Frost-free zone on macrotextured surfacesYuehan Yao (姚岳瀚)a, Tom Y. Zhaob, Christian Machadob

, Emma Feldmanc, Neelesh A. Patankarb,and Kyoo-Chul Parkb,1

aDepartment of Materials Science and Engineering, Northwestern University, Evanston, IL 60208; bDepartment of Mechanical Engineering, NorthwesternUniversity, Evanston, IL 60208; and cDepartment of Chemical and Biological Engineering, Northwestern University, Evanston, IL 60208

Edited by Howard A. Stone, Princeton University, Princeton, NJ, and approved February 18, 2020 (received for review September 23, 2019)

Numerous studies have focused on designing functional surfacesthat delay frost formation or reduce ice adhesion. However, solu-tions to the scientific challenges of developing antiicing surfacesremain elusive because of degradation such as mechanical wear-ing. Inspired by the discontinuous frost pattern on natural leaves,here we report findings on the condensation frosting process onsurfaces with serrated structures on the millimeter scale, which isdistinct from that on a conventional planar surface with microscale/nanoscale textures. Dropwise condensation, during the first stage offrosting, is enhanced on the peaks and suppressed in the valleys,causing frost to initiate from the peaks, regardless of surface chem-istry. The condensed droplets in the valley are then evaporated dueto the lower vapor pressure of ice compared with water, resulting ina frost-free zone in the valley, which resists frost propagation evenon superhydrophilic surfaces. The dependence of the frost-free arealfraction on the geometric parameters and the ambient conditions iselucidated by both numerical simulations based on steady-state dif-fusion and an analytical method with an understanding of bound-ary conditions independent of surface chemistry. We envision thatthis study would provide a unified framework to design surfacesthat can spatially control frost formation, crystal growth, diffusion-controlled growth of biominerals, and material deposition over abroad range of applications.

condensation frosting | antifrosting | bioinspiration | diffusion

Ice accretion on surfaces can cause serious energy waste andsafety threats in many practical scenarios (1–3). Air drag builds

up when ice accumulates on aircraft and wind turbines and,hence, disturbs the smooth airflow around them (4). Ice forma-tion can result from the freezing of subcooled liquid water, or thefrosting of moisture in the ambient air (5). The frosting mech-anism accounts for a significant portion of icing problems underhigh humidity. For example, frost accrues on the condensingunits of refrigerators and air conditioners, and the thermal effi-ciency can be compromised due to the low thermal conductivityof ice (6). In aviation, the frost layer has a relatively thin thick-ness, and its color is similar to that of the aircraft surface, makingit likely to be undetected (3). Numerous research efforts havebeen devoted to the development of antiicing (defined as mini-mizing ice formation) and deicing (defined as facilitating iceremoval) strategies. Superhydrophobic surfaces, which incorpo-rate an extremely low surface energy coating with microscale/nanoscale surface roughness, are effective in delaying ice for-mation because of their few heterogeneous nucleation sites inaddition to their minimized water−solid heat transfer by thesuspended Cassie−Baxter wetting state (7–9). However, frostingis thermodynamically inevitable on such surfaces in the long termsince water vapor in the humid air can easily diffuse between thesmall asperities (10, 11). This so-called Wenzel ice, which leadsto the ice−solid contact, interlocks with the surface roughness,has a high removal strength, and potentially degrades the surfacecoating and microtextures/nanotextures (12, 13). To facilitate iceremoval, an additional layer of fluid that is not miscible withwater and has a low melting point (lower than the operatingtemperature) is often introduced between the ice and solid sur-face. Slippery liquid-infused porous surfaces, which use a low

surface tension oil as the fluid to achieve molecular-levelsmoothness, not only are able to delay freezing but also areable to reduce ice adhesion strength by up to two orders of mag-nitude compared to noncoated surfaces (14–16). Alternatively,with the presence of a hygroscopic material coating such as poly-ethylene oxide brushes, the frozen ice can also be self-lubricated bya thin layer of water at the ice−solid interface (17, 18). Based onthese discoveries, other functional surfaces, including magneticfield-driven ferrofluids and oil-infused polymeric materials, havealso been investigated (19–21). However, a robust surface designthat is able to show a long-term resistance to ice accretion, espe-cially when harsh operating conditions such as high humidity andmechanical wearing are present, still remains a challenge (22–25).Frosting is an interfacial process that is ubiquitous in nature

when the surface temperature drops below a critical level (26–28).Fig. 1 shows the frost patterns on both natural and artificially threedimensional(3D)-printed leaves. The frost patterns on the leavesin Fig. 1 A and B were observed under natural frosting conditionswhich were not artificially controlled. The air temperature plum-mets at night, and the water saturation increases. The surfacetemperature of leaves further decreases below 0 °C because ofradiative cooling (27). Despite the different species of the twoplants in Fig. 1 A and B, the frost patterns of both leaves showsimilarities. The number density of ice crystals is noticeably higheron the leaf veins which are topographically convex (blue dashedcircles in Fig. 1A), while the flat regions between the veins showmuch less frost coverage. This correlation is further evidenced bythe almost complementary frost pattern on the front (concave, reddashed circles) and back (convex, blue dashed circle) sides of the

Significance

In spite of a decades-long surface science research endeavor toreduce frost coverage, most state-of-the-art surfaces de-veloped based on micrometer/nanometer and molecular-levelsurface treatment still suffer from surface degradation andresult in full frost coverage under continuous frosting condi-tions. Based on our fundamental understanding of discontin-uous frost patterns found on the leaf vein structure on thescale of millimeters, we elucidate the thermodynamic correla-tion between the frost-free area and two major surface systemparameters—macroscopic surface geometry and ambient hu-midity. This systematic study on the frost formation mecha-nism allows us to demonstrate a ∼50% of frost coverage evenfor superhydrophilic surfaces and provides a quantitativeguideline for further reducing frost coverage.

Author contributions: Y.Y., N.A.P., and K.-C.P. designed research; Y.Y., C.M., and E.F.performed research; Y.Y., T.Y.Z., and K.-C.P. analyzed data; and Y.Y. and K.-C.P. wrotethe paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1To whom correspondence may be addressed. Email: kpark@northwestern.edu.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1915959117/-/DCSupplemental.

First published March 10, 2020.

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same leaf in Fig. 1B. Similar suppression of frosting is also ob-served in the laboratory environment (see Controlled FrostingMeasurement) on the concave vein features of an artificial leafwhich is 3D-printed with methacrylic acid esters as shown in Fig.1C and Movie S1. The convex and concave surface topography onleaves coupled with their discontinuous frost patterns inspired usto investigate the impact of similar millimetric surface features onthe frosting process. Specifically, in this work, we focus on thecondensation frosting process observed on artificially fabricatedaluminum surfaces with simple millimeter-scale serrated featuresas shown in Fig. 2A. In contrast to previous attempts to addressthe icing problem from a microengineering/nanoengineering per-spective, here we demonstrate an approach that is based on surfacefeatures about three orders of magnitude greater, which is easy tofabricate and can be readily combined with other state-of-the-artantifrosting technologies based on materials design (9, 14, 17, 20).

Results and DiscussionUsing the hydrophobic surfaces and the experimental setupshown in Fig. 2 A and B, a multistage frosting process is observedon the serrated surfaces by direct visualization under a macrolens (Fig. 2C) and a digital microscope (Fig. 2 D and E; seeMaterials and Methods for more details). Both surfaces wereuniformly hydrophobic and examined under the same experi-mental conditions, except that the vertex angle α was 60° underthe macro lens, and 90° under the microscope lens for betterimaging. The ambient humidity and temperature were keptconstant at 25 ± 1% and 23.5 ± 0.5 °C, respectively. The surfacetemperature was maintained at −12 ± 0.3 °C in both cases. Fig.2F schematically demonstrates the stages observed, includingcondensation, frost initiation and propagation, and evaporation.First, dropwise condensation happens due to the supersaturatedambient air. A uniform initial contact with the supersaturated airresults in uniform water nucleation as shown in Fig. 2C (t = 80 s)(SI Appendix, Fig. S1). Different from the uniform condensation ona topographically flat surface, the growth rate of the supercooleddroplets is clearly affected by the surface geometry, as evidenced bythe distribution of droplet sizes shown in Fig. 2 D and E (t = 300 s;yellow dashed circles). The observation of a disparity betweendroplet sizes on the peak (∼40 μm) and those in the valley(<10 μm) supports our previous studies on the effect of macro-scopic surface topography on the local droplet growth (29, 30). Thefirst freezing event of the supercooled droplets is then initiated atthe peaks, followed by the quick propagation of the frost fronttoward the valley. It was revealed, in recent studies, that the vaporpressure difference between the frost front and its neighboringnonfrozen droplets establishes a local diffusion field (31), such thatvaporous water molecules transport from the liquid phase anddeposit onto the frost front as an “ice bridge” (9, 31, 32). If the “icebridge” successfully connects the frost front and the nonfrozen

droplet, the liquid can quickly freeze as a result of heterogeneousnucleation. The frozen droplet then becomes the new frost front.The other potential outcome is that the “ice bridge” fails to reachto the nearest droplet because the droplet completely evaporatesdue to diffusion, in which case the ice bridging stops, and the frostfront consequently becomes dendritic (33). The clear gap betweenthe frost front and droplets in the valley (Fig. 2E; t = 420 s) sug-gests the depletion of the condensed liquid, which leads to for-mation of the frost-free zone in Fig. 2C (t = 2,270 s) and Fig. 2E(t = 3,300 s). It is worth noting that the rate with which the frostfront propagates becomes significantly slower after the evaporationstage. The frost-free zone therefore can be considered to have along-term resistance to frosting. By using a serrated surface with avertex angle α = 40° (defined in Fig. 2A), the areal frost-freefraction (1− f = h=H in Fig. 2F) is still above 50% after 5 h underthe aforementioned experimental conditions (see SI Appendix, Figs.S2 and S3 for more details). Similar frost-free zones can be observedon surfaces with characteristic traits such as superhydrophilicity, lowthermal conductivity, and feature size similar to those found onleaves (visualized in SI Appendix, Fig. S4).

Initial Condensation and Frost Formation. For supersaturated watervapor subject to a surface below 0 °C, water can exist in two phases: 1)solid ice via nucleation and 2) supercooled liquid water (31). Underthe experimental conditions of this study, a substantial supersaturationof ambient water vapor pressure (p) can result in both the conden-sation of supercooled water (p/psat,w > 1.3) and the desublimation ofice (p/psat,i > 1.5) (34). Condensation is considered to be the kineticallypreferred pathway for phase change due to a lower energy barrier fornucleation resulting from a lower surface energy of water comparedwith ice (SI Appendix, Figs. S5–S7) (33, 35–37).Following the nucleation of liquid water, the difference be-

tween the partial pressure of water vapor in the ambient air (p)and that near the liquid phase (psat,w) drives water molecules todiffuse, contributing to droplet growth. To explain the fastergrowth rate of droplets on the peaks than those in the valleys asshown in Fig. 2 D and E, we numerically simulated the concen-tration field of water vapor near the serrated features in thecondensation stage. The boundary conditions are shown in Fig.2G. A steady-state diffusion model is employed such that theLaplace equation is satisfied. The thickness of the diffusionboundary layer ξ is assumed to be 1 cm according to previousreports (28, 38). The cw is set to be the equilibrium vapor pres-sure of supercooled water at the surface temperature (−12 °C),that is, psat,w = 223 Pa, or, equivalently, 0.091 mol/m3 at ambienttemperature (23.5 °C), and c0 is set to be the water vapor con-centration in the ambient air, which is 0.29 mol/m3 (or, equiva-lently, P = 707 Pa) if RH = 25% at 23.5 °C (34, 39). Bothconcentrations are calculated by using the ideal gas law. Solvingthe Laplace equation with these boundary conditions yields the

-

A B C

Fig. 1. Discontinuous frost pattern on (A and B) the natural leaves and (C) the 3D-printed artificial leaf (the area within the red dashed circle in the Leftimage is magnified in the Right image). The blue and red dashed circles represent the preferred/suppressed frosting on the convex/concave vein features,respectively.

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concentration field of water vapor shown in Fig. 2H. The con-centration isolines gradually curve toward the serrated features.The local condensation rate is characterized by the amount ofwater molecules diffusing onto the surface per unit time, or themagnitude of diffusion flux J, which is governed by Fick’s first lawof diffusion J =Dj∇cj, where D ≈ 0.28 cm2·s−1 is the mass dif-fusivity of water vapor in air at 23.5 °C (39), and j∇cj is themagnitude of local concentration gradient determined by theconcentration field at steady state. The distribution of J nearthe serrated features after being normalized by J0 = (c0 – cw)/ξ,which maximizes at the two peaks and minimizes in the valley, isshown in Fig. 2I, meaning that more water vapor diffuses to thepeaks than to the valley, although the concentration is constantregardless of the local topography, as shown in Fig. 2H. Thisinterpretation agrees with the observation of droplet sizes de-creasing from the peaks to the valley, and, as another example ofthe convex/concave topography, it supports our previous resultsof dropwise condensation on the bump/dimple structures (29,30). The abundance of supercooled water on the peaks is

beneficial for triggering freezing events by either homogenous orheterogenous nucleation, since the occurrence of homogenousnucleation events is proportional to the liquid mass, and theprobability of a nonfrozen droplet touching a neighboring het-erogenous nucleation site is higher by increasing the water−solidcontact area (see the schematic in SI Appendix, Fig. S8).

Evaporation. For a quantitative analysis of how the system vari-ables would affect the areal fraction of the frost-free band, whichis of both scientific and practical interest, we chose the surfacegeometry, represented by the vertex angle α (defined in Fig. 2A),and the ambient humidity level RH to be two independent var-iables. Experimental results in Fig. 3 A and B show that the frostcoverage (f, the areal fraction of area covered by ice between thepeaks, defined in Fig. 2F) increases as α or RH increases, whileall other variables remain fixed. To explain these results, theboundary conditions shown in Fig. 3C were used to describe thediffusion of water vapor near the serrated features. The serratedsurface is divided into two domains: the liquid domain in the valley

A D E

B

C

F

HG I

Fig. 2. Condensation frosting on serrated surfaces inspired by leaves. (A) Schematic of an aluminum serrated surface. The images show an apparent contactangle θ* = 114° on hydrophobic surface, and 18° on superhydrophilic surface. (Scale bars, 1 mm.) Droplet volume is 5 μL. (B) Schematic of the experimentalsetup for controlled frost growth (not drawn to scale). A humidity sensor connected to an external humidity controller is not drawn, for simplicity. (C) Timelapse images showing the condensation (80 s), fast propagation (200 s), evaporation (910 s), and ice-free (2,270 s) bands. The surface has a vertex angle of 60°.The regime between the yellow dashed lines indicates the frost-free zone. (Scale bars, 1 mm.) (D and E) Micrographs of the frosting process at the (D) peakand (E) valley. The surface has a vertex angle of 90°. (Scale bars, 0.2 mm.) The red and blue dashed lines indicate the relative positions of the peak and valley,respectively. Yellow dashed circles show the distribution of droplet sizes condensed near the peak and valley regions. Both surfaces are hydrophobic. Theambient humidity is 25% at T = 23.5 °C, and surface temperature is −12 °C. (F) Schematic of the frosting mechanism. (G) Boundary conditions for simulatingthe diffusion of water vapor during the condensation stage. The schematic is not to scale. (H) The concentration field of water vapor and (I) the normalizedflux magnitude, both near the serrated surface features.

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(denoted by “L” in Fig. 3D) where the concentration on theboundaries is cw = 0.091 mol/m3 and the ice domain (denoted by“S” in Fig. 3D) where a lower concentration, ci = 0.080 mol/m3, isused. Both concentrations used for the boundary conditions wereobtained from the equilibrium vapor pressure of ice (psat,i = 196Pa) and supercooled water (psat,w = 223 Pa) at a surface temper-ature of −12 °C, using the ideal gas law. The vapor diffusion flux inthe domain near the condensed liquid in the valley is determinedby two competing mechanisms: the incoming flux from the am-bient air due to the high ambient humidity c0 > cw and the out-going flux toward the frost front due to ci < cw. As the frostcoverage increases, the low-pressure−induced evaporation domi-nates diffusion in the liquid domain. A critical frost coverage f p1exists, above which all fluxes onto the wall in the “liquid phasedomain” become negative (i.e., going outward). Physically, thismeans that all water droplets will evaporate until they vanish whenf > f p1 . This shift in the direction of flux is shown in Fig. 3D. Usingf p1 as an estimation of the areal frost coverage seen after depletionof droplets in the valley, the predicted values are plotted againsteach of the two independent variables (α and RH) in Fig. 3 A andB (evaporation model, open circles). A good agreement with the

experimental results is observed. The predicted frost coverages areslightly lower than the experimental values, indicating that theboundary layer thickness of 1 cm used for simulations could belarger than that in reality (see SI Appendix, Fig. S9 for the simulatedresults by using a range of ξ values) (38, 40, 41). It is worth noticingthat, in the limiting case, f → 0 as α→ 0°, while f → 1 as α→ 180°.This is because the iced peaks are close enough to each otherwhen α→ 0°, such that the entire surface can be protected bythe low vapor pressure of any small amount of ice at the peaks,due to their proximity. However, when α→ 180°, the serrated sur-face becomes flat, and a 100% frost coverage is thermodynamicallyinevitable.

Stability of the Frost-free Zone after Evaporation. The rate of frostinvasion toward the valley becomes negligible once the evapo-ration is completed. SI Appendix, Figs. S2 and S3 plot the frostcoverage as a function of time. The curves are asymptotic to zeroslope after evaporation, indicating frosting is not favored. Herewe discuss the stability of the frost-free zone after evaporationagainst frosting due to condensation and desublimation. To un-derstand the reason why no condensation is present in the valley,

Fig. 3. Formation of the frost-free band. Increasing (A) the vertex angle of the serrated features α or (B) the ambient humidity RH results in a wider arealfrost coverage. Experimental measurements of the frost coverage (crosses) agree with the predictions made by simulations using the evaporation model(open circles) and the no-condensation model (open diamonds). (C) Boundary conditions for simulating evaporation of droplets in the valley. (D) Diffusionflux in the valley is positive (water vapor diffuses onto the wall) when frost coverage is low (50%), and negative (water vapor diffuses out from the wall) whenfrost coverage is high (62%). The arrows indicate directions and do not represent their magnitudes. (E) Boundary conditions for simulating diffusion of watervapor after evaporation. (F) Water vapor in the valley is supersaturated when frost coverage is low (42%), and undersaturated when frost coverage is high(58%). S, L, and Dry stand for solid, liquid, and dry domain, respectively.

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diffusion of water vapor is simulated by using the boundaryconditions shown in Fig. 3E. We consider the walls in the valleyto be impermeable to water vapor following complete evapora-tion of the liquid water. Fig. 3F shows that the water vaporconcentration in the valley is higher than cw at a narrow frostcoverage, while the vapor phase becomes undersaturated forcondensation to happen once the frost coverage exceeds acritical value f p2 . This simulation result suggests that the frost-free zone is thermodynamically stable against condensationfrosting if f > f p2 ; f

p2 is plotted versus α and RH in Fig. 3 A and B

(no-condensation model, open diamonds). It is clear that theevaporation model predicts higher frost coverage values thanthe no-condensation model does. This means that the frostcoverage is wide enough to prevent condensation once evaporationis completed.However, the concentration of water vapor in the valley pre-

dicted by the no-condensation model is still greater than ci, anddesublimation may take place. To understand the stability against

frosting by desublimation, we estimate the concentration of watervapor in the valley by using the boundary conditions for the no-condensation model together with the critical frost coverage pre-dicted by the evaporation model. For α= 60°, RH = 25%, and f =60%, the water vapor concentration on the wall maximizes at thevalley in the dry domain, which indicates a supersaturation of c/ci(or, equivalently, p/psat,i) around 1.06 for ice nucleation (34). Thenucleation rate of ice ðrÞ in the valley can then be predicted byusing the classical nucleation theory,

r= r0 exp

0B@−NA

RT4πγ3

3�RTVm

ln�

ppsat

��2 f ðθÞ

1CA, [1]

where r0 is a kinetic constant, NA is the Avogadro constant, R isthe ideal gas constant, T is the surface temperature, p is the pressureof water vapor in air, θ is the equilibrium contact angle of ice nuclei

A B

C D

Fig. 4. Analytical model with the boundary conditions for predicting the areal frost coverage on the serrated surface. (A) Boundary conditions in each phaseregime between two peaks. (B) Scaling relation that connects the concentration at the boundary of the vapor and solid domain (cb, defined in A) with thegeometric parameter (α, defined in Fig. 2A) and the ambient water vapor concentration (c0, defined in A). (C) Scaling relation between the frost-free zone(1 − f, defined in Fig. 2F) and the combined parameter composed of cb and α. Simulated data points are based on the evaporation model using 20° ≤ α ≤ 90°,and 20% ≤ RH ≤ 100%. (D) Map of frost-free zone as a function of the vertex angle (α) and ambient relative humidity (RH) with experimental results. Whilestate-of-the-art microtextured/nanotextured surfaces show no frost-free zone (i.e., 100% frost coverage), the vein of the artificial leaf (α = 45°) in Fig. 1C anda serrated surface with α = 40° show the predicted frost-free zone (i.e., 50% frost coverage) for 5 h (SI Appendix, Fig. S3G) under a similar frost condition (RH =25 to 50%, surface temperature is <−10 °C). The Inset shows the frost-free coverage on the veins of an artificial leaf. The frost coverage that corresponds withvertex angle and ambient humidity further corroborates the results of the map.

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on the substrate, and γ, Vm, and psat are the surface energy, molarvolume, and saturation vapor pressure, respectively, of the ice nu-clei. The shape factor f ðθÞ= ð2+ cos θÞð1− cos θÞ2. Pluggingp/psat,i = 1.06 and T = 261 K into Eq. 1 yields r=r0 to be around10−4,100 (see SI Appendix for the detailed values of each parameterused for this calculation). This effectively makes the nucleationrate r in the valley become negligible, and the frost-free zone canbe considered to be antifrosting over a great length of time.

Dependence of the Frost Coverage on Surface Topography andAmbient Conditions. The first-order approximation of the frostcoverage when the vertex angle is small is of particular interest tous because it shows a chance to achieve a high frost-free fraction(1 − f → 1). Using the coordinates and boundary conditions setby the evaporation model in Fig. 4A and assuming steady-statediffusion ðΔc= 0Þ, here we estimate the water vapor concentra-tion field above and between the surface serrations (42). Mostimportantly, we are interested in how fast the concentrationdecays along the y axis, that is, c(x = 0, y). To do so, we divide thespace of interest into three sections (vapor, solid, and liquiddomain, denoted by V, S, and L, respectively) and try to seek forsolutions in each section by applying proper conditions at theV−S and S−L boundaries. For the V−S boundary, we first as-sume the concentration distribution is an arbitrary functiongðxÞ= cðx, y=HpÞ, and gðx= 0Þ= cb. This yields the solutions inthe V and S domains in the form of Fourier series. By invokingthe continuity of ∂c/∂y at the V-S boundary, one can determine cbin terms of the surface geometry ðαÞ and the ambient watercontent ðc0Þ in the limit of α→ 0,

cb − cic0 − cb

∝ tanα

2. [2]

The linearity is captured in Fig. 4B. We then plug this relationinto the solution to the S domain. By keeping only the basebandof the series, we found that, along the y axis, the concentration ofwater vapor in the S domain decays as

cðx= 0, yÞ≈ ci +GLHp

ðc0 − cbÞ�ξ− yH

� π2 tan α2, [3]

where G is a constant, L is the peak-to-peak distance, and ξ is thetotal thickness of the boundary layer. To ensure the flux in theL domain is negative, that is, evaporation, the water vapor con-centration at the S−L boundary is chosen to be the satura-tion vapor pressure of water ðcwÞ, that is, cðx= 0, y=HppÞ= cw.This yields an estimation of the frost coverage defined asf = ðHpp −HpÞ=H,

lnð1− f Þ= 2πtan

α

2ln�cw − cicb − ci

�− lnðGÞ. [4]

Fig. 4C shows this scaling relation between lnð1− f Þ andtanðα=2Þln½ðcw − ciÞ=ðcb − ciÞ�, where the fitted slope (0.68) isclose to the value of the constant coefficient of 2=π ≈ 0.64 inEq. 4 (see SI Appendix for detailed derivation). Eqs. 2 and 4suggest that the ambient humidity (c0) contributes to the criticalfrost coverage f by affecting cb, which is a result of diffusion inthe vapor phase domain, while the surface topography (α) notonly restricts the magnitude of cb but also accelerates the con-centration decay in the solid phase domain. Eqs. 2 and 4 are usedto estimate the required geometry (α) for a desired fraction offrost-free zone (1 − f) given the ambient conditions (c0 or RH),and the results are shown in Fig. 4D. As clearly dictated by thecontour map, reduced frost coverage is possible by decreasingthe vertex angle of the serrations, α at even RH = 100%.

Although the condensation frosting map in Fig. 4D shows that,as α→ 0, the frost coverage reduces to 0 regardless of the am-bient conditions, it is important to acknowledge the practicallimitations that prevent us from achieving a completely non-frosted surface using surface serrations. Firstly, our model onlydiscusses the in-plane growth of the frost coverage; in reality, how-ever, frost also grows perpendicular to the plane (i.e., out-of-planegrowth that increases the thickness of frost layer). Decreasing thevertex angle decreases the peak-to-peak distance, and therefore in-creases the chances for the frost to connect out of the plane. Sec-ondly, our model is strictly diffusion based, and does not considerthe effects of various types of convection. Outdoor conditions areoften complex with forced convection, which may make a 0% frostcoverage nonviable (43).In those field conditions, the environment involves forced

convection such as winds, which thins the diffusion-controlledboundary layer above the serrated surface, and changes the direc-tion of the mass fluxes overall. Although the main focus of thisstudy is the mechanism of diffusion-limited processes, we reportsome results associated with frost formation on the serrated surface(hydrophobic, α = 40°, Tsurf = −10 °C) under forced convection(Tambient = 23.5 °C, RH = 25%) in this context. Airflow was gen-erated parallel to the surface texture, and the air velocity away fromthe surface was measured to be around 0.4 m/s. The frost formationprocess and results in Movie S2 and SI Appendix, Fig. S10 show thatthe frost coverage (f) is higher compared to that under the naturalconvective condition (fforced = 83% vs. fnatural = 50%). However,because the air motion is deflected by the surface serrations, theconvective effects in the valley regions are minimized. The masstransfer in the valley appears to still be diffusion dominated. Aclear frost-free band is still visible in the valley, further supportingthe effectiveness of the surface serrations in antifrosting applica-tions under forced convection conditions.

ConclusionWe have comprehensively studied the condensation frostingprocess on macrotextured surfaces inspired by the frost patternon natural leaves. Our results show that frosting initiates fromthe peak and undergoes a multistage process on chemically ho-mogenous surfaces including a superhydrophilic surface. A frost-free zone forms in the valley and is found to be able to resistfurther frosting for a long period of time. The spatial span of thenonfrosted area increases when the vertex angle of the serratedfeature and the ambient relative humidity decreases. Numericalsimulations of the concentration field near the serrated surfacesuggest that diffusion plays a key role in the formation of thefrost-free zone, and show good agreement with the experimentalresults. An analytical model has been developed to correlate thefrost-free fraction with the surface geometry and ambient con-ditions. Finally, the stability of the frost-free zone is discussed byusing classical nucleation theory. The low supersaturation in thevalley makes nucleation of ice by desublimation physically im-possible for a long period of time. These results suggest that themacroscopic surface topography found on natural leaves or ar-tificial surfaces that possess a similar length scale of surface texturesor other physicochemical properties (e.g., wettability or thermalconductivity; see SI Appendix, Fig. S4) can be applied to realizespatial control of frost formation in a wide range of applicationssuch as aviation, wind power generation, and infrastructure.

Materials and MethodsFabrication of Aluminum Surfaces with Serrated Patterns. The serrated sur-faces with various vertex angles defined in Fig. 2A (α = 40°, 60°, 90°, and100°) were fabricated by a simple molding procedure. The molds with cor-responding geometric designs were first printed out by a 3D printer (Form 2,Formlabs, clear resin). The surface patterns were then transferred to a thinaluminum sheet (Al 1100, 0.127 mm in thickness; McMaster-Carr) by pressingit between the molds. The height of the peaks is kept to 5 mm, and the

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length and width of the samples are 50 and 40 mm, respectively. The arti-ficial leaf with concave vein structures was fabricated by the same 3D printer(Form 2, Formlabs, black resin). The vein features were 5 mm wide and 6 mmdeep, and were concave. The leaf structure was designed to be hollow with0.4-mm wall thickness such that the surface could be readily cooled down bycirculating liquid coolant.

Fabrication of Hydrophobic and Superhydrophilic Leaf-like Serrated Surfaces.The patterned aluminum surfaces are intrinsically hydrophilic, with a watercontact angle of 81°. The patterned samples were cleaned by oxygen plasma(SBT PC 2000 Plasma Cleaner) for 1 min to remove the organic contaminants.To obtain hydrophobicity, the cleaned samples were immersed in 1 wt %solution of the fluoroaliphatic phosphate ester fluorosurfactant (FS-100;Pilot Chemical) in ethanol at 70 °C for 30 min. Superhydrophilic sampleswere made by boiling the cleaned surfaces in water for 30 min to undergothe boehmitization process. The apparent contact angles on the super-hydrophilic and hydrophobic surfaces are measured to be 18° and 114°,respectively.

Controlled Frosting Measurement. The visualization of the frosting processwas done inside a customized chamber (107 cm × 38 cm × 30 cm, L × W × H).The relative humidity level inside the chamber was maintained by using anelectronic humidity controller (Model 5100-240; Electro-Tech Systems Inc.)connected with an ultrasonic humidifier (Pure Enrichment) and an externalsource of dry air. The air velocity inside the chamber was kept at a low level(<0.1 m/s) to minimize the effects of forced convection. Samples were taped(3M Scotch Double-Sided Conducted Copper Tape, 12.7 mm wide and0.04 mm thick) onto a 3D-printed cooling unit, one side of which has thesame pattern as the aluminum sample to maximize the thermal contact. Asmall wall thickness of 0.5 mm of the cooling unit was used to minimize thetemperature difference across the aluminum surface. The cooling unit was

connected to an external circulating liquid chiller (7L AP; VMR) to keep thesurface temperature of the samples at −12 ± 0.3 °C, which was measured bya digital thermometer (HH66U; OMEGA). The ambient temperature was23.5 ± 0.5 °C. The samples were vertically positioned, and a plastic cover wasused to isolate the surfaces from the ambient air before the temperature ofthe sample surface was stabilized, at which point t = 0 is defined. Thefrosting processes were recorded using a Nikon D5500 camera with a macrolens (Nikon AF-S DX Micro NIKKOR 40 mm f/2.8G) and a handheld digitalmicroscope (Dino-Lite Premier AF3113T). Fig. 2B shows a schematic of theexperimental setup for the controlled frosting experiments. Frosting on the3D-printed leaf was performed at the same surface temperature of −12 ±0.3 °C, and the relative humidity was kept to 50% at the ambient temper-ature of 23.5 ± 0.5 °C to test the effects of the distinct geometric features(such as macroscopic concave texture) of natural leaves.

Numerical Simulation of Diffusion by Using COMSOL Multiphysics©. Models forsteady-state transport of dilute species were used to numerically simulate thediffusion of water vapor near the serrated features; two-dimensional coor-dinates were employed for all simulations. The models were built using thecross-sectional geometries of the serrated surfaces to be studied. Boundaryconditions were chosen depending on the specific stage to be simulated. SeeFigs. 2G, 3C, and 3E for the boundary conditions used for each simulation.

Data Availability. All data presented in the figures and SI Appendix will bemade available to readers upon request.

ACKNOWLEDGMENTS. We are grateful to C. Li and X. Zhang for helpfuldiscussions on the mathematical analysis. This work was partially supportedby the Water Collaboration Seed Funds program of the Northwestern Centerfor Water Research.

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