General Frost Growth Mechanism on Solid Substrates with Different

Article pubs.acs.org/Langmuir

General Frost Growth Mechanism on Solid Substrates with Different Stiffness Julien Petit* and Elmar Bonaccurso Center of Smart Interfaces, Technische Universität Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany S Supporting Information *

ABSTRACT: Preventing or delaying frost formation on surfaces is of significant importance in many aspects of our daily life. Despite many efforts and improvements recently achieved in the design of new icephobic materials and substrates, not all proposed solutions are universally applicable and frost formation still remains a problem in need of further flexible solutions. In this respect, we propose to take benefit from the tunable viscoelastic properties of soft polymer gel substrates, since they are known to strongly influence the dropwise condensation process of water, and to investigate condensation frosting on them. Using polymer gels with different stiffness and a hard substrate as a reference, we demonstrate their ability to delay frost formation compared to recent results reported in the literature on other solid substrates and in particular on superhydrophobic surfaces. By investigating the frost front propagation we singled out a general behavior of its dynamic evolution consisting of two processes presenting two different time scales. This general growth appears to be independent of experimental conditions as well as substrate stiffness.

1. INTRODUCTION Frost formation and propagation on surfaces generally lead to undesirable consequences on many aspects of our daily life, from the safety of aircraft1,2 and other transportation vehicles to the power lines transmission3 and efficiency of wind turbines4,5 or on heating, ventilation, and air conditioning (HVAC) systems and telecommunication apparatus.6 The strategies usually employed to prevent or delay frost formation consist of using active chemical, thermal, and/or mechanical processes that are energetically and economically demanding.7−9 Thus, passive anti-icing methods, i.e., working without the use of chemicals or electrical power, are of great interest. During the past decade, numbers of scientific studies have been devoted to development of new types of materials or coatings for preventing frost formation and accretion. In particular, high efficacy was expected from superhydrophobic surfaces (SHS).10,11 The strong water-repellent properties of SHS combined with the low contact angle hysteresis induce a high-energy barrier for ice nucleation and low ice adhesion.6,12−18 Furthermore, some theoretical and experimental works have highlighted that if the micro/nanoscale structure of the surfaces is well chosen19 these surfaces are able to repel impacting supercooled liquid droplets (SLD).20,21 Nevertheless, the losses of the superhydrophobic character in particular under condensation frosting conditions (which are usually encountered in nature) have recently generated debate about SHS as best candidates for icephobic material.18,22−29 Indeed, it has been demonstrated that in conditions where condensation is favorable, the liquid water is no longer in a Cassie−Baxter state but penetrates (partially or completely) the surface structure. © 2014 American Chemical Society

Thus, during freezing of liquid water, the ice and surface become mechanically interlocked. This results in an increase of the ice adhesion strength. However, from the thermodynamic definition of an icephobic material, established recently by Hejazi et al.,27 such a material must exhibit low ice adhesion strength which is not the case when SHS undergoes condensation frosting. Very recently, some attempts were done to circumvent the initial Wenzel state in which nucleated condensed droplets appear on SHS by additionally nanostructuring the surface, but the conclusion was that formation of frost remained “inevitable”.19 In parallel, researchers have taken inspiration from nature to synthetize specific liquid-repellent surfaces. They consist of the combination of micro/nanoporous substrates imbibed or infused with a lubricating liquid (generally oil immiscible with water). These types of surfaces, called “slippery liquid-infused porous surfaces” (SLIPS), have been demonstrated to have omniphobic, self-cleaning, self-healing, and “exceptional” icephobic properties.30−32 However, recent work has revealed that the self-healing properties of SLIPS can be lost after few icing and deicing cycles, due to migration of the lubricating liquid (perfluorinated oil) from the wetting ridge.33 This oil depletion, driven by capillary forces, implies the use of liquid reservoirs to refill the porous substrate and conserve the anti-icing performance of SLIPS. Therefore, it appears that SLIPS also Received: October 22, 2013 Revised: January 8, 2014 Published: January 13, 2014 1160

dx.doi.org/10.1021/la404084m | Langmuir 2014, 30, 1160−1168

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suffer from drawbacks to prevent frost formation and frost accretion. Thus, further investigations are essential to find new materials able to prevent frost formation or at least delay it to a large extent. One of the main problems usually related to the previous studies is the way water is deposited or grows on the substrate before freezing starts (in particular, in case of water condensation). Actually, different modes of condensation exist (from filmwise to dropwise), depending on the wettability of the substrate, and can influence the freezing process.14,34 A few years ago, Sokuler et al. reported that condensation differs significantly whether the substrate is “soft” or “hard”.35 They employed soft elastic polymer substrates based on poly(dimethyl siloxane) (PDMS) with different densities of crosslinking in order to vary the deformability of the substrate. One major result from this work was that the softer the substrate was (i.e., by decreasing the cross-linking density), the higher the density of nucleated droplets and the faster the condensation rate were. They explained these experimental observations by a reduction of the activation barrier for nucleation. Indeed, condensed droplets on soft materials can deform the substrate and so reduce the free energy of nucleated droplets, because the more energetic water−air interface is reduced. At the same time, the Laplace pressure compresses the substrate underneath the drop, whereas the surface tension of the liquid pulls the substrate up. These result in formation of a wetting ridge along with the perimeter of the nucleated droplet and can, on lowshear modulus substrate, be so high that it prevents contact between two adjacent droplets.36−39 This interferes with the coalescence process usually described in dropwise condensation processes,40−42 and more area is available for new droplets to nucleate, thereby increasing the surface area covered by liquid drops. In the present study, we propose using the control of the condensation process offered by these soft polymer gel substrates on the investigation of frost formation. Using a homemade microscope device, we quantify the microscale antiicing performance of a variety of PDMS substrates with different cross-linking densities and one “hard” polymer substrate consisting of poly(methyl methacrylate) (PMMA).

Table 1. Young’s Modulus, CA of Water, and Roughness Measurements on PDMS and PMMA Samples sample Young’s modulus (MPa) (Error of ∼10%) CA (deg) mean roughness (nm)

PDMS 10:1

PDMS 20:1

PDMS 30:1

PDMS 40:1

PMMA

5.4

1.8

1.1

0.7

3000.0

111 ± 2 0.7

119 ± 2 0.7

121 ± 2 1.5

124 ± 2 1.1

72 ± 2 1.1

performed by analyzing the profile of several sessile water drops on each substrate with the drop shape analysis system DSA 100 (Krüss GmbH, Hamburg, Germany). The CA for the PDMS substrates was almost constant with an average value of θPDMS = 119 ± 2°. The PMMA used was Plexiglas Superclear (0.80 mm thickness, Evonik Röhm GmbH, Darmstadt, Germany), and CA was θPMMA = 72 ± 2°. We used a MFP-3D (Asylum Research, Santa Barbara, CA) atomic force microscope for surface roughness measurements in noncontact mode. The cantilever used was a Tap190-G (BudgetSensors, Innovative Solutions Bulgaria Ltd., Bulgaria) with a tip radius < 10 nm (manufacturer information). The average deviations from a flattened plane over an area of 10 × 10 μm2 did not exceed 1.5 nm; thus, all substrates can be considered smooth on a nanometer scale (Table 1). All experiments were performed with a homemade microscope setup including a cooling system. It consisted of a powerful cooling plate (AHP-1200CAS associated with CCP-31, TECA Corp., Chicago, IL) and visualization from the top with a CCD camera (1 fps, 1280 × 1024 pixels, PL-E531MU, PixeLINK, SVS-VISTEK GmbH, Seefeld, Germany) equipped with an adapter tube (2× Zoom 6000 system, Navitar, Rochester, NY) and a microscope objective (10× M Plan Apo, Mitutoyo, Kawasaki, Japan) which allowed a resolution of 0.36 μm/pixel. The substrate was illuminated by a cold light source (KL 1600 LED, Schott AG, Mainz, Germany) and enclosed in a custommade sealed experimental chamber where the relative humidity (rH) was controlled by means of two mass flow controllers (MFC 358, Analyt-MTC GmbH, Müllheim, Germany) and monitored by a humidity/temperature sensor (SHT75, Sensirion AG, Staefa, Switzerland) built in under the top window of the experimental chamber (i.e., ∼10 mm above the surface of the substrate). An incident nitrogen gas stream was first separated in two channels, one called “dry” and the other “wet”. One channel was thus at rH = 100% and the other at 0%. By adjusting the flow rate of each mass flow controller and with the humidity sensor we controlled the rH in the experimental chamber (accuracy ±1.8% between 10% and 90%). We assured a homogeneous rH in the experimental chamber due to the larger inlet gas stream flow rate applied (200 cm3/min) compared to the volume of the experimental chamber (∼ 106 cm3). The ambient temperature in the experimental chamber is also monitored by the sensor and considered homogeneous in the chamber. Before starting the cooling, the temperature and rH are maintained constant. Finally, to avoid possible contamination of the sample by dust from the ambient room atmosphere, the experimental chamber was enclosed in a large ambient chamber. The temperature in this ambient chamber was maintained almost constant using a homemade water circulation at fixed temperature regulated by a chiller (Ministat 125, Peter Huber Kältemaschinenbau GmbH, Offenburg, Germany). A scheme of the setup is shown in Figure 1. All data were analyzed by combination of a self-written image processing MATLAB script (R2011b, MathWorks, Natick, USA) and the open source Image J (1.46r, National Institute of Health, USA).

2. MATERIALS AND METHODS Realization of soft substrates based on a PDMS gel (Sylgard 184, Dow Corning, Wiesbaden, Germany) consisted of mixing a defined ratio of oligomer (base) and cross-linker (curing agent). The ratio PDMS:cross-linker determined the elasticity of the substrate. Here, we used PDMS:cross-linker ratios of 10:1, 20:1, 30:1, and 40:1 presenting Young’s moduli ranging from 0.3 to 2.5 MPa. The Young’s moduli were obtained via nanoindentation measurements with a scanning probe microscope and evaluated with the Hertz model for elastic deformations.43 After mixing the two components, the mixtures were degassed for 30 min under vacuum and then PDMS substrates with a height of approximately 1 mm were cast. Substrates were cured overnight (12 h) in an oven at 70 °C under vacuum. In order to remove all non-cross-linked PDMS (free oligomers) from the substrates after curing, we immersed them in tetrahydrofuran (THF, ROTISOLV, HPLC grade, Carl Roth GmbH, Karlsruhe, Germany) for 3 days. THF was daily exchanged. Afterward, substrates were rinsed with new THF and finally carefully dried over a period of 3 days to avoid surface wrinkle formation. Before starting experiments, each substrate was rinsed with ethanol and distilled water and dried in a nitrogen stream to avoid any contamination. Then, each substrate was deposited on a microscope glass slide (1 mm thickness, Carl Roth GmbH, Karlsruhe, Germany). Static contact angles (CA) for the different substrates are reported in Table 1. Measurements were 1161


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3. RESULTS AND DISCUSSION We studied frost formation from dropwise water vapor condensate on cold soft and hard substrates in controlled environment. 3.1. Dropwise Condensation. As recently reported by Sokuler et al.,35 condensation on soft viscoelastic substrates differs considerably from condensation on hard nondeformable substrates. The condensation patterns obtained for different ratios between base and cross-linker are illustrated Figure 2a. Thereby, we confirm the previous results35 by showing that (i) the softer the substrate is (i.e., here PDMS 40:1) the higher the nucleation density is and (ii) the softer the substrate is the higher the condensation rate is. These trends are also quantified by the graph in Figure 2b representing the surface coverage defined as the ratio between the surface area covered by condensed water droplets and the total surface areaover time. The argument advanced by Sokuler and collaborators to explain the increase of nucleation density with the softness of the substrate was that condensation on such substrate includes complex processes due to the triple phase contact line (TPCL) deforming the substrate and moving over it. Indeed, the increase of the nucleation density was due to the decrease of the energy barrier for nucleation on soft substrates. Following a thermodynamics approach, Eslami et al.44 demonstrated that nucleation is easiest on a soft substrate where deformation of the substrate by the condensed droplets plays a major role on the decrease of the free energy barrier for heterogeneous nucleation. In addition, deformation of the substrate by condensed drops induces formation of a ridge at the TPCL, delaying (or preventing) coalescence of adjacent drops,

Figure 1. Scheme of the setup for the study of frost formation on soft substrates.

Figure 2. (a) Optical microscope images of the density of condensed water droplets on soft PDMS substrates. Substrates used are of PDMS:crosslinker ratios ranging from 10:1 (“hardest”) to 40:1 (“softest”). Images correspond to the situation after ∼1 h at a substrate temperature of Tset = −10 °C and under rH = 90%. (b) Dynamic evolution of the surface coverage corresponding to experiments presented in a. (c) Behavior of the dew point temperature with respect to rH (full and open symbols with “+” and with “×” correspond to Tset = −10, −20, −30, and −40 °C, respectively). Solid line corresponds to a fit of all data by an approximation of the Magnus−Tetens formula given by eq 11 from ref 45 (R2 = 0.70). 1162


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microscale study discussed here allows highlighting a particular and recurring process by which frost propagates along the substrate. Actually, freezing starts from the edges of the substrate where more defects are localized and constitute natural nucleation sites because of the geometric singularity and the low free energy barrier for heterogeneous ice nucleation. Then, frost propagates from the first “edge nucleus” and forms a frost front growing parallel to the substrate. This propagation occurs from droplet-to-droplet in a well-organized manner. First, frost grows isotropically from the first nucleus and at a certain point starts to feel the SLD in the near environment. At this point, the growing of the frost front becomes oriented toward one or several target SLD, forming dendrites that we will call hereafter “ice bridges”. At that time, the target SLD starts to evaporate during the progression of the ice bridge. When the ice bridge reaches the target SLD, the latter immediately freezes and becomes a new ice nucleus from which the frost front will begin another propagation. Thus, this process is a chain reaction involving two major mechanisms which are formation of ice bridges associated with evaporation of target SLD as depicted in Figure 3 (this process can be clearly observed on the experimental movie in the Supporting Information). Surprisingly, this description of the frost front propagation from droplet to droplet has been only very recently reported in the literature on SHS46,47 and on smooth hard hydrophobic substrates.48 Since this mechanism of frost front propagation appears to be similar on different substrates undergoing condensation frosting, we have chosen to present the optical microscope images from our experiments in Figure 3 according to a similar representation used in previous studies by other groups.46,48 It is worth noting that under “extreme” conditions of rH and substrate temperature, some SLD can spontaneously freeze. As we will see, this characteristic can influence the dynamics of the freezing process. 3.3. Characterization of the Freezing Process on Soft Substrates. 3.3.1. Freezing Time and Duration. Characterization of the freezing process, qualitatively described above, begins by investigating the freezing time and the duration of it. The freezing time is the time needed to observe the first ice nucleus from the time we observed the first liquid condensed droplet, and the freezing duration is the duration needed to observe complete freezing of all droplets in the region of interest. We report in Figure 4a the dependence of the freezing time on the substrate used, and we conclude that the substrate does not play a major role on the freezing time because the latter is weakly dependent on the stiffness of the substrate. Besides, as we have already discussed above, the stiffness of the substrate greatly influences the surface coverage (Figure 2b), and naturally this would be influencing the freezing process afterward. In Figure 4b, we show the evolution of the final surface coverage (i.e., the surface coverage at the end of the condensation part and just before freezing starts in the region of interest) with respect to rH for all substrates at Tset = −30 and −40 °C. We demonstrate that when rH and the PDMS:cross-linker ratio increase, the final surface coverage also increases. Furthermore, we establish that the final surface coverage is slightly higher on the PMMA substrate. This is due to the stronger wettability of this substrate compared to the soft PDMS ones. Indeed, the CA of the PMMA is smaller than on PDMS substrates, and so water-condensed droplets spread more on PMMA than on PDMS, thus increasing the surface area covered by liquid. We report in Figure 4c and 4d the impact of the final surface coverage (and so, intrinsically, the

promoting nucleation of new droplets, and so increasing the surface coverage. In Figure 2c, we report the evolution of the dew point temperature (i.e., substrate temperature needed to observe the first nuclei of water condensed droplets) with respect to rH. The dew point temperature decreases by decreasing rH, and the general trend follows a logarithmic behavior characteristic of an approximation of the Magnus− Tetens formula usually used to determine the dew point temperature of a system.45 The fit realized on all data, Figure 2c, results from eq 11 in ref 45 as Td = [1 − (T ln(rH/100)/ (L/Rw))]−1, where Td is the dew point temperature in Kelvin, T is the temperature in Kelvin, L is the enthalpy of vaporization (considered constant over the temperature range investigated and taken as L = 2.5 × 106 J kg−1), and Rw is the gas constant for water vapor (Rw = 461.5 J K−1 kg−1). Since the condensation process on soft substrates is not the core interest of the current study, we will not go into further detail in the present paper. 3.2. Frost Formation. 3.2.1. Influence of the Stiffness of the Substrate and Experimental Conditions (Relative Humidity and Substrate Temperature) on Frost formation. We investigated a large number of different experimental conditions with five different substrates. We used four “soft” substrates based on various PDMS:cross-linker mixing ratios (10:1, 20:1, 30:1, and 40:1) and one “hard” solid PMMA substrate as a reference. The experimental protocol was the same for all substrates. For each substrate, we studied four rH (30%, 50%, 70%, and 90%), and for each rH, the substrate was cooled from about 20 °C to four different substrate temperatures (Tset = −10, −2, −30, and −40 °C). An overview of the main results obtained 1 h after the beginning of the cooling is presented in Table 2. Table 2. Overview of the Results of Freezing Induction for All Substrates and Depending on the Experimental Conditions (rH and set temperature), 1 h after the Beginning of the Coolinga rH \Tset

−10 °C

−20 °C

−30 °C

−40 °C

30% 50% 70% 90%

× PMMA PMMA PMMA

PMMA PMMA and PDMS 40:1 all substrates all substrates

PMMA all substrates all substrates all substrates

PMMA all substrates all substrates all substrates

× corresponds to a situation where no ice was observed in the region of interest.

a

From Table 2, it appears that at low humidity (i.e., rH = 30%) and/or moderately low substrate temperature (Tset = −10 °C) frosting is not observed on soft substrates but only on the hard one (PMMA). It is worth noting that for the experimental conditions of rH = 30% and Tset = −10 °C, no frosting was observed in the region of interest of the substrate (i.e., its center, the field of view of our microscope) whatever the substrate used. The first soft substrate on which frost formation was observed was the more viscous one (i.e., PDMS 40:1). We will discuss afterward this particularity, but this seems to be related to the larger surface density of supercooled liquid droplets (SLD) compared with the other soft substrates. 3.2.2. Identification of the Mechanisms Involved in the Freezing Process. As we can see in Table 2, although we consider soft substrates, frosting can be induced under very low substrate temperatures and/or very high rH. Nevertheless, the 1163


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Figure 3. Sequence of images showing the propagation of the frost front along a soft substrate. Substrate shown here is PDMS 30:1 under rH = 70% and substrate temperature Tset = −30 °C. In this sequence, ice propagation starts from the SLD labeled 0 and freezes droplet 1 via an ice bridge. Then two ice bridges develop from the SLD 1 toward the SLD 2 and 3 that freeze upon contact with the ice bridge and develop in turn other ice bridges toward the SLD 4 and so on. Colored dashed circles superimposed on three different SLD demonstrate evaporation of them during the progress of the frost front. Scale bar represents 20 μm.

external source is involved. To answer this question we, in first approximation, considered that advancement of the ice bridges is linear in time, as previously reported.48 We estimated the average velocity from n different ice bridges, for each experiment, as ⟨V⟩ = 1/n∑i n= 1Li/ti (where Li corresponds to the total distance separating the frozen droplet and the target SLD i and ti is the time related to the propagation of the ice bridge i). The results for a substrate temperature Tset = −30 °C are presented in Figure 5. Although the substrate PDMS 40:1 is the most viscous one, the average velocity of the ice bridge is the fastest. Energy dissipation during self-propagation of an ice bridge on a viscous substrate should be a brake for the advancement because of additional friction in the motion. Due to this effect, known as “viscoelastic braking”35,38,39 we should observe a global decrease of the velocity when the viscosity of the substrate increases. Therefore, we can conclude that the propagation of the ice bridges results from an accumulation of matter from an external source and is not due to a self-sustained mechanism. As discussed above, propagation of the frost front is associated to growth of ice bridges and evaporation of SLD. As first identified by Guadarrama-Cetina et al.,48 we confirmed that water vapor comes from evaporating SLD feeding ice bridges via the well-known Wegener−Bergeron−Findeisen (W−B−F) effect.52−54 The ice bridges act as sinks and the SLD as sources of water vapor. This results in an oriented growth of the ice bridges toward the SLD. The W−B−F effect occurs during phase transitions in mixed phase clouds, where ice crystals and SLD are in coexistence. Because the saturation vapor pressure over ice is lower than over liquid water at the same temperature (e.g., the difference in saturation vapor pressure over the two species is about 25% at Tset = −30 °C) a

impact of the stiffness of the substrate) on the duration of the freezing process for Tset = −30 and −40 °C, respectively. As far as the final surface coverage is increased, the freezing process is faster. In particular, the fastest freezing duration observed on soft substrates is obtained for the most viscous one (PDMS 40:1), and this property is closely related to the greater final surface coverage of this substrate. Whereas it is clearly identified, in Figure 4c at Tset = −30 °C, that the freezing process is faster when the final surface coverage increases, this global trend is less well defined at a lower substrate temperature (Figure 4d). At Tset = −40 °C, generally the freezing duration is slightly faster. Actually, under this “extreme” condition some SLD can spontaneously freeze. We suppose, on one hand, that these spontaneous events occur due to the greater thermal gradient between the SLD and the substrate. This thermal gradient induces more pronounced convection in SLD that spontaneously freeze without connection with the frost front. On the other hand, close to this substrate temperature, we approach the temperature range promoting homogeneous ice nucleation from SLD.49−51 Therefore, the probability to observe spontaneous freezing events becomes larger. These features result in a faster global freezing process because they add to the frost front propagation. 3.4. Characterization of Ice Bridges. 3.4.1. Frost Front Propagation. We analyzed the propagation of the frost front on our viscoelastic substrates according to previous studies reporting this process on other types of substrates (SHS46 and hard smooth surfaces48). One interesting question raised by Guadarrama-Cetina et al.48 concerned advancement of the ice bridges (called “dendrites” in their case). Indeed, we can try to work out if this ice bridge growth is self-sustained or if an 1164


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Figure 4. (a) Freezing time on various substrates used (closed and open symbols correspond to Tset = −30 and −40 °C, respectively). (b) Final surface coverage versus rH (closed and open symbols correspond to Tset = −30 and −40 °C, respectively). (c) Freezing duration versus final surface coverage at Tset = −30 °C. (d) Freezing duration versus final surface coverage at Tset = −40 °C. (c and d) Symbols closed, open, and with “+”correspond to rH = 50%, 70%, and 90%, respectively.).

evaporating completely before being reached by an ice bridge. Therefore, we will not consider this limitation of the frost front propagation mechanism in our later analysis. 3.4.2. Growth Dynamics of Ice Bridges. In order to more precisely investigate the frost front propagation on soft polymer gels, we report in Figure 6a the dynamic evolution of the ice bridge lengths for all substrates. We have chosen to present the results for rH = 70% and Tset = −30 °C because they are the most representative of the freezing process described above. Indeed, at higher Tset, icing is a little more difficult to induce on all substrates, and at lower Tset some spontaneous events modify the global process (as discussed above). In fact, growth of the ice bridge in time is not simply linear, but two distinct dynamic behaviors can be identified. First, bridge length evolves linearly in time until a certain couple time and length, which we will call hereafter critical time tc and critical length Lc. Second, all curves deviate from this linear behavior and become nonlinear, reflecting an acceleration of the ice bridge growth when the target SLD gets closer. Concerning the first linear part of the bridge growth (before tc, L = αt with α being the slope of the linear growth) we can estimate the order of magnitude of the growth velocity of the ice bridges and find V ≈ 0.18 − 0.64 μm/s, which is in accordance with the results from Guadarrama-Cetina et al.48 Furthermore, by comparing the linear part of the curves to the final accelerations, we establish that on average ice bridges evolve linearly for about 80% of the total distance separating the frost front and the target SLD and for about 65% of the total duration of the

Figure 5. Evolution of the average velocity of ice bridges versus the substrate used at Tset = −30 °C. Dashed line is only a guide for eye.

subsaturated environment arises over the liquid and a supersaturated environment over the ice, inducing vapor diffusion from SLD to ice. In addition to the mechanism discussed above, in some specific cases other features can occur. Indeed, as shown by Guadarrama-Cetina et al.,48 when the distance separating the ice bridge and the target SLD is too large or when the diameter of the target SLD is too small, the target SLD can completely evaporate before the ice bridge can reach it. This depletion phenomenon, depending also on the surface coverage, is not so significant in our case. In fact, the surface coverage on our viscoelastic substrates with varying stiffness is sensibly higher, and we observe only very few SLD 1165


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the ice bridge approaches the SLD, resulting in a possible acceleration of the evaporation rate of the target SLD. 3.5. Evaporation of Target SLD. In the previous section, we demonstrated that ice bridges start to accelerate when the distance separating the tip of the frost front from the target SLD represents about 20% of the total length. Our view is that at this “point” the evaporation rate of the target SLD starts also to increase in order to feed more the ice bridge, globally accelerating the ice bridge growth process. To support this hypothesis, we report for each tc a critical target droplet diameter (noted Dc). We represent in Figure 7 the normal-

Figure 7. Target droplet diameter squared normalized by the critical target droplet diameter squared (corresponding to the diameter at t = tc) over the time normalized by the critical time. Solid and dashed lines correspond to linear regressions before and after t/tc = 1, respectively. Experimental conditions are rH = 70% and Tset = −30 °C.

Figure 6. (a) Dynamic evolution of the ice bridge length L for all substrates under the experimental conditions of rH = 70% and Tset = −30 °C. Two different dynamics can be identified, first a linear evolution and then a deviation in a nonlinear trend. Solid lines are linear regressions (R2 = 0.99). (b) Representation of the ice bridge length normalized by the critical length (Lc) with respect to the time normalized to the critical time (tc). All experiments presented have been performed at Tset = −30 °C (symbols closed, open, and with “+”correspond to rH = 50%, 70%, and 90%, respectively). Solid line corresponds to a linear regression of all the data for t/tc ≤ 1 (R2 = 0.99), and the dashed curve is a power law fit on all data for t/tc > 1 as L/L = 0.97(t/tc)1.97 (R2 = 0.80).

ization of the target drop surface area by the critical target drop surface area with respect to the time normalized to the critical time as (D/Dc)2 = f(t/tc) (at Tset = −30 °C and rH = 70%). Evaporation of a SLD is very sensitive to its nearby environment. In particular, if other ice spots develop in the surrounding, the latent heat released during freezing of these neighboring droplets induces a thermal gradient between them and the target SLD that can also enhance drop evaporation. This environmental influence is reflected by the nonuniform slopes of the different linear regressions in Figure 7 before and after t/tc = 1. We also take into consideration the fact that the elasticity of the substrate plays a role on the condensation as well as evaporation processes. Indeed, Lopes et al. reported recently that the softer the substrate is, the shorter the evaporation duration and higher the drop evaporation rate.55,56 These conclusions make the ice bridge growth mechanism even more complicated. Nevertheless, even if it is difficult to single out all involved processes quantitatively, data show that (D/ Dc)2 evolves with two different linear decays and with a higher slope (i.e., a higher evaporation rate) after t/tc = 1. In particular, the evaporation rate is higher on softer substrates (due to the stronger drop pinning), and we have already discussed that the average ice bridge velocity increases when the stiffness of the substrate increases (Figure 5). Therefore, we could confirm that ice bridge growth is faster on softer substrates due to the higher evaporation rate and that the final acceleration of ice bridges results from a larger evaporation rate of the target SLD as well. This is related to the increase of the saturation vapor pressure gradient existing between ice and SLD.

bridge growth. Therefore, in order to accurately describe the dynamics of the freezing process, it is important to take into account this final acceleration. After extracting for each case tc and Lc, we normalize the evolution of the ice bridge growth with respect to this couple of parameters as L/Lc = f(t/tc). In Figure 6b, we plot the normalized results for all experiments performed and demonstrate that all data rescale in a master trend composed of two processes. This master trend corresponds to a linear evolution for t/tc ≤ 1 and to a power-law after t/tc = 1 with an exponent very close to 2. Since for a number of different substrates of different stiffness and for a number of different experimental conditions we observed that all data rescale in a master trend composed of two dynamic processes, we can deduce that the growth mechanism of individual ice bridges has a general character, that is, the freezing process evolves via combination of the ice bridge growth and evaporation of the target SLD with two different dynamic behaviors: one slow before tc and the second faster after tc. This appears to be independent of the properties of the substrate or the experimental conditions. As done in refs 46 and 48 we could also interpret the final acceleration of the process by the W−B−F effect. Indeed, the saturation vapor pressure gradient between the ice bridge and the target SLD increases as 1166


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4. CONCLUSIONS We used soft polymer gel substrates made of poly(dimethyl siloxane) to investigate frost formation from condensed supercooled liquid droplets. We demonstrated that considering such soft substrates allows one to delay (or prevent in some cases) frost formation with respect to other solid substrates, like hydrophilic, hydrophobic, superhydrophobic (SHS), or slippery liquid-infused porous surfaces (SLIPS). For example, whereas on these latter substrates frosting is induced at Tset = −10 °C under rH ≈ 65%, no ice is observed on soft PDMS substrates. Thus, control of the condensation pattern allowed by the tunable viscoelastic character of the PDMS substrates opens a new route in designing icephobic materials. The most significant findings are as follows. First, condensation frosting proceeds via combination of two major mechanisms that are propagation of the frost front via ice bridges (between frozen droplets and target SLD) associated with evaporation of the target SLD. This process is closely related to a well-known effect occurring in atmospheric phenomena, the Wegener−Bergeron−Findeisen effect. Second, we demonstrated that growth of ice bridges is composed of two processes, linear in the beginning and nonlinear later, reflecting an acceleration of the freezing process when the tip of the frost front has reached about 80% of the total distance separating the frozen droplet and the SLD. By normalizing the ice bridge length with respect to this critical length for a number of different substrates (viscoelastic and hard) and for a number of different experimental conditions, we could point out the general character of the ice bridge growth processwhatever the experimental conditions as well as the properties of the substrates. These findings are of great importance in the characterization and design of new types of anti-icing substrates and contribute to our global understanding of the complex mechanism of frost formation and possibly of way(s) to circumvent it.

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REFERENCES

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ASSOCIATED CONTENT

* Supporting Information S

Video S1 shows the frost front growth on soft polymer gel substrate; condensation frosting is observed on a soft PDMS substrate 30:1 at Tset = −30 °C and rH = 70% (it results from association of ice bridges growth and evaporation of supercooled liquid droplets (scale bar represents 20 μm)). This material is available free of charge via the Internet at http:// pubs.acs.org.

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Article

AUTHOR INFORMATION

Corresponding Author

*Phone: +49 6151 16 70825. Fax: +49 6151 16 2048. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Longquan Chen, Antonio Criscione, and Marcus C. Lopes for fruitful discussions. We thank Andreas Plog for the Young’s moduli measurements. We thank Lars O. Heim for the roughness measurements with the atomic force microscope. This research was supported by the German Research Foundation (DFG) within the cluster of Excellence 259 “Smart Interfaces−Understanding and Designing Fluid Boundaries”. 1167


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General Frost Growth Mechanism on Solid Substrates with Different

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