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## Creator

Pritam Kumar Roy, Yui Takai, Rui Matsubara, [Mizuki Tenjimbayashi](https://orcid.org/0000-0002-8107-8285), Timothée Mouterde

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[Hot liquid marbles](https://mdr.nims.go.jp/datasets/8a91e2e3-80d8-4790-9aca-72c1bccbdf69)

## Fulltext

1  1  2  3 Hot liquid marbles 4  5 Pritam K. Roy1, Yui Takai1, Rui Matsubara1, Mizuki Tenjimbayashi2 and Timothée Mouterde1* 6 1. The University of Tokyo, Department of Mechanical Engineering, School of Engineering, 7 Hongo 7-3-1, Bunkyo, 113-0033, Tokyo, Japan. 8 2. Research Center for Materials Nanoarchitectonics (MANA), National Institute for Materials 9 Science (NIMS), 1-1 Namiki, Tsukuba, Ibaraki, 305-0044, Japan. 10  11 Timothée Mouterde 12 Email:  mouterde@g.ecc.u-tokyo.ac.jp  13 Author Contributions: T.M. conceived the project. T.M. and P.K.R designed the project. M.T. 14 provided particles and contributed to the discussions. P.K.R. performed most experiments to 15 which Y.T. and R.M. also participated. P.K.R., T.M. and Y.T. performed the analyses. T.M. built 16 the model. T.M. and P.K.R. wrote the manuscript with inputs from Y.T. and M.T. 17 Competing Interest Statement: T.M. and P.K.R declare that they are listed as an inventor on a 18 patent application, Japanese Patent Application No. 2024-099815. that relates to the findings 19 described in this manuscript. 20 Classification: Physical science, Applied Physical Sciences. 21 Keywords: drops; liquid marbles; static friction; condensation. 22 This PDF file includes: 23 Main Text (with Figures) 24 Figures 1 to 5 25   26 mailto:mouterde@g.ecc.u-tokyo.ac.jp  2  Abstract 27  28 In the insect realm, liquids become traps due to capillary and viscous forces dominant at their 29 scale. Yet, aphids handle the highly viscous honeydew droplets they secrete, by coating them 30 with hydrophobic wax powder which maintains an air layer between their body and the liquid. 31 These coated droplets, known as liquid marbles, exhibit low friction and high mobility, enabling 32 manipulation of small liquid volumes which is useful for biomedical analysis where sample 33 volumes are limited, chemistry to reduce chemical waste, or digital microfluidics for large-scale 34 cell culturing and drug testing. For such applications—including exothermic reactions or biological 35 studies typically conducted above room temperature—the ability to carry hot liquid is important 36 but remains unexplored. This article investigates the stability and static friction of hot liquid 37 marbles placed on a substrate cooler by ∆T. We show that for large ∆T, the core liquid 38 evaporates and condenses within the air layer below the marbles creating liquid bridges resulting 39 in marble rupture on hydrophilic substrates and increased static friction on hydrophobic ones. The 40 temperature difference modifies the static friction nature, solid friction dominates at small ∆T and 41 is replaced by a liquid pinning force at larger ∆T due to the increased liquid bridge density 42 resulting from condensation. Finally, our study provides ways to avoid the rupture and increased 43 static friction of hot liquid marbles due to the bridge formation by increasing the particle size, 44 decreasing the liquid volatility, or using nanostructured superhydrophobic substrates. 45 Significance Statement 46 Droplets coated with hydrophobic particles, known as liquid marbles, exhibit ultra-low friction as 47 an air layer separates liquid from solid. This enables manipulation of small liquid volumes without 48 losses, with applications in biomedical analysis, digital microfluidics, and chemistry. Yet, their 49 capacity to carry hot liquids remains unexplored. This research examines the stability and static 50 friction of hot liquid marbles placed on cooler substrates. We show that, on hydrophilic surfaces, 51 temperature differences cause rupture due to condensation bridging the core liquid with the 52 substrate, while on hydrophobic surfaces, bridging increases static friction, shifting its nature from 53 solid to liquid. Our model provides strategies to prevent rupture and friction, with larger particles, 54 lower liquid volatility, or superhydrophobic substrates, broadening liquid marbles’ potential. 55 Introduction 56  57 Below the millimeter scale, droplets are generally pinned due to the dominance of viscous and 58 capillary forces [1-4]. Limiting their pinning is essential for applications that require manipulating 59 small amounts of liquids, such as chemistry—where reducing chemical waste is crucial—or in 60 biomedical analysis and biology, where sample volumes are limited. Low static friction and high 61 mobility can be achieved with superhydrophobic materials on which hydrophobic structures 62 stabilize an air layer between the liquid and the solid, limiting the effect of surface tension and 63 viscosity [5-6]. Another approach consists of forming liquid marbles by covering the drop with 64 hydrophobic powder [7]. The hydrophobic structure is then directly embedded at the liquid 65 surface, making the non-wetting state independent of the substrate. While liquid marbles are 66 sometimes regarded as a simple variation of superhydrophobic surfaces; fixing the hydrophobic 67 textures on a solid or leaving them mobile at a liquid surface leads to distinct wetting behavior 68 compared to superhydrophobic surfaces. For example, drops on superhydrophobic materials and 69 liquid marbles have different static and dynamic frictions [8-9] and bouncing properties [10-14]. 70 The particles’ shell also allows for bringing additional functions to liquid marbles by modifying the 71 particles’ properties, thus offering them a strong potential for applications [15-17]. Notably, 72 magnetite particles allow the positioning and opening of liquid marbles with magnetic fields [18], 73 while silver particles permit ultrasensitive electrochemistry and surface-enhanced Raman 74 spectroscopy [19], suitable for lab-on-a-chip applications. The literature on liquid marbles 75 continues to expand in domains as diverse as biology [20-22], chemistry [23], sensing [24-25] or 76 materials science [26]. Among these applications, temperature plays an important role in enabling 77   3  various functionalities. For instance, thermal Marangoni flows allow to move marbles placed on 78 liquid bath [27-30], while the on-demand release of the marble content can be achieved through 79 the melting of wax particles upon heating [31]. When liquid marbles are placed on a hot surface, 80 their particle shell can limit evaporation and prevent boiling similar to the Leidenfrost effect [32]. 81 Conversely, upon freezing, the properties of the particle shell control the final shape of the liquid 82 marbles, which can deviate from that of drops on superhydrophobic surfaces [33-34]. Additionally, 83 cooling down liquid marbles provides a noninvasive refilling mechanism through the condensation 84 of atmospheric water across the porous shell [35]. Condensation effects may also occur in the 85 opposite scenario, when the core liquid is warmer than its surroundings. This raises the broader 86 question of marbles’ ability to contain and transport hot liquids, an aspect that remains largely 87 unexplored. In this study, we investigate the stability and static friction of hot liquid marbles 88 placed on a cooler substrate. This situation is particularly relevant for biology and chemistry, 89 where cell cultures, analyses at 37°C or exothermic reactions can create temperature differences 90 between the liquid and the substrate. 91 Hot liquid marble stability  92 We first investigate the stability of hot liquid marbles in contact with hydrophobic or hydrophilic 93 silicon wafers kept at room temperature To = 24 ± 1°C. Hydrophobicity is obtained by chemical 94 vapor deposition of 1H,1H,2H,2H-perfluorodecyltrichlorosilane (see Methods section ‘Substrate 95 preparation’), which provides water advancing and receding contact angles of a = 126 ± 96 2°r  = 99 ± 2°, respectively. This surface is compared with hydrophilic pristine silicon, a = 23 ± 97 2°; r = 12 ± 2° (see SI Appendix, Text S1 and Fig. 1). We form the hot liquid marbles by coating 98 water drops with temperature To + ∆T, density  and volume  = 50 µL with lycopodium grains of 99 typical size d ≈ 30 µm (Fig. 1a), so that there is an initial temperature difference ∆T between the 100 marble and its substrate (see Methods section ‘Hot liquid marbles preparation’ and SI Appendix, 101 Fig. S2). We observe the evolution of the hot liquid marble and show its state on Fig. 1b around 3 102 minutes after deposition for ∆T = 0°C and ∆T = 16°C, respectively. Without temperature 103 difference (∆T = 0°C, left), the liquid marble shape and apparent contact angle are independent of 104 the substrate wettability, as expected due to the absence of contact between the liquid and the 105 substrate. Conversely, when the water is warmer than the substrate (∆T = 16°C, right) the liquid 106 marble shape strongly depends on the surface wettability. In the hydrophilic case, the liquid 107 marble ruptures and spreads on the substrate, resulting in a decreased apparent contact angle 108 ~ 34 ± 2° and an increased contact radius ~ 12 mm. In contrast, the temperature difference in the 109 hydrophobic case only results in a minimal change of shape and apparent contact angle when 110 compared with the isothermal situation, ∆T = 0°C. We quantify the shape evolution by measuring 111 the final apparent contact angle f of the hot liquid marble as a function of ∆T (Fig. 1c). On the 112 hydrophobic substrate (red data), f slightly decreases from 157 ± 2° to 144 ± 2° when increasing 113 ∆T from 0 to 29°C. In contrast, with the hydrophilic substrate (blue data), when the temperature 114 exceeds a critical value ∆Tc ≈ 1.5 ± 1°Cf drops from a plateau at 154 ± 2° to a lower plateau at 115 18° ± 2°. The values of the two plateaus correspond to the isothermal liquid marble apparent 116 contact angle for (∆T < ∆Tc) and to the water contact angle on the pristine silicon wafer (∆T > ∆Tc) 117 (SI Appendix, Fig. S1).  118 To understand the rupture above ∆Tc, using a high-speed camera (Fastcam Nova S6, Photron), 119 we record the evolution of the apparent contact angle  with time t (taking t = 0 when the marble 120 first touches the substrate) for different initial ∆T and we report the measurements in Figure 1d 121 (Movie S1). The graph reveals two types of curves: for ∆T < ∆Tc, the marble is stable, and the 122 contact angle remains constant at 154 ± 2° as expected from the results of Figure 1c. For ∆T > 123 ∆Tc, the liquid marble is initially stable with an apparent contact angle value close to the 124 isothermal case (∆T = 0°C) but ruptures after a time tr (Fig. 1d, vertical dashed lines) when its 125 apparent contact angle drops within a few tens of milliseconds to the small final value shown in 126 the Figures 1b and 1c. This graph also reveals that the rupture time tr decreases with increasing 127 temperature differences, which we now further explore. We plot in Figure 1e the rupture time tr as 128 a function of ∆T, the graph contains two main information: (i) tr is a decreasing function of ∆T, (ii) 129   4  slightly above ∆Tc, the rupture time is very sensitive to the temperature difference as increasing 130 ∆T from 3 to 14°C results in a 170-fold decrease of tr from  6 s to  36 ms. 131 Origin of the hot liquid marbles’ rupture  132 The hot liquid marble stability depends on the substrate wettability which implies that the core 133 water contacts with the substrate. To unveil this mechanism, we place the liquid marble on a 134 hydrophilic glass substrate (see Methods section ‘Substrate preparation’ and SI Appendix, Fig. 135 S1) and observe the bottom air layer with a total internal reflection (TIR) microscope (see Fig. 136 2a). The TIR microscope consists of illuminating the interface with a collimated light beam with an 137 angle of incidence between the critical angle for glass/air and glass/water interfaces so that light 138 is only transmitted at the glass/water interface (see Methods section ‘Total Internal Reflection 139 Microscope’). The images reflected at the interface between the bottom of the liquid marble and 140 the glass slide are captured with a high-speed camera (Fastcam Nova S6, Photron) at a frame 141 rate of 500 or 4000 fps for ∆T = 1.6°C and 16°C, respectively. The TIR movies reveal the 142 dynamic of the water/substrate contacts. In the isothermal case (∆T = 0°C), all the light is 143 reflected to the camera (Fig. 2b). This demonstrates the absence of water/substrate contact due 144 to the air layer formed by the particles and explains why, for ∆T = 0°C the liquid marble shape is 145 independent of the substrate wettability (Fig. 1b, 1c). On the contrary, for hot liquid marbles 146 (∆T > 0), shortly after deposition the contact area darkens due to the nucleation of condensation 147 droplets below the marble which water/glass interface transmits light (Fig. 2c). For ∆T < ∆Tc when 148 the marbles are stable, the condensation evaporates after a few seconds to reach a final state 149 where all the dark spots disappear, identical to that of isothermal liquid marbles (Fig. 2c top, 150 Movie S2). For larger ∆T ≈ 16°C (Fig. 2c bottom, Movie S2), condensation forms and grows 151 quickly until t ≈ 0.46 s, when a darker spot appears and rapidly expands on the substrate. This 152 corresponds to the marble rupture with the spreading of the core liquid on the hydrophilic 153 substrate.  154 These observations allow us to explain and model the origin of the rupture mechanism: when the 155 hot liquid marble is brought in contact with the colder substrate, the water vapor contained in the 156 hot air surrounding the marble nucleates and condenses within the air layer separating the hot 157 core liquid from the substrate. The difference of saturated vapor mass concentration ∆csat(∆T) = 158 csat(To + ∆T) − csat(To) between the hot evaporating interface at temperature To + ∆T and the 159 colder condensation droplet kept at To, creates a diffusive vapor flux D∆csat/h responsible for the 160 condensate growth, where D ≈ 20 mm2 s-1 is the diffusion coefficient of water vapor in air, and h 161 the distance between the core liquid and the condensation droplet (Fig. 2d). After a typical time 162 tcond  h2/D∆csat(∆T), the condensation height reaches h and forms a liquid bridge between the 163 core liquid and the substrate resulting in the liquid marble rupture. The TIR images reveal that the 164 first bridging event (Fig. 2c, t ≈ 0.46 s) corresponds to the core liquid spreading on the hydrophilic 165 substrate. Consequently, the relevant length h for rupture should be the minimum of the air layer 166 thickness (Fig. 2d) which is on the order of the particle diameter h  d, hence the rupture time: 167 (1)     tr  d2/D∆csat(∆T) 168 Taking the values previously mentioned for D and , and using the Rankine formula for csat(T) 169 (see Methods section ‘Saturated vapor concentration’), d is the only free parameter in Eq. (1), 170 which we use as an adjustment parameter. The best fit is obtained for d = 19.7 ± 2.1 µm (95% 171 interval of confidence) and plotted in black in Figure 1e. We find that our model captures well the 172 evolution of the marble rupture time, and the value for d is in excellent agreement with the typical 173 lycopodium diameter of  30 µm (Fig. 1a and SI Appendix, Fig. S3a). However, this model 174 predicts that even for small ∆T marbles should rupture while experiments show that they remain 175 stable.  176 To understand the stability of marbles for ∆T < ∆Tc, we need to consider the liquid marble cooling 177 due to the surrounding atmosphere. The condensation process only occurs while there is a 178   5  temperature difference between the liquid and the substrate, and the liquid marble remains stable 179 if the condensation time tr is larger than the cooling time  which we now evaluate. Considering 180 that the marble cools down due to heat transfer with the outside, the energy conservation of the 181 marble writes Cd∆T/dt = − S∆T, where C is the specific heat capacity of water,  the water/air 182 heat transfer coefficient and S is the liquid marble surface area. The temperature difference 183 decays exponentially ∆T(t) = ∆To exp(−t/ with a characteristic cooling time S)*C. In the 184 range of volume explored, liquid marbles are flattened by gravity and can be approximated as 185 cylinders of base contact radius Rc and height 2a, where a = (/g)1/2 is the capillary length, with  186 the effective marble surface tension [36] which we approximate as that of pure water at room 187 temperature  ≈ 72 mN/m and g ≈ 9.81 m/s2 the acceleration of gravity. With this approximation, 188 the volume-to-surface ratio takes a simple expression: S Rca/2Rc  a, hence the cooling 189 time Ca Taking  ≈ 1000 W/m2, ≈1000 kg/m3 and C ≈ 4100 J/kg/K, we find  to be 190 around 10 s. This value is in excellent agreement with the time evolution of ∆T measured by 191 inserting a thermocouple in liquid marbles for different initial ∆T and plotted in Figure 2e, where all 192 the data are well fitted by an exponential decay with cooling time of  = 11 s (plain line).  193 The cooling dynamics allow us to predict the transition from stable to rupturing marble, which 194 should occur for the temperature difference ∆Tc where the condensation time equals the cooling 195 time tr(∆Tc) = . For ∆T < ∆Tc, (tr > the marble cools faster than the condensation time required 196 to form the first liquid bridge which explains the marble stability as the core liquid remains 197 disconnected from the substrate. Conversely, for ∆T > ∆Tc condensation is faster and a liquid 198 bridge forms before thermalization, resulting in the marble rupture. For small ∆T, we approximate 199 ∆csat(∆T) with a Taylor expansion of the Rankine formula: ∆csat(∆T) = csat(To)5120∆T/To2, this 200 leads to a simplified expression of Eq. (1) tr = d2To2/5120Dcsat(To)∆T, which we plot on Fig. 1e in 201 red and is found to also nicely fits the data with d ≈ 18.3 ± 2.6 µm. This approximation also 202 provides an explicit formula for ∆Tc: ∆Tc = d2To2/5120Dcsat(To). Using the previously defined 203 values, our model predicts ∆Tc = 1.2 °C, which indeed separates the stable from unstable 204 marbles (Fig. 1c, 1e).  205 Hot liquid marbles static friction 206 The air layer between the substrate and the liquid is responsible for both the large apparent 207 contact angle and the low friction of the liquid marbles. The invasion of the air layer by 208 condensation results, on hydrophilic substrates, in the marble rupture characterized by a low 209 apparent contact angle and high static friction. Conversely, on hydrophobic substrates, the hot 210 liquid marble shape is only marginally affected by the temperature difference (Fig. 1c), here we 211 explore how condensation affects the static friction of hot liquid marbles.  212 We deposit lycopodium hot marbles ( = 50 µL) on a hydrophobic silicon substrate, we wait for a 213 minute—a time larger than the cooling time —to avoid transient effects (see SI Appendix, Text 214 S2 and Fig. S4) and we then increase the tilt angle between the surface and horizontal (see 215 Methods section ‘Static friction experiments’) until we reach the critical value * where the marble 216 departs from the substrate (Fig. 3a, left images and Movie S3). This slope provides us with the 217 static friction F which is equal to the projected weight mg sin *, where m =  mpis the liquid 218 marble mass with  and mp the liquid and particles mass, respectively (SI Appendix, Tables S1 219 and 2). We repeat this experiment for different ∆T and we plot in red in the Figure 3b, the static 220 friction F as a function of ∆T. For temperature differences smaller than 5°C, the friction remains 221 constant around 13.5 ± 1.5 µN which corresponds to only 3% of the marble weight, thus marbles 222 depart from their substrate for a small tilt angle 1.6 ± 0.5°. When increasing ∆T from 5 to 15°C the 223 static friction rises by 25-fold before reaching a plateau at  330 µN (70% of the marble weight). 224 The hot marbles are then strongly pinned and move only for large tilts around 45° (Fig. 3a left, 225 Movie S3). To understand the static friction mechanism, we perform a separate experiment where 226 we observe the wet contact, with the TIR setup previously described,  5 minutes after deposition 227 on a hydrophobic glass substrate (see Methods section ‘Substrate preparation’). We show the 228   6  TIR images corresponding to each ∆T of Figure 3a (right images for each ∆T and SI Appendix, 229 Fig. S5 and Movie S4). The images reveal two points: (i) contrary to the hydrophilic case, when 230 condensation forms a bridge between the substrate and the core liquid, the marble does not 231 spread on the substrate and the air layer below the marble is only partially invaded by water. We 232 quantify this partial invasion of the air layer by defining the wetted fraction  as the liquid/solid 233 contact area normalized by the liquid marble contact area Rc2 where the contact radius Rc is 234 measured from side view images (SI Appendix, Fig. S6). (ii) The wetted fraction increases with 235 the temperature difference ∆T from no contact = (∆T = 3°C) to a very wet contact  ≈ 0.9 at ∆T 236 = 29°C.  237 We plot in Figure 3b (blue data, right-side vertical scale), the wetted fraction  as a function of ∆T, 238 the evolution of (∆T) is very similar to the friction F(∆T) with a low plateau value below 5°C, 239 followed by a rapid increase up to 15°C from which  plateaus. To confirm this observation, the 240 temperature-induced static friction ∆F = F(∆T) – F(∆T = 0) is plotted as a function of , Figure 3c, 241 this reveals the linearity between the two quantities ∆F   242 We further explore the nature of the static friction of hot liquid marbles by changing their size; for 243 a given ∆T, we measure the evolution of the friction as a function of the marble volume . The 244 results, plotted in Fig. 3d with a log-log scale, reveal a change in the friction nature between low 245 and high-temperature differences. At small ∆T (blue data), the static friction is small and seems 246 linear with the volume F  , while for larger ∆T (red data), the friction is typically 20 times larger 247 and the slope close to 0.5, F  0.5.  248  Model of hot liquid marbles static friction 249 We now model the static friction of hot liquid marbles. At small ∆T, the data suggest a solid/solid 250 friction as described by Jin et al. [8], the contact between the liquid marble and the substrate 251 remains dry ( = 0), and the Coulomb friction applies. The static friction is proportional to the force 252 normal to the substrate through a static friction coefficient µ: F = µ g cos  which balanced 253 with the horizontal component of the weight leads gives the critical tilting angle * = atan µ. At the 254 sliding angle, neglecting the mass of the particles (mp/ ≈ 0.02 << 1, see Methods section 255 ‘Particles’ weight measurements’ and SI Appendix, Tables S1 and S2the static friction can be 256 rewritten as:  257  (2)               Fdry = ρΩ𝑔 µ √1+µ2  258 The equation (2) is plotted in blue in Figure 3d with µ = 0.023 obtained from the sliding angle for 259 ∆T = 0°C. It describes well the static friction of marbles in the low-temperature regime and 260 explains the linear dependency with the volume, i.e. F  .  261 We then model the case where ∆T is sufficiently large for condensation to partially fill the air layer 262 ( > 0). The liquid bridges create an additional force: moving the receding contact line by dx 263 requires replacing the liquid/solid interface of area Rcdx with an air/solid and a liquid/air 264 interface. The required energy corresponds to the static friction force work and writes 265 Fdx = Rcdx(1 + cos o) with o the Young contact angle of the liquid on the subjacent 266 substrate. As previously mentioned, for the studied volumes, liquid marbles are flattened by 267 gravity so that their volume  is that of a cylinder of base Rc2and height 2a. Thus, the contact 268 radius writes Rc ≈/2a and we obtain the static friction force:  269 (3)       Fwet ≈ (1 + cos o)a270 Qualitatively, our model predicts that static friction should be proportional to the wetted surface 271 area F  , which is consistent with the force and TIR measurements shown in Figures 3b and 3c. 272 Furthermore, the model also captures the increase of the friction with the square root of the 273   7  volume F   observed for large ∆T in Fig. 3d. Quantitatively, taking  = 50 µL,  = 72 mN/m, a 274 = 2.7 mm, and the measured value of o = 99°, Equation 3 is best fitted to the experimental data 275 in Fig. 3c with a prefactor 1.19 ± 0.12 (95% confidence level) as represented on Fig. 3c by the red 276 curve and the blue shaded area.  Using the same prefactor, we plot our model Eq. (3) in 277 Figure 3d, taking for the wetted area  ≈ 0.9 measured at large ∆T (Fig. 3b). We also find an 278 excellent agreement with the experimental data and the value of the adjustment coefficient very 279 close to unity confirming that our model describes quantitatively the static friction of hot marbles. 280 The small deviation observed may stem from the approximation of the contact radius, Rc 281 ≈/2a, which remains accurate within 10% over the explored volume range or from the 282 small relative variation (13% on average), of Rc with ∆T, which we did not account for (Fig. S6). 283 Alternatively, it could arise from the contact angle uncertainty , which introduces a relative force 284 uncertainty: |Fwet(o) – Fwet (o)|/|Fwet(o)| = | sin o|/(1 + cos o). For o = 99° and °, this 285 results in an error of only 4%, though it increases for superhydrophobic substrates where o 286 approaches 180°. 287 These results clarify the friction transition, as the temperature difference ∆T increases, the 288 formation of liquid bridges transforms the nature of the static friction from the low dry friction Fdry 289 [Eq. (2)] characteristic of liquid marbles, to a significantly larger wet pinning force Fwet [Eq. (3)]. 290 For 50 µL marbles, the crossover occurs when the wetted area  is as small as 4% which further 291 demonstrates why avoiding liquid/solid contact is key to the low friction of non-wetting states in 292 general. This value of the liquid/solid contact fraction is typically that of superhydrophobic 293 materials which explains the similar pinning strength despite their difference of nature.  294 Dry to wet static friction transition temperature – effect of particle size and liquid volatility 295 Our model predicts the amplitude of the static friction of hot liquid marbles depending on the 296 evolution of the fraction of wetted area , we now focus on understanding the critical temperature 297 at which  (hence F) increases leading to the dry-to-wet friction transition. The transition depends 298 on the dynamics of the liquid bridge formation, a mechanism similar to the one discussed for the 299 hot liquid marble stability on hydrophilic substrates where both the thickness of the air layer and 300 the liquid volatility controlled by ∆csat play a crucial role. We first investigate the effect of the 301 average air layer thickness H on the static friction. In addition to the lycopodium previously used 302 (d ≈ 30 µm), we test two other kinds of particles with different sizes: fumed silica nanoparticles 303 (d ≈ 40 nm), and hydrophobic glass beads (d ≈ 100 µm) (SI Appendix, Fig. S3 and Methods 304 section ‘Hydrophobic glass beads preparation’). While the minimum thickness of the air layer h is 305 determined by the particle size d, the average thickness H may differ from it due to particle 306 aggregation (Fig. 2d). To measure H, we dye the marble water core and a glass substrate with 307 Rhodamine B (see Methods section ‘Air layer thickness measurement’), a fluorescent amphiphilic 308 molecule, and we take 3D images of the bottom of the liquid marbles (Fig. 4a) with a confocal 309 microscope (SP8, Leica). Rhodamine B at the substrate/air and liquid/air interfaces allows us to 310 visualize the air layer and to measure its thickness H, and we obtain 30 ± 3 µm; 56 ± 2 µm and 311 242 ± 8 µm, for fumed silica nanoparticles, lycopodium and glass beads, respectively (SI 312 Appendix, Fig. S7).  313 We plot in Figure 4b, the static friction force as a function of ∆T for the fumed silica nanoparticles 314 (blue triangles) and glass beads (green squares) that we compare with the previous data for 315 lycopodium (red disks). The static friction for smaller air layer thickness (H ≈ 30 ± 3 µm, blue 316 triangles), has three notable features: (i) the shape of the curve is similar to that of lycopodium: 317 the static friction is constant at low ∆T (dry friction) before rapidly increasing to reach a plateau 318 (wet pinning). (ii) The transition temperature between the two regimes is around ∆T ≈ 5°C a value 319 smaller than that of lycopodium (∆T ≈ 12°C). (iii) The plateau value of the static friction at large ∆T 320 is identical to that measured for lycopodium, which confirms that the particles’ size does not 321 change the maximum friction as predicted by our model. In contrast, the static friction curve for 322 thicker air layer notably differs (glass beads, H ≈ 242 ± 8 µm, green squares); the increase 323 between low and high temperature is much smaller, only twofold, compared to the 25-fold 324   8  increase observed with lycopodium and silica particles. We note that for all particles, the static 325 friction varies with the wetted fraction measured with the TIR microscope (SI Appendix, Fig. S5 326 and S8 and Movies S5 and S6). For ∆T = 29°C higher frictions for lycopodium and silica 327 nanoparticles correspond to larger wetted area (Fig. 4c, blue and red frames), while the quasi 328 absence of wet contact coincides with the low friction for glass beads (Fig. 4c, green frame).  329 We now test the effect of the liquid volatility which should also affect the liquid bridge formation. 330 To this end, we measure the static friction of lycopodium hot liquid marbles made with glycerol, a 331 liquid with low volatility (vapor pressure about 10,000 times lower than that of water) due to its 332 strong hydrogen bond network [37]. The static friction of hot glycerol marbles is plotted in 333 Figure 4d as a function of ∆T (black diamonds), the difference with water marbles (red disks) is 334 striking: the friction of glycerol marbles always remains in the dry friction regime below 10 µN for 335 the whole range of explored temperature. This observation is confirmed by the absence of 336 liquid/solid contacts observed with TIR microscopy even for large ∆T = 29°C (Fig. 4c, black frame 337 and SI Appendix, Fig. S9).  338 We now model the occurrence of the dry-to-wet static friction transition. The transition happens 339 due to the liquid bridge formation, governed by the cooling and condensation dynamics. The 340 liquid cools down with a characteristic time Ca during which the temperature difference 341 maintains a water vapor flux D∆csat(∆T)/ causes condensation to grow within the air layer to a 342 typical maximum height  D∆csat(∆T)/. We note here that confocal images revealed that the 343 interface is not covered by a monolayer of particles, which makes the air layer thickness non-344 uniform. On hydrophilic substrates, the minimum of the air layer thickness, on the order of the 345 particle size d, will be the rupture point as a single liquid bridge will let the core liquid spread on 346 the surface. Conversely, on hydrophobic substrates, a single liquid bridge only marginally affects 347 the static friction. The bridge forming between hydrophobic particles as a typical size on the order 348 of the particle diameter d which increases the wetted area  by a quantity on the order of  d2/Rc2, 349 this gives with Eq. (3) and for a 50 µL lycopodium marbles, an increase of the friction force 350 around  0.3 µN, a value much smaller than the dry friction  13.5 ± 2 µN. Thus, the static friction 351 increases significantly only when the condensation height  reaches the average distance 352 between the substrate and the liquid interface given by H that we previously measured (Fig. 4a). 353 Hence, we expect the static friction to jump at ∆Tc when the ratio between the average air layer 354 thickness and the maximum condensation height given by: 355 (4)            H/HD∆csat(∆Tc)  356 becomes of order 1. To test our model, for each data point of Figures 4b and 4d, we plot in 357 Figure 4e, the value of H/ predicted by Equation (4) as a function of ∆T and we classify the data 358 into two categories by comparing the increase of static friction from the dry state: ∆F = F –359  F(∆T = 0), with the maximum force Fmax = Fwet(), which is around 330 µN. The low static 360 friction points, defined as ∆F < Fmax/2, are marked in green, while the high static friction points, ∆F 361 > Fmax/2, are marked in red, where Fmax = Fwet() ≈ 330 µN. For glycerol data, we approximate 362 the saturated vapor mass concentration with a modified Clarke and Glew equation [38], the 363 diffusion coefficient by the Chapman-Enskog kinetic theory [39], (see Methods section ‘Glycerol 364 vapor diffusion coefficient’) and the cooling time is estimated from direct measurements (SI 365 Appendix, Fig. S10). We find that the low (green) and high (red) static friction states are 366 separated by a line H/≈ 0.35;a value close to H/ ≈ 1 for which our model predicts the transition 367 between dry and wet friction to occur.  368 Static friction prevention 369 Our findings give two practical ways to prevent the static friction of hot liquid marbles by avoiding 370 the liquid bridge formation: (1) with low volatility liquids or (2) by increasing the thickness of the air 371 layer with larger particles. These friction prevention methods limit the range of liquids and 372 particles available for practical applications, but our model, Eq. (3), suggests a third way to limit 373   9  the hot marble static friction by increasing the substrate hydrophobicity. Indeed the friction 374 proportional to (1 + cos ) should vanish on superhydrophobic substrates where the contact 375 angle tends to  We test this hypothesis with superhydrophobic samples made by coating a 376 silicon wafer with a solution of hydrophobic silica nanoparticles (Glaco Mirror Coat Zero, Soft 99, 377 Japan) which forms a nanoscale roughness on the surface (see Methods section ‘Substrate 378 preparation’ and SI Appendix, Fig. S1). The nanoscale roughness is selected on purpose as it 379 maintains its superhydrophobicity even with hot water [40]. The advancing and receding contact 380 angles for water on this surface are a ≈ 161 ± 2° and r ≈ 152 ± 2°, respectively (see SI 381 Appendix, Fig. S1). We compare in Fig. 5a, F(∆T) for lycopodium liquid marbles on the 382 superhydrophobic (green triangles) and hydrophobic (red disks) substrates. On the 383 superhydrophobic surface, the static friction is strongly reduced and remains always below 70 µN 384 even for high-temperature differences, a value 5 times smaller than that of the hydrophobic 385 substrate. This decreased friction is in qualitative agreement with our model that predicts the 386 force to be proportional to (1 + cos o) a quantity that decreases when increasing the water 387 contact angle on the substrate. To further test our model, we subtract from the total static friction 388 the dry component F(∆T = 0) which does not depend on the contact angle, and we plot in Fig. 5b 389 the temperature-induced static friction ∆F = F – F(∆T = 0) normalized by (1+cos o) as a function 390 of ∆T. We take for o the receding contact angle measured on the hydrophobic H = 99° and 391 superhydrophobic substratesSH = 152°. We find that (i) the increase of friction for hydrophobic 392 ∆FH and superhydrophobic ∆FSH surfaces are close once normalized by (1+cos o), and that 393 using the superhydrophobic substrate reduces the static friction by 394 ∆FH/∆FSH ≈ (1 + cos H)/(1 + cos SH) which is around 7. (ii) For both the hydrophobic and 395 superhydrophobic cases the friction transition occurs around ∆T ≈ 12°C, which our model also 396 explains as the transition depends only on the air layer thickness and the liquid volatility which are 397 unchanged for these two cases (lycopodium with water). (iii) Finally, we note that while our model 398 predicts a saturation for the superhydrophobic case, the static friction still increases with ∆T. This 399 observation could deserve a separate study but should be related to condensation invading the 400 nanostructures of the superhydrophobic coating leading to a partial Wenzel state which 401 decreases SH and increases the hot liquid affinity with the substrate [40].  402 Conclusion 403 In this study, we investigated the possibility to manipulate hot droplets using liquid marbles. Our 404 results demonstrated that when placed on cooler substrates, hot liquid marbles exhibit distinct 405 behaviors depending on the substrate wettability: they rupture on hydrophilic ones, while adhering 406 to hydrophobic ones. This transition is due to condensation filling the air gap between the liquid 407 and the substrate previously responsible for the low static friction and high mobility of the marbles. 408 Remarkably, this condensation-induced transition not only increases the marble static friction, but 409 it shifts its nature from solid (at thermal equilibrium) to liquid for hot marbles. These temperature-410 controlled rupture and static friction have potential for application in open millifluidics systems for 411 bioanalysis or chemistry. Indeed, by creating a temperature difference between the marble and 412 the substrate (for example by cooling the substrate with a Peltier or by heating up the liquid with a 413 light source), one can selectively secure a marble position or disperse its content onto surface 414 sensors for analysis.  415 Our model also provides ways to avoid hot liquid static adhesion, this can be achieved by tuning 416 the liquid marble properties to delay the condensation-bridging with larger particles or lower liquid 417 volatility, or by using nanostructured superhydrophobic substrates to limit the bridges’ adhesion. 418 While our study focused on marbles hotter than their substrate, it would be worth investigating the 419 case of liquid marbles deposited on hot substrates and the practical case of rolling or impacting 420 hot liquid marbles where in addition to the cooling and condensation time, the dynamic contact 421 time should play a role.  422  423   10   424 Materials and Methods 425  426 Substrate preparation. The hydrophilic substrates are pristine silicon wafers (As ONE Corp., 427 Japan) or glass slides. Using the same materials, we make the substrate hydrophobic by a 40 s 428 activation step inside a plasma cleaner (Harrick Plasma, PDC-32G) followed by chemical vapor 429 deposition of 1H,1H,2H,2H-perfluorodecyltrichlorosilane (Fluorochem Ltd.) in a reduced pressure 430 atmosphere for 40 minutes. The superhydrophobic substrates are prepared by spraying a 431 commercial solution of hydrophobic silica nanobeads dissolved in isopropyl alcohol (Glaco Mirror 432 Coat Zero, Soft99). The substrates are then dried vertically and once the solvent evaporated, 433 heated on a hot plate at 200°C for 30 minutes, we repeat this process three times. The substrates 434 are renewed regularly to ensure their quality.  435 Hot liquid marbles preparation. To form hot liquid marbles, we heat up liquids (pure water or 436 glycerol) and particles (lycopodium - Sigma Aldrich, fumed silica nanoparticles - Aerosil RY50, or 437 glass beads - ASGB-320, As ONE Corp.) with a hot plate (SI Appendix, Fig. S2a). Using a 438 micropipette, we deposit a droplet of hot liquid of volume  on the particles bed, the droplet is 439 rolled until fully covered with particles and transferred on the substrate using a spatula made 440 superhydrophobic with a Glaco coating. We measure the initial marble temperature ∆T as a 441 function of the liquid bath temperature Tb by inserting a thermocouple inside the marbles. ∆T 442 varies linearly with the plate temperature, and we use the best linear fit to estimate the 443 temperature difference in the experiments: ∆T = 0.44(Tb – 23.9) with Tb in Celsius. The same 444 method is used for glycerol and gives ∆T = 0.45(Tb – 23.9) (SI Appendix, Fig. S2b). 445 Total Internal Reflection Microscope. The TIR microscope consists of a collimated 446 monochromatic LED light (M660L4, Thorlabs) shined on a dove prism (PS994, Thorlabs) with an 447 angle of incidence , adjusted to be larger than the total reflection angle for the glass/air interface 448 air = asin(nair/ng) ≈ 41° but smaller than the total reflection angle for the glass/water w = 449 asin(nw/ng) ≈ 62° or glass/glycerol interfaces gly = asin(ngly/ng) ≈ 77° with nair = 1.00, ng = 1.51, 450 nw = 1.33, ngly = 1.47, the refractive index of air, glass, water and glycerol, respectively. The glass 451 substrates are placed on top of the prism where a thin layer of immersion oil (Olympus IMMOIL-452 F30CC) is used to avoid light reflection at the prism/glass interface. The reflected light is captured 453 with a high-speed camera (Fastcam Nova S6, Photron) and the movie’s width deformed in the 454 optical path is stretched back to the correct physical with a Matlab code. 455 Saturated vapor concentration. The water and glycerol saturated vapor concentration csat at a 456 temperature T are derived from Dalton’s law: csat(T) =air(Mliq/Mair)(Psat(T)/Po) with air the air 457 density, Mliq and Mair the molar masses of water (resp. glycerol) and air, respectively, Psat(T) the 458 saturated vapor pressure of water (resp. glycerol) at temperature T, and Po is the atmospheric 459 pressure. The saturated vapor pressure of water is estimated with the Rankine formula: Psat(T) = 460 Po exp (13.7 – 5120/T), while that of glycerol is given by a modified Clarke and Glew equation 461 proposed by Verevkin et al. [38]: Psat(T) = Po exp[(–23263/388.7 + 82273(1/388.7–1/T) – 462 83(388.7/T –1+log(T/388.7)))/R], with R the molar gas constant.  463 Static friction experiments. The substrates are placed on a brass plate which stands on two lab 464 jacks, the substrate is tilted by manually changing the height of one of the lab jacks. The angle is 465 measured with a digital goniometer accurate to 0.1° (Monotaro, Japan). Images and movies are 466 recorded with a Nikon D3200. Each point is obtained by averaging the static friction of five liquid 467 marbles.  468 Particles’ weight measurements. To estimate the particles’ mass at the liquid interface, using a 469 precision balance (Shinko Denshi Co., LTD., Model XFR–225W, accuracy of 0.01 mg), we 470 measure the average weight m of 10 liquid marbles of volume Ω to which we subtract the 471 average weight ρΩ of ten water droplets of same volume. The volume is controlled with a 472   11  micropipette (M200, Monotaro). The particle mass is calculated by subtracting the droplet to the 473 marble weights mp = m - ρΩ. The weights and particle to total mass ratio are summarized in SI 474 Appendix, Tables S1 and S2.  475 Hydrophobic glass beads preparation. The glass beads (ASGB-320, As ONE Corp.) are 476 rendered hydrophobic through chemical vapor deposition of Trichloro(1H,1H,2H,2H-tridecafluoro-477 n-octyl)silane (Tokyo Chemical Industry Co., Ltd.,) on its surface. The glass bead surface is first 478 cleaned using plasma cleaner (PIB-10, Vacuum Device Inc.). Then, 5g of glass beads are sealed 479 in a glass container (diameter 90 mm and height 20 mm) with 30 μL of the silane and heated at 480 80°C for 12 hours. 481 Air layer thickness measurement. The air layer thickness between the marble and the 482 substrate is measured using a confocal microscope (Leica, SP8). To obtain the position of the 483 substrate and the marble, they are marked with Rhodamine B (Sigma-Aldrich), a fluorescent 484 molecule with a peak of excitation wavelength around 542.8 nm and emission at 565 nm. The 485 substrate, a 24 mm × 60 mm × 0.13 mm glass slide, is coated with a 20 µL droplet of 14 µM of 486 Rhodamine B in ethanol, which once evaporated provides a thin fluorescent layer which marks 487 the solid/air interface. The liquid marbles (50 µL) are made with a 10 µM Rhodamine B/water 488 solution, a concentration at which surface tension remains close to that of water [41]. Due to its 489 amphiphilic nature, Rhodamine B tends to adsorb at the liquid/air interface which reveals its 490 position. The fluorescent marbles are placed on the coated substrate (SI Appendix, Fig. S7a) and 491 the 3D structure of the air layer is obtained by taking 2D images at 200 different vertical positions 492 spaced by 1 to 4 µm using 552 nm excitation wavelength. The intensity I of the light emitted by 493 the Rhodamine B is averaged for each vertical position, it shows a first peak corresponding to the 494 substrate and a second one for the air/liquid interface of intensity Io, which we use to normalize 495 the intensity. The air layer thickness H is defined as the distance between the substrate and the 496 bottom of the air-liquid interface given by I/Io = 0.18 (SI Appendix, Fig. S7b). 497 Glycerol vapor diffusion coefficient. The diffusion coefficient of glycerol vapor is estimated by 498 the Chapman-Enskog kinetic theory in the rigid spheres approximation [39]: 499 D = (3/8) [(NA/(2)(1/Mg+1/Mair)]1/2kT3/2/(PoAB2), with NA the Avogadro number, Mg and Mair the 500 molar mass of glycerol and air, respectively, k the Boltzmann constant, Po the atmospheric 501 pressure, and AB is approximated to the diameter of glycerol molecules d  = 5.47 Å. This gives a 502 typical value at room temperature of D ≈ 9.3 mm2 s-1. 503   504   12  Acknowledgments 505  506 We thank David Quéré and Syuji Fujii for fruitful discussions. We thank Kazumasa Takeuchi for 507 kindly allowing us to use his confocal microscope. We thank Kanata Hashimoto for the 508 lycopodium photograph. TM and MT acknowledge the financial support provided by the Japan 509 Society for the Promotion of Science (JSPS) - Grant-in-Aid for Scientific Research (B), 24K01341. 510 TM and PKR gratefully acknowledge the financial support provided by the JSPS Grant-in-Aid for 511 JSPS Fellows 23KF0106. MT appreciates support from WPI-MANA. YT is grateful for the 512 financial support from JST SPRING (Grant Number JPMJSP2108) and the Leadership 513 Development Program for Ph.D. students at the University of Tokyo, School of Engineering. 514   515   13  References 516  517 1. E. B. V. Dussan, R. T. P. 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Smith McWilliams, S. Ergülen, M. M. Ogle, C. A. de los Reyes, M. Pasquali, and A. A. 607 Marti, Fluorescent surfactants from common dyes – Rhodamine B and Eosin Y, Pure Appl. 608 Chem. 92, 265–274, (2020). 609 Figure Legends. 610 Figure 1. Hot liquid marbles stability. a, Schematic of the experiment: a liquid marble of volume 611  = 50 µL and temperature To + ∆T is placed on a silicon wafer kept at To. Inset: optical 612 microscope image of the lycopodium grains of diameter 30 µm coating the liquid. b, Images of hot 613 liquid marbles 3 minutes after being placed on the hydrophilic substrate (top) and hydrophobic 614 substrate (bottom), for ∆T = 0°C (left) and ∆T = 16°C (right). c, Final apparent contact angle f of 615 liquid marbles placed on hydrophilic (blue data) or hydrophobic (red data) substrates as a 616 function of the temperature difference ∆T. Plain lines are guides for the eyes. d, Time evolution of 617 hot liquid marbles’ apparent contact angle when placed on hydrophilic substrates for different ΔT. 618 The vertical dashed lines mark the rupture time tr. e, Rupture time of hot liquid marble placed on 619 hydrophilic substrates as a function ΔT (red data). The black line represents Eq. (1) with d ≈ 19.7 620 ± 2.1 µm, the blue shaded area corresponding to the 95% interval of confidence. The red line 621 represents the first-order Taylor expansion approximation (d ≈ 18.3 ± 2.6 µm) of Eq. (1) and the 622 green area corresponds to the stable marble regime (tr > ; ∆T < ∆Tc).  623 Figure 2. Condensation-induced rupture mechanism of hot liquid marbles on hydrophilic 624 substrates. a, Schematic of the Total Internal Reflection setup. The hot liquid marble is placed on 625 a glass substrate and the illumination incidence angle is between the total internal reflection angle 626 for the glass/air and glass/water interfaces. When water contacts the interface, the light goes 627 through the substrate and appears black on the high-speed camera image. b, TIR images below 628 a liquid marble placed on a hydrophilic glass substrate in the isothermal case (∆T = 0°C). c, High-629 speed TIR images revealing the condensation process below a hot liquid marble (∆T = 1.6°C and 630 ∆T = 16°C) deposited on a hydrophilic glass substrate. The liquid marble ruptures from t ≈ 0.46 s 631 and rapidly spreads on the surface. d, Schematic of the air layer below the liquid marble showing 632 the average air layer thickness H (top), and the condensation process at the minimum of the air 633 layer thickness h. e, Evolution of the temperature difference ∆T as a function of time for different 634 initial ∆T. The temperature is well fitted by an exponential decay ∆T(t = 0) = ∆To exp(−t/) with a 635 characteristic cooling time  = 11 s (plain lines).  636   16  Figure 3. Static friction of hot liquid marbles on hydrophobic substrates. a, Images of 50 µL 637 lycopodium hot liquid marbles deposited on a hydrophobic surface just before rolling due to slope 638 (left images), along with corresponding TIR images (right images), for ∆T = 3°C, 7°C and ΔT = 639 25°C from left to right. b, Evolution of the static friction force (red disks, left vertical axis) and the 640 fraction of wetted area  (blue triangles, right vertical axis) as a function of ΔT. c, Temperature-641 induced static friction ∆F = F – F(∆T = 0) plotted as a function of the fraction of wetted area . The 642 red line represents the best linear fit obtained by multiplying Eq. (3) by a prefactor 1.19 ± 0.12, 643 the 95% confidence interval is represented by the blue shaded area (R2 ≈ 0.96). d, log-log plot of 644 the static friction of lycopodium hot liquid marbles as a function of their volume Ω for different ∆T. 645 For small ∆T (blue data) the static friction varies linearly with the marble volume as described by 646 the dry friction model of Eq. (2) plotted with a blue line. For larger ∆T, the friction nature changes 647 to a wet static friction which increases with Ω1/2 as captured by our model, Eq. (3), which is 648 plotted as a red line with a prefactor of 1.19. 649  650 Figure 4. Static friction of hot liquid marbles on hydrophobic substrates. a, Confocal images 651 of the air layer of average height H between a glass substrate coated with dried Rhodamine B 652 and liquid marbles filled with an aqueous solution of Rhodamine B and made with three kinds of 653 particles: lycopodium, fumed silica nanoparticles and glass beads. b, Variation of static friction 654 with ΔT for marbles made with fumed silica (H = 30 ± 3 µm, blue triangles), lycopodium (H = 56 ± 655 2 µm, red disks), and glass beads (H = 242 ± 8 µm, green squares). c, TIR images of water hot 656 liquid marbles with ∆T = 29°C for: fumed silica (blue), lycopodium (red), and glass beads (green) 657 and hot glycerol made with lycopodium (black). d, Evolution of the static friction as a function of 658 ∆T for water (red disks) and glycerol (black diamonds) showing the effect of the liquid volatility on 659 static friction. e, Plot of the ratio between the average air layer thickness H and the predicted 660 maximum condensation height  (Eq. 4) as a function of ΔT. We mark in green (resp. red) the low 661 (resp. high) friction marbles defined by ∆F < Fmax/2 = Fwet( = 1)/2  (resp. ∆F > Fmax/2). Our model 662 predicts the friction transition to occur for H/ which is in excellent agreement with the line 663 separating the two regimes H/ ≈ 0.35. 664 Figure 5. Static friction of hot liquid marbles on nanostructured superhydrophobic substrates. 665 a, Static friction of 50 µL lycopodium liquid marbles plotted as a function of the temperature 666 difference for hydrophobic (red disks) and superhydrophobic (green triangles) substrates. b, 667 Temperature-induced static friction ∆F = F – F(∆T = 0) normalized by (1 + cos ) plotted as a 668 function of ∆T for the hydrophobic and superhydrophobic substrates.  669  670 0 1 2 30501001500 10 20 30024681012E0 10 20 300501001502 mmhydrophobichydrophilicT = 0°C T = 16°CA BCT (°C)f (°)θhydrophilichydrophobict (s) T (°C)rupture time (s) 0111672950 µmToTo + TT (°C)D (°)θrupture time (s) 0 20 40 600510152025CA DET = 0°C2 mmT (°C)t (s)B24 ºC11 ºC20 ºCT (t = 0 s)  LED lightdove prismhigh-speed camera oilsubstratedCsat(To + T)Csat(To)h dToTo + THhydrophilict = 0.16 sT = 16°CT = 1.6°Ct = 1.0 s0.31 s4.0 s0.46 s7.0 s2 mmtime5 mm14.9 s0.50 srupture10 1001010010000 0.2 0.4 0.6 0.8 101002003004000 10 20 30010020030040000.20.40.60.81T = 3°C T = 25°CA* = 42.0°α2 mm * = 1.6°α 2 mmfraction of wetted area ϕT (°C)B C (µL)DF (µN)F (µN)072016325T (°C)wet static frictiondry static friction 111/21fraction of wetted area ϕF (µN)static frictionfraction of wetted areaT = 7°C* = 15.1°α*α0 10 20 300.011100100000 10 20 3001002003004000 10 20 3001002003004000 10 20 300100200300400lycopodiumfumed silicaglass beads100 µmF (µN)AF (µN) low static friction high static friction T (°C)H/δT (°C)T (°C)BDE ect of particle size E ect of liquid volatility EAir layer thickness3056242H (µm) waterglycerol56 (gly)3056242H (µm)  = 50 µL = 50 µLT = 29°C2 mm56 µm | glycerol242 µm | water56 µm | water30 µm | waterC0 10 20 3001002003004005006000 10 20 300100200300400∆T (°C)F (µN)A Bhydrophobicsuperhydrophobic  (µN)ΔF/(1+cosθo)∆T (°C)100°θo =153°θo =Ω = 50 µL Manuscript File Figure 1 Figure 2 Figure 3 Figure 4 Figure 5