by designing functionalized surfaces that can spatially control nucleation sites with desired interdroplet distances [108, 110, 113, 116, ]. One possible mechanism for doing so is to have chemical micropatterns that are hydrophilic on a hydrophobic background (Figure 6b). However, note that spatial control of nucleation is not enough to control interdroplet distances, as the nucleated liquid droplets keep growing larger and larger over time until the first freezing event happens. In fact, in a recent study with precise temporal control over the first freezing event, it was shown that at a fixed temperature T w ¼ 10 C, the longer the delay in ice nucleation, the faster the ice bridge propagation rates [50]. It was also demonstrated that an intentional triggering of a very early freezing event can cause a global failure of ice bridge connections resulting in a global dry zone (Figure 7c). The extent of such a dry zone was accurately modeled 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 an isolated 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 ice bridging shows very specific preferential growth in the direction of the nearest neighboring water droplet that is harvested. The percolation dynamics of interdroplet ice growth across a population of droplets has yet to be studied in detail; we encourage more in-depth studies of this topic as a future direction of research. Once the global freeze front has propagated though the entire population condensate, a network of interconnected frozen droplets provides the foundation upon which out-of-plane frost growth can happen. The thermodynamics of frost densification has been studied in extensive detail for decades, which has produced many excellent reviews and articles [2, 80 99].

13 NANOSCALE AND MICROSCALE THERMOPHYSICAL ENGINEERING 13 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: partial crystal dissolution, (III) t ¼ 30 s: condensation dry zone forming around the salty water droplet under humid conditions, (IV) t ¼ 34 s: frost invades the upper-left corner, but a dry zone δ I develops between the ice and the salt crystal while a condensation dry zone δ W remains between the salt and the remaining liquid condensate, (V) t ¼ 49:6 s: 40 ms before the growing ice bridge can connect to the salty water droplet (white circle), and (VI) t ¼ 50 s: upon contact, the salty droplet immediately freezes. Reprinted with 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]. Copyright 2015, American Chemical Society. (c) Using ice itself to prevent frost growth. The freezing of a film of water (visible at the bottom of the images) at T s ¼ 5 C and subsequent cooling to T w ¼ 12:5 C evaporates the condensate around it to cause a global failure of ice bridge connections. The condensate in these experiments has been grown on hydrophilic stripes and freezing causes evaporation along these stripes to halt interdroplet frost growth. Reprinted with permission from [50]; copyright 2016, Nature Publishing Group. Anti-frosting strategies It should now be clear that the physics of incipient frost growth are quite distinct from the accretion of ice onasurfaceduetodepositedwater.butdothesedifferencesalsoextendtoanti-icingstrategies?here,we summarize recent findings regarding the fabrication of passive anti-icing surfaces and discuss which techniques also apply to anti-frosting surface technology. In the respects where anti-icing and anti-frosting differ, 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. When both of these features are in place, supercooled water impacting the surface is able to bounce/slide off the substrate before the onset of heterogeneous ice nucleation [66, ]. At first glance, this strategy may also seem amenable for the promotion of anti-frosting, considering that the dynamical removal of condensate can be achieved by gravity at millimetric length scales [66, 130] or by

14 14 S. NATH ET AL. coalescence-induced jumping at micrometric length scales [46, 100]. However, even when supercooled condensate is continually removed from the substrate by sliding and/or jumping, frost inevitably forms due to the heterogeneous nucleation of ice at edge defects, which subsequently propagates frost across the entire surface via interdroplet ice bridging [46, 150]. If the edge defects are shielded for example, by using a rubber gasket ice nucleation will still occur due to surface defects caused by dust particles and other unavoidable imperfections [149]. Even on liquid-impregnated surfaces [156], where both hysteresis and surface defects are minimal, ice nucleation is followed by spontaneous oil migration onto the frozen droplets, causing irreversible damage to the self-healing properties of the surface [131]. Because heterogeneous ice nucleation will eventually occur somewhere on any real-life surface, promoting a delay in freezing onset is actually harmful in the context of anti-frosting efforts, because the rate of interdroplet ice bridging propagating from this point source will significantly increase due to the increased size and surface coverage of the supercooled condensate [46, 50]. Thus, we see that anti icing techniques do not typically translate to anti frosting except for the unlikely case where the delay in ice nucleation achieved for all supercooled 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 is the primary mechanism for in-plane frost growth, we suggest that the only viable anti-frosting strategies are to (1) promote the failure of interdroplet ice bridges (i.e., water droplets evaporate before bridge connects) or (2) prevent the nucleation and growth of any nearby supercooled droplets so that bridges cannot grow at all. Both of these approaches to creating a dry zone, free from condensation and frost, are best 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 Lopez et al. [160] in 1993 and Aizenberg et al. [161] in 1999 that established how patterned surface functionalization can be utilized for spatial control of nucleation. However, these dry zones were reported as regions where nucleation was inhibited because the Gibbs free energy requirement had been increased by surface functionalization. Such nucleation dry zones are different from flux dry zones that emerge from the cooperative diffusion mechanism between condensing droplets [162] (see Stage IV), where growth is suppressed by the presence of neighboring humidity sinks that evaporate any condensing embryos. The first 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 glycol droplets 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, in order to help suppress condensation frosting. Guadarrama Cetina et al. showed that a salt crystal can promote 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 dry from condensate and frost even under supersaturated conditions (Figure 7b). When using traditional humidity sinks such as salt crystals or glycols, their hygroscopic properties are gradually lost as they continually harvest water vapor from the ambient and become diluted. As a result, the dry zones eventually break down and the frost proceeds to invade across the surface after only a few minutes have passed [51, 52]. It follows that the applicability of such an approach is constrained to the (rather impractical) case where the time required to dilute the humidity sinks exceeds the desired operation time of the system. Therefore we argue here that the best choice of hygroscopic material is, ironically enough, ice itself. After all, as ice harvests water vapor from the ambient (and any neighboring supercooled droplets), it remains pure ice. Therefore, the depressed vapor pressure of ice with respect to water continues unabated, regardless of how much water has been harvested. In Figure 5d, we see a stable dry zone around ice, which is analogous to the dry zones we see around hygroscopic droplets [105]. Figure 7c shows the dynamic evolution of such a dry zone around a droplet that has just been frozen [50]. This shows a global failure of all ice bridge connections from the frozen droplet. Thus, it seems only logical to have sequential arrays of ice itself to uniquely promote stable and overlapping dry zones in between. This should in principle severely