## Abstract

The thermal transport to drops that roll or slide down heated superhydrophobic surfaces is explored. High-speed infrared imaging is performed to provide time-resolved measurement of the heat transfer to the drop. Data are obtained for drops moving along smooth hydrophobic and structured superhydrophobic surfaces. Both post and rib style structures with surface solid fractions ranging from 0.06 to 1.0 are considered. The inclination angle of the surfaces was varied from 10 deg to 25 deg, and the drop volume was varied from 12 to 40 μL. The measurements reveal that the drop speed is a strong function of both the inclination angle and the surface solid fraction. Further, the heat transfer is strongly affected by the surface solid fraction and the drop speed. Surfaces with low solid fraction result in a decrease in the initial heat transfer compared to the behavior observed for drops on a smooth surface. At the smallest solid fractions explored the reduction in heat transfer is nearly 80%. For rib structured surfaces, drop motion both along and perpendicular to the rib direction was considered and the heat transfer is larger for drops moving in the parallel rib configuration. This variation is likely caused by the increased rolling speed that prevails for the parallel rib case. Over 130 unique conditions were explored, and the results from all cases were used to develop correlations that enable prediction of the heat transfer to drops rolling or sliding down smooth hydrophobic and superhydrophobic surfaces.

## 1 Introduction

Sliding or rolling of drops on surfaces occurs in many systems and is ubiquitous in condensing applications. Drop-wise condensation is advantageous when considering the overall condenser heat transfer. Superhydrophobic (SH) surfaces have emerged as a promising solution to facilitate drop-wise condensation and have exhibited improved heat transfer characteristics in condensation applications. The heat transfer to drops on nonwetting surfaces is also important for glass that is so-called self-cleaning, as in the case of solar panels and anti-icing surfaces [1–3]. The thermal transport from a hot surface to a rolling drop is a strong function of the surface characteristics and the angle at which the surface is inclined. The focus of this paper is to explore the influence that SH surfaces exert on the drop speed and heat transfer to the drops as they roll or slide. Both drop speed and heat transfer are also functions of drop volume and surface inclination angle.

An important measure of hydrophobicity is the contact angle between a water drop and the surface. Hydrophobic surfaces can be made SH by adding a micro- or nanoscale texturing. When this is done, static contact angles larger than 120 deg can be realized [4]. The left image of Fig. 1 illustrates a sessile drop on a SH surface. The size of the microscale posts has been exaggerated. In reality, thousands of posts would exist under the droplet. The level of hydrophobicity that prevails for a drop on a surface is determined by both the static contact angle and the contact angle hysteresis (i.e., the difference between the receding and advancing contact angles) [5–8]. The center and right images of Fig. 1 illustrate the static contact angle ($\theta c$) of a drop on a horizontal surface and the advancing ($\theta A$) and receding ($\theta R$) contact angles of a drop on an inclined surface. The receding contact angle represents the minimum angle the liquid can maintain contact with the surface before it starts to pull away and the advancing contact angle is the maximum angle the liquid can maintain contact with the surface before the drop starts to spread.

Superhydrophobic surfaces experience several significant near wall effects due to the combination of hydrophobic chemistry and micro/nanoscale structures. For liquid pressures lower than the Laplace pressure, the drop does not penetrate the cavities between structures and only contacts the tops of the structures. Consequently, the working fluid encounters alternating liquid–air and liquid–solid interfaces. The no-slip condition is maintained at the liquid–solid interface. However, at the liquid–air interface, the velocity need not be zero, and the shear imposed by the air on the liquid is negligibly small. The aggregate effect of the alternating slip and no slip behavior is an apparent slip velocity at the wall.

The presence of air-filled microcavities at SH surfaces also alters thermal transport. The thermal conductivity of the post or rib structures for a metallic substrate is several orders of magnitude greater than the thermal conductivity of the air. Thus, the liquid–air interfaces may be considered adiabatic while heat conducts primarily through the microfeatures and into the liquid via the liquid–solid interfaces. The aggregate effect of these alternating boundary conditions is an apparent temperature jump at the wall, $\Delta Tw$ [9,10]. It has been shown that $\Delta Tw$ is proportional to the wall heat flux, $qw\u2033$, and may be expressed $\Delta Tw=qw\u2033\lambda T/kw$, where $\lambda T$ is a wall temperature jump length and *k _{w}* is the liquid thermal conductivity. In other words, the ratio $kw/\lambda T$ can be thought of as a thermal resistance.

*f*is entered as a decimal value between 0 and 1

_{s}Rolling drop dynamics have been considered previously for small drops moving down inclined nonwetting surfaces [11,12]. Mahadevan et al. [11] developed a model to allow estimation of droplet speed. Here, droplets smaller than the capillary length were considered. Yang et al. [12] employed an energy balance to model the motion of a drop on a flat surface where the drop moved as a result of spatial variations in surface energy. The drop volumes considered by both of these investigators were less than 10 $\mu $L.

An important question related to the motion of a drop moving along a surface is whether the drop moves by rolling or by sliding. For a drop immersed in a gas shear flow, the static contact angle and contact angle hysteresis were shown by Xie et al. [6] to be the most important factors that determine if sliding or rolling will be the resulting motion. Further, Yilbas et al. [7] and [8] showed that the contact angle hysteresis is the most important factor that affects the rotational speed of a drop moving on a surface. They also demonstrated that as the drop's linear speed increases the contact angle hysteresis also increases and sliding behavior will exist. Further, their results indicate that as the drop moves along the surface temporal variations in drop diameter and height are likely to occur.

Backholm et al. [13] considered drops moving along SH surfaces. They concluded that when the ratio of the drop speed to the radius of the drop is small the drop will roll and at higher speeds a critical rotational rate is achieved and above this rate the motion transitions to a combination of sliding and rolling. In a related investigation, Sakai et al. [14] compared drop motion on SH and hydrophobic surfaces. They noted that on hydrophobic surfaces drops move with an internal rolling dynamic. Conversely, for drops on SH surfaces, the motion was more similar to an object sliding down a frictionless plane. In another study, Sakai et al. [15] experimentally characterized the influence of surface roughness factor on the drop motion. Here, they observed that the drop acceleration was nearly constant for motion on surfaces with high roughness. The implications of the above observations are that drops on SH surfaces or more likely to slide rather than roll and drops on hydrophobic surfaces are more likely to roll rather than slide. In this study, the heat transfer to drops on both SH and hydrophobic surfaces is explored and both rolling and sliding motions are expected; depending on the contact angle, surface inclination angle, and the degree of superhydrophobicity.

This paper is the first to provide experimental data that quantifies the heat transfer to drops moving on SH surfaces under the influence of gravity. It is known that the thermal transport to a liquid contacting a SH surface decreases as the solid fraction of the surface decreases. One reason for this is that the air filled cavities between the structures represents an added thermal resistance. Hays et al. [16] experimentally confirmed this phenomenon for stationary drops on SH surfaces.

When a drop is placed on a heated nonwetting surface it may experience natural and Marangoni convection. Phadnis et al. [17] showed that the heat transfer coefficient could be enhanced by 10–40% due to Marangoni flow. They considered surfaces with contact angles ranging from 90 to 120 deg. Tam et al. [18] quantified the internal drop velocity for a stationary drop and showed that the drop velocity is dominated by Marangoni convection. Static drops that are restrained from rolling but are placed on inclined nonwetting surfaces (ranging from 0 to 25 deg) also exhibit Marangoni convection [19]. This study showed that the inclination angle affects the overall Nusselt number when the drop volume is greater than 15 $\mu $L, with the Nusselt number decreasing as drop volume increases.

The heat transfer to drops that impinge on SH surfaces has also been explored. In one study Guo et al. [20] considered drops that impacted SH surfaces that were maintained at temperatures well below the boiling point. They performed experiments to measure the total heat transfer to the drop during the impingement event. They also presented an analysis modeling the instantaneous thermal transport as a function of the important fluid dynamic and SH surface parameters. The most important variable influencing the drop heat transfer was the solid fraction, and as the solid fraction decreased the drop heat transfer decreased as well. Interestingly, the volume of the drop diameter was only of secondary importance. An earlier study was conducted by Pasandideh-Fard et al. [21] who computationally considered drop impact on smooth surfaces. While their results did not consider SH surfaces, the work did establish benchmark conditions for the drop heat transfer process. Superhydrophobic surfaces yield a thermal resistance between the drop and the solid due to the trapped air in the cavity regions. This trapped air leads to decreased heat transfer and a modestly smaller total drop-surface contact time. It is expected that for rolling/sliding drops on SH surfaces a decreased heat transfer and shorter drop residence time will prevail as the solid fraction decreases.

The impact of SH surfaces on laminar channel flow heat transfer has also been previously explored. Maynes et al. [9] and [22] considered fully developed laminar flow in a parallel plate style duct with rib structured SH surfaces. Their results showed a lower average Nusselt number as the solid fraction decreases. It was also shown that the rib-to-rib spacing affects the heat transfer, with larger pitch (relative to tip channel size) yielding smaller heat transfer. Cowley et al. [10] performed a similar study, but for poststructured SH surfaces and they reported similar findings.

The goal of this paper is to present experimental measurements of the thermal transport to a drop moving down an inclined SH surface. Specifically, the paper aims to quantify how drop speed, contact angle hysteresis, drop volume and SH surface parameters affect the heat transfer. Experiments were conducted with SH surface solid fractions ranging from 0.06 to 0.5 and at surface inclination angles ranging from 10 to 25 deg. The volume of the drop was varied from 12 to 40 $\mu $L and the surface temperature was varied from 50 to 80 °C. Both post and rib structured SH surfaces were considered. The remainder of this paper is organized as follows. Section 2 describes the experimental method and data process approach. Subsequently, the results from the experiments are shown and then conclusions of the work are presented. The results of the study are of particular importance to high-efficiency condensers that promote drop-wise condensation where the drops move along the walls of the condenser while transferring heat. The results are also important for spray cooling applications where far from the impingement point the liquid jet has broken up into drops that move along the surface and transfer heat from the surface.

## 2 Methodology

### 2.1 Surface Manufacturing.

In this study, SH surfaces that were fabricated with both post and rib structures were considered. The SH structures were fabricated in a class 10 cleanroom using 100 mm diameter silicon wafers of 525 $\mu $m thickness. Photolithography was employed to pattern photoresist and the resulting rib or postpatterns. Subsequently, the structures were etched to depths ranging from 10 to 15 $\mu $m. To enhance Teflon adhesion, chromium was deposited on the etched wafers. Finally, a Teflon solution was applied to the surfaces, followed by the evaporation of the solvent, resulting in a hydrophobic surface chemistry. Equation (1) or (2) was utilized to compute the solid fraction of the surfaces based on post or rib thickness, height, and pitch. Figure 2 shows schematics and SEM images of typical post (left) and rib (right) structured surfaces. The dimensions of the surface structures were measured using a three-dimensional profilometer manufactured by Zeta Instruments with an instrument uncertainty of ± 0.1 μm. Experiments were conducted on each surface at several discrete locations to allow determination of average behavior. Table 1 shows all surface structure dimensions, $fs$, $\theta c$, $\theta A$, and $\theta R$ for all surfaces considered here. The measurement uncertainty for all of the surface parameters and the static contact angles are also shown in the table.

Surface | $w$ ($\mu $m) ± 0.1 | $b$ ($\mu $m) ± 0.5 | $h$ ($\mu $m) ± 0.5 | $fs\u2009$± 0.03 | $\theta c\u2009$± 2 | $\theta A\u2009$± 2 | $\theta R\u2009$± 2 | |
---|---|---|---|---|---|---|---|---|

Smooth | N/A | N/A | N/A | 1.00 | 118.6 deg | 118.8 deg | 109.6 deg | |

Post-SH | 24.2 | 17.6 | 15.8 | 0.40 | 145.7 deg | 146.5 deg | 125.3 deg | |

8.0 | 4.7 | 9.8 | 0.27 | 151.6 deg | 152.5 deg | 132.7 deg | ||

12.2 | 6.6 | 12.4 | 0.23 | 154.2 deg | 154.4 deg | 135.3 deg | ||

24.1 | 6.6 | 12.2 | 0.09 | 159.6 deg | 160.9 deg | 145.3 deg | ||

32.1 | 4.9 | 13.4 | 0.06 | 156.4 deg | 162.2 deg | 147.7 deg |

Surface | $w$ ($\mu $m) ± 0.1 | $b$ ($\mu $m) ± 0.5 | $h$ ($\mu $m) ± 0.5 | $fs\u2009$± 0.03 | $\theta c\u2009$± 2 | $\theta A\u2009$± 2 | $\theta R\u2009$± 2 | |
---|---|---|---|---|---|---|---|---|

Smooth | N/A | N/A | N/A | 1.00 | 118.6 deg | 118.8 deg | 109.6 deg | |

Post-SH | 24.2 | 17.6 | 15.8 | 0.40 | 145.7 deg | 146.5 deg | 125.3 deg | |

8.0 | 4.7 | 9.8 | 0.27 | 151.6 deg | 152.5 deg | 132.7 deg | ||

12.2 | 6.6 | 12.4 | 0.23 | 154.2 deg | 154.4 deg | 135.3 deg | ||

24.1 | 6.6 | 12.2 | 0.09 | 159.6 deg | 160.9 deg | 145.3 deg | ||

32.1 | 4.9 | 13.4 | 0.06 | 156.4 deg | 162.2 deg | 147.7 deg |

Rib SH | Parallel | Perpendicular | ||||||
---|---|---|---|---|---|---|---|---|

60.0 | 30.0 | 14.4 | 0.50 | 137.7 deg | 146.7 deg | 141.9 deg | 120.6 deg | |

40.0 | 20.0 | 15.5 | 0.50 | 138.1 deg | 147.2 deg | 141.9 deg | 120.6 deg | |

16.0 | 4.8 | 14.4 | 0.30 | 140.1 deg | 148.1 deg | 151.9 deg | 130.9 deg | |

40.2 | 7.0 | 13.7 | 0.18 | 148.0 deg | 152.1 deg | 156.7 deg | 138.7 deg | |

40.0 | 3.8 | 13.3 | 0.09 | 156.7 deg | 154.1 deg | 160.9 deg | 145.3 deg |

Rib SH | Parallel | Perpendicular | ||||||
---|---|---|---|---|---|---|---|---|

60.0 | 30.0 | 14.4 | 0.50 | 137.7 deg | 146.7 deg | 141.9 deg | 120.6 deg | |

40.0 | 20.0 | 15.5 | 0.50 | 138.1 deg | 147.2 deg | 141.9 deg | 120.6 deg | |

16.0 | 4.8 | 14.4 | 0.30 | 140.1 deg | 148.1 deg | 151.9 deg | 130.9 deg | |

40.2 | 7.0 | 13.7 | 0.18 | 148.0 deg | 152.1 deg | 156.7 deg | 138.7 deg | |

40.0 | 3.8 | 13.3 | 0.09 | 156.7 deg | 154.1 deg | 160.9 deg | 145.3 deg |

The static contact angle for each surface was measured using a DSLR camera and custom image processing algorithm. Measurements were made at five different locations and an average value was computed for each surface. Advancing and receding contact angles were determined using Eqs. (3) and (4) presented by Humayun et al. [4] for poststructured surfaces. The value of $\theta c$ for rib-structured surfaces differed based on whether the ribs were aligned parallel or perpendicular to the camera viewing direction. For drops placed in the parallel rib orientation the drop spreads out more than in the perpendicular directions, resulting in an oval-shaped contact area. The result of this is lower measured values of $\theta c$ in the parallel-oriented direction compared to the perpendicular orientation.

### 2.2 Experimental Setup.

Figure 3 shows a schematic of the test apparatus. An IR camera was oriented to view the plane aligned with drop motion, and an ice bath was placed directly behind the surface on which the drop rolled to provide higher temperature contrast between the drop boundary and the background. Each test surface was placed on top of a heated inclined flat metal block of aluminum and the drop motion was caused by gravity. The inclination angle, $\alpha $, was measured using an angle cube and was varied from 5 to 25 deg. The surface temperature was maintained by heating an aluminum block with two embedded cartridge heaters that were controlled with a feedback loop using three thermocouples positioned just below the block's surface. The three thermocouples were spread out to ensure the surface temperature was uniform along the entire test area. Once the aluminum block was heated to the desired temperature, a test surface was placed on the heated block and left there until thermal equilibrium was reached. Surface temperatures ($Tw$) of 50, 65, or 80 °C were considered, with the majority of tests being performed at $Tw$ = 50 °C. A high-speed IR camera operating at 240 fps with resolution of 120 × 640 pixels was utilized to measure the instantaneous drop temperature. An emissivity of 0.96 was specified in the IR video acquisition software to match the emissivity of water.

Video of the transient heating process was acquired in the following manner. First, a drop was placed on the surface using a micropipette with an uncertainty of $\xb1$0.05 $\mu $L. When the drop contacted the surface, the IR camera was triggered to acquire video until the drop had rolled out of the viewing window. This process was repeated for each test condition considered. The total number of unique experimental conditions was 133 and at each condition, at least 10 repeat experiments were conducted. For each repeat experiment, the drop was placed in a slightly different location on the surface and the results of all repeat experiments were averaged. Table 2 shows the parameters tested for each surface.

Surface type | $fs$ | $w$ ($\mu $m) | Volume ($\mu $L) | $Tw$ (°C) | $\alpha $ (deg) |
---|---|---|---|---|---|

Smooth | 1.00 | N/A | 12, 20, 30, 40 | 50, 65, 80 | 10, 15, 20, 25 |

Post | 0.40 | 24.2 | 12, 20, 30, 40 | 50, 65, 80 | 20,25 |

0.27 | 8.0 | 12, 20, 30, 40 | 50, 65, 80 | 10, 15, 20, 25 | |

0.23 | 12.2 | 12, 20, 30, 40 | 50, 65, 80 | 10,15 | |

0.09 | 24.1 | 12, 20, 30, 40 | 50, 65, 80 | 5, 10, 15, 20 | |

0.06 | 32.1 | 12, 20, 30, 40 | 50, 65, 80 | 10 | |

Rib | 0.50 | 60.0 | 20, 40 | 50 | 15, 20 |

0.50 | 30.0 | 20, 40 | 50 | 15, 20 | |

0.30 | 16.0 | 20, 40 | 50 | 15, 20 | |

0.18 | 40.2 | 20, 40 | 50 | 15, 20 | |

0.09 | 40.0 | 20, 40 | 50 | 15, 20 |

Surface type | $fs$ | $w$ ($\mu $m) | Volume ($\mu $L) | $Tw$ (°C) | $\alpha $ (deg) |
---|---|---|---|---|---|

Smooth | 1.00 | N/A | 12, 20, 30, 40 | 50, 65, 80 | 10, 15, 20, 25 |

Post | 0.40 | 24.2 | 12, 20, 30, 40 | 50, 65, 80 | 20,25 |

0.27 | 8.0 | 12, 20, 30, 40 | 50, 65, 80 | 10, 15, 20, 25 | |

0.23 | 12.2 | 12, 20, 30, 40 | 50, 65, 80 | 10,15 | |

0.09 | 24.1 | 12, 20, 30, 40 | 50, 65, 80 | 5, 10, 15, 20 | |

0.06 | 32.1 | 12, 20, 30, 40 | 50, 65, 80 | 10 | |

Rib | 0.50 | 60.0 | 20, 40 | 50 | 15, 20 |

0.50 | 30.0 | 20, 40 | 50 | 15, 20 | |

0.30 | 16.0 | 20, 40 | 50 | 15, 20 | |

0.18 | 40.2 | 20, 40 | 50 | 15, 20 | |

0.09 | 40.0 | 20, 40 | 50 | 15, 20 |

### 2.3 Image Analysis.

After IR video acquisition for a single rolling droplet scenario, the video was imported into matlab frame by frame for subsequent image processing. The first frame of each video included a physical scale of known length that allows determination of a calibration conversion factor from pixel values to physical length. A typical conversion factor was nominally $9.03\xb10.07$ pixels/mm. The video was analyzed from the time when the micropipette left the viewing window until the droplet had rolled out of the camera field of view. All frames were processed using the following method. First, all images were normalized using the minimum and maximum temperatures to rescale the temperature values to range from 0 to 1. This was done to utilize a common threshold when determining the droplet edge for all acquired videos. Then, an initial frame that showed only the background was subtracted from all subsequent frames to provide a sharper contrast between the drop and the background. At this point, filtered images were converted to binary, with all pixels within the droplet region set to yellow and the background set to blue. Using this binary image, all edges of the drop and the drop centroid were determined. This process is shown schematically in Fig. 4.

### 2.4 Instantaneous Drop Velocity and Average Drop Temperature Calculation.

The instantaneous temperature and velocity of the drop were determined for each scenario using the acquired IR images. Instantaneous drop position was determined by quantifying the centroid of the drop for each frame from the binary images. The initial centroid position was then subtracted from this location for all subsequent images to determine the change in drop position. The drop velocity was calculated by using the difference between successive centroid positions and the camera frame rate. Shown in Fig. 5 is the drop position (left panel) and velocity (right panel) as a function of time for a typical scenario. The data represent an average of 10 unique tests for the same surface, temperature, volume, and test surface. In addition to the instantaneous drop velocity, an average drop velocity, $V\xaf$, was also computed. This was done by dividing the total distance the drop traveled along the surface by the total elapsed time. Results here will be reported using a normalized average velocity, where the velocity is normalized by gravity, $g$, and the total distance traveled, $L$, along each surface, ($V\u0302=V\xaf/gL$). The distance *L* was nominally the same for all surfaces (60 ± 1 mm), but exact values were determined for each case.

The instantaneous average drop temperature ($T\xafD,t$) was determined by spatially averaging the IR temperature values inside the drop up to a maximum radius of 80% of the total radius, since the drop curvature presents a challenge when using IR imaging at high viewing angles. Cheng et al. [23] showed that, when viewing a sphere with an IR camera at viewing angles smaller than 60 deg (with 0 deg corresponding to a perpendicular view of the center of the drop), the measured temperature is not affected by the viewing angle. However, at angles greater than 60 deg (near the outer radius of the drop), the observed IR temperature deviates from the actual temperature, and the amount of deviation increases as the edge of the sphere is approached. For a sphere, a 60 deg viewing angle corresponds to a radial location that is at 80% of the total radius. Thus, the average drop temperature here was determined by spatially averaging all temperature values correlating with pixels located within 80% of the total drop radius. A surface area weighted temperature average was also computed, resulting in values that were always within $\xb1$ 0.1 °C of the values computed using the simple average.

*ε*, was calculated.

*ε*is defined to be the total instantaneous heat that has been transferred to the drop divided by the maximum possible heat transfer as expressed below

$T\xafD,i$ is the initial average drop temperature, $T\xafD,t$ is the instantaneous drop temperature, $Tw$ is the measured wall temperature, $md$ is the mass of the drop and *C _{p}* is the water specific heat. A value of 0 for the heating effectiveness corresponds to no heating of the drop. Conversely, when

*ε*= 1, the drop has increased in energy and matches the temperature of the surface.

Shown in Fig. 6 is $\epsilon $ as a function of time, averaged over 10 distinct scenarios for the same surface, at the same surface temperature and drop volume. $\epsilon $ increases linearly during the initial stages of heating and then levels off and this behavior was consistent with all scenarios explored. Here, we consider two measures of the normalized heat transfer to a drop; an initial heating rate ($d\epsilon /dt\u0302$) and an average heating rate ($\Delta \epsilon /\Delta t\u0302$). Time was normalized using the gravitational constant and the equivalent diameter*, D*, of the drop as it is placed on the surface, $t\u0302=tg/D$. $d\epsilon /dt\u0302$ was determined using the initial linear slope of the $\epsilon $ versus $t$ curve for a given scenario and $\Delta \epsilon /\Delta t\u0302$ was determined by dividing the final value of $\epsilon $ by the total elapsed time. For all scenarios, the *ε* versus *t* curve exhibited a linear regime at early times and $d\epsilon /dt\u0302$ was determined by fitting a line to this linear portion and determining the slope of the line.

Error bars are shown on the data in Fig. 6 and all errors were tracked during the experiments. The total error for a given parameter consists of the uncertainty in each of the measured quantities, based on the accuracy of each of the individual measuring devices, and the statistical variation that exists between repeat experiments. At least 10 repeat experiments were conducted for each scenario explored, and the 95% confidence interval was determined by computing the standard deviation in all measured quantities. The propagation of error method was employed here to compute a total uncertainty for all quantities reported and error bars are shown with the results presented in the following section, with all error bars corresponding to a 95% confidence level.

## 3 Results and Discussion

Shown in Fig. 7 is a series of IR images of drops moving down an inclined surface at several different times during the transient heating process. The images show the drops as they move from left to right and provide qualitative understanding of the influence that solid fraction and inclination angle exert on drop heating. Four scenarios are shown, and droplets are shown at nine distinct times in the process as they move along the test surface. The wall temperature was $Tw$ = 50 °C, and the drop volume was 40 *μ*L for all scenarios shown. The top two rows of images correspond to a smooth hydrophobic surface (*f _{s}* = 1) inclined at angles of $\alpha $ = 15 and 25 deg. The bottom two images correspond to an inclination angle of $\alpha $ = 15 deg and postpatterned SH surfaces with solid fractions of

*f*= 0.27 and 0.09. The location of each drop at

_{s}*t*=

*0.15 s is denoted to demonstrate the drop displacement at a consistent time after the drop has been placed on the surface. Other locations of the droplets that are shown were selected to prevent overlap of the drop images, and are not indicative of the frame rate. The bottom three sets of images show that the drop moves faster as*

*f*decreases. This is illustrated by the fact that at

_{s}*t*=

*0.15 s the drop is further to the right for the*

*f*= 0.27 and 0.09 cases. Unsurprisingly, the drop velocity also increases with $\alpha $ as shown by the top two series of images with

_{s}*f*= 1.

_{s}The images of Fig. 7 also reveal that as the solid fraction decreases, the heat transfer also decreases. The hottest portion of the drop for the *f _{s}* = 0.09 case is much lower than for the smooth surface (

*f*= 1). Further, for all cases shown, the temperature of the drop is highest around its periphery, with regions of higher temperature along the top of the drop and at the lower left corner of the drop where it is touching the surface. This behavior is more pronounced for the smooth surface, but exists for the lower

_{s}*f*cases as well. The images also show that the amount of contact between the drop and surface is greater for the smooth surface than for the superhydrophobic surfaces (

_{s}*f*= 0.27 and 0.09). Further, the droplets on the smooth surfaces are more skewed (toward the right or downhill direction) and less spherical than for the SH surfaces with lower solid fractions. Although not evident from the still images, the corresponding videos from which the series of still images were taken reveal that the drops on the smooth surface exhibit a rolling type motion and those on the SH surfaces exhibit more of a sliding motion. Comparing the images for the two smooth surface cases, it is evident that the drop moving faster (

_{s}*α*= 25 deg) has a drop temperature that is increasing at a higher rate.

We now consider measurements of the heating effectiveness and the instantaneous drop velocity as functions of time and displacement from the initial position for a few representative scenarios. Figure 8 shows the instantaneous drop velocity and the heating effectiveness as functions of time (panels a and c) and drop displacement (panels b and d). For all data shown in Fig. 8, the drop volume was 30 $\mu $L, *T _{w}* = 50 °C, and

*α*= 15°. Each panel includes data corresponding to the smooth hydrophobic surface (

*f*= 1) and two postpatterned SH surfaces with

_{s}*f*= 0.27, and 0.09. Shown on each data marker are error bars that quantify the total uncertainty for the measured quantities. Also shown on each plot is a dashed line that corresponds to the speed of an object that is accelerating down a frictionless surface inclined at the same angle. For a frictionless surface, the instantaneous velocity of a sliding object increases linearly with time and can be expressed in terms of time as

_{s}*V*=

*gt*sin

*α*and in terms of displacement,

*x*, as $V=2xgsin\alpha $. Reyssat et al. performed experiments for a similar droplet volume (35 mL) rolling down a microstructured superhydrophobic surface with

*θ*= 165 deg, comparable to the

_{c}*f*= 0.09 surface here, inclined at 13 deg. They found the droplet motion follows that of free fall for the first 0.1 s quite well then begins to deviate, and similar results can be seen in Fig. 8. [24]. Comparisons with other prior works are not made here as the terminal velocities presented in other studies are not reached in this work.

_{s}The data of panels a and b clearly show that decreasing *f _{s}* results in drops with higher velocity. Note that the length over which the drops roll/slide is approximately the same for all surfaces (illustrated in panel b). In contrast, the total time that each drop is on a surface decreases as the drop speed increases (panel a). Compared to the frictionless surface scenario, the drop speed at the end of the viewing window is approximately 50% lower for the smooth surface, 40% lower for the

*f*

_{s}= 0.27 surface, and 20% lower for the

*f*= 0.09 surface. As expected, as

_{s}*f*decreases so does the resistance to drop motion and a consequence of this is less internal convective motion. This is further illustrated in the images of Fig. 7 where the drops are much less skewed for the

_{s}*f*= 0.27 and 0.09 surfaces.

_{s}The data of panels c and d show that at a given time or displacement, *ε* decreases with decreasing *f _{s}*, corresponding to lower heat transferred. At the end of the viewing window, the total heat transferred to the drop on the smooth surface yields

*ε*≈ 0.32 and

*ε*decreases to 0.18 and 0.12 for the

*f*= 0.27 and 0.09 surfaces, respectively. These reductions correspond to 38% and 56% reductions in the amount of heat that is transferred to the surface. The

_{s}*ε*versus time data initially shows a linear variation in

*ε*with increasing time. An important implication of this linear behavior is that d

*ε*/d

*t*and the initial heat transfer to the drop remain constant for some time. Further, this linear regime (constant heat transfer regime) extends to a longer time for the smooth surface than for the two SH surfaces. At larger times (greater displacement) the rate of increase in

*ε*with time begins to decrease.

The reason for the linear variation in *ε* with time exists due to the following physical behavior. For the *f _{s}* =1 (smooth) surface, the drop moves in a rolling motion, and as it rolls the liquid in contact with the surface is continuously replaced with unheated liquid that was previously in contact with the surrounding air. Thus, the temperature difference between the surface and the bottom of the drop, and the associated heat transfer, remain constant. As time progresses, eventually the liquid that was in contact with the surface rotates completely around the drop and returns to the surface again. When this occurs, the heat transfer to the drop begins to decrease and the

*ε*versus time data begins to depart from the initial linear behavior. As the drop continues to roll and to increase in temperature the heat transfer will continue to decrease, leading to

*ε*beginning to level off. In contrast, for the perfectly superhydrophobic surface (

*f*→ 0) the drop will move strictly in a sliding motion. Here, the liquid near the surface will remain near the surface (in the absence of free convection), while heating, and so the instantaneous temperature difference between the surface and the drop will be smaller. In this scenario, a linear regime of

_{s}*ε*versus time would not exist. For real SH surfaces, with

*f*values in the ranges considered here (0.06 – 0.5), a combination of sliding and rolling exists with the relative amount of sliding increasing as

_{s}*f*decreases. Thus, as

_{s}*f*decreases the linear portion of the

_{s}*ε*versus time curve will decrease as well, as demonstrated by the data.

In addition to the behavior described above, there are two other important factors that lead to decreased heat transfer with decreasing solid fraction. First, all contact angles (static, advancing, and receding) are much larger for SH surfaces than for the smooth hydrophobic surface and this yields a smaller contact area between the drops and the plane of the surface. Second, while all of the liquid at the plane of the smooth surface is in contact with the surface, the solid–liquid contact is reduced for the SH surfaces due to the trapped air in the cavity regions between the microstructures.

The data of Fig. 8 and the images of Fig. 7 provide an overview of the influence SH surfaces exert on drop velocity and thermal transport to rolling/sliding drops on nonwetting surfaces. The following sections will present data for all scenarios considered, where the specific impacts of solid fraction, drop volume, inclination angle, and substrate temperature will be demonstrated.

### 3.1 Average Drop Velocity.

The measured instantaneous drop velocity was shown in Fig. 8 for a few scenarios to illustrate general behavior. Here we consider the normalized average drop velocity, $V\u0302=V\xaf/gL$. Shown in Fig. 9 is $V\u0302$ as a function of *f _{s}* and surface inclination angles of $\alpha $ = 10, 15, 20, and 25 deg for postpatterned SH surfaces. The drop volume was 40 $\mu $L, the surface temperature was 50 °C, and linear trend lines are included with the data to highlight general behavior. Some scatter in the data exists for each inclination angle and this is likely due to the inherent variability in the surface coating for each of the surfaces. For all inclination angles, $V\u0302$ decreases as $fs$ increases, as expected. The percentage decrease in velocity is much larger at lower values of

*α*. As

*f*increases from 0.09 to 1, $V\u0302$ decreases by nominally 85% for the

_{s}*α*= 10 deg scenario and the decrease is closer to 20% for the

*α*= 25 deg case. As expected, $V\u0302$ increases with increasing

*α*for all solid fractions. Here, the percentage increase is greatest for the

*f*= 1(smooth) surface, where $V\u0302$ increases by approximately 175% as

_{s}*α*increases from 10 deg to 25 deg. In contrast, at

*f*= 0.09, $V\u0302$ increases by approximately 70% for the same change in

_{s}*α*.

The influence of *α* is further illustrated in Fig. 10, which provides $V\u0302$ as a function of *α* (left panel) and as a function of drop volume (right panel). Data are shown for postpatterned SH surfaces with *f _{s}* = 1, 0.41, 0.27, and 0.09 and at a fixed surface temperature of 50 °C. For the data shown in the left panel, the drop volume was constant at 30 $\mu $L and for the data in the right panel

*α*was held constant at 20 deg. The dashed lines shown correspond to the average speed of a freely accelerating object on a frictionless surface, $V\u0302=sin\alpha /2$. As noted above, and evident again in the data of Fig. 10, as

*f*decreases the average drop speed moves closer to the frictionless surface line. For all surfaces the shape of the $V\u0302$ versus

_{s}*α*curve is similar to that of the frictionless surface, with greater deviation from the frictionless surface line as

*f*increases.

_{s}The impact of drop volume on $V\u0302$ is illustrated in the right panel of Fig. 10 for a surface inclination of $\alpha $ = 20 deg. The general behavior is an increasing average drop speed as drop volume increases. However, this behavior diminishes as *f _{s}* decreases, and at

*f*= 0.09 the increase in $V\u0302$ is vanishingly small (similar to the behavior of an object accelerating down a frictionless surface). In contrast, for

_{s}*f*= 1, $V\u0302$ increases by approximately 30% as the drop volume increases from 12 to 30 μ

_{s}*L*. For the

*f*= 0.27 surface, $V\u0302$ increases by approximately 10% as the drop volume increases from 12 to 30

_{s}*μ*L. Interestingly, for the smooth surface (

*f*= 1), the average speed actually decreases as the drop volume increases from 30 to 40 μ

_{s}*L*and the reason for this is not entirely clear. It is possible that this decrease is caused by the drop deformation that occurs at high drop volumes as the capillary length scale is exceeded, which would act to provide greater resistance to drop motion as the relative interfacial contact area increases. Drop motion results from the gravitational force, while adhesion caused by contact angle hysteresis and viscous drag resist motion. Drop inertia is negligibly small compared to the applied and resisting forces. As the drop volume increases, the gravitational force and drop inertia both increase concomitantly. However, the resisting shear and adhesion forces do not scale with volume and, on a relative basis, they exert less influence on the overall dynamics resulting in an increased average drop speed during the early times explored here.

Experiments were also conducted at various wall temperatures ranging from 50 to 80 °C. For these experiments, there was no observable difference in either the droplet speed or the heating effectiveness as the wall temperature was increased. It is important to note that the time period of the entire event is short (less than 0.3 s) and as a consequence, the total drop heating was insufficient to yield changes in the drop viscosity that could affect the dynamics of the drop motion.

We now consider the speed of drops moving on SH surfaces with rib-patterned structures. Experiments were performed for drops moving both parallel and perpendicular to the direction of the rib structures. Shown in Fig. 11 is $V\u0302$ as a function of *f _{s}* for both rib orientations. The drop volume for all data shown was 20 $\mu $L and

*T*= 50 °C. Results are shown at inclination angles of $\alpha $ = 15 deg (a) and $\alpha $ = 20 deg (b). The dashed lines shown in the figures represent curve fits of the data and are included to visually delineate between the two sets of data and to illustrate the overall trends. Of course, at

_{w}*f*= 1 (smooth surface) the results are identical. As $fs$ → 0 the influence of the orientation of the rib structures is diminished and at $fs$ ≈ 0.09 the results have essentially converged. Further, at

_{s}*f*≈ 0.09, the values of $V\u0302$ for the two inclination angles are approximately 0.34 ($\alpha =15\u2009deg)\u2009and\u20090.39\u2009(\alpha =20\u2009deg)$. These two $V\u0302$ values are very nearly the same as those observed in Fig. 9 for the postpatterned surfaces at the same inclination angles. Thus, the data clearly show that at very low solid fractions the nature of the SH surface structuring exerts negligible influence on the average drop speed. However, this is not the case for intermediate values of

_{s}*f*, where $V\u0302$ is consistently lower for the perpendicular oriented ribs. The largest difference in the data was observed for a solid fraction of

_{s}*f*≈ 0.5, where $V\u0302$ is between 25% and 33% lower for the perpendicular oriented rib surface. Interestingly, the values of $V\u0302$ for the poststructured surface (shown in Fig. 9) lie between the sets of data for the two rib patterned surfaces, with the values being just modestly lower than the parallel rib results.

_{s}*V*/d

*t*) scales with the average drop speed and the time over which acceleration is occurring,

*t*∼ $V\xaf/L$. Thus, the total acceleration of the drop scales as $V\xaf2/L$. The total force exerted on the drop can be expressed as

*F*=

_{g}*m*sin

_{d}g*α*, where

*m*is the mass of the drop. The adhesion force

_{d}*, F*, scales with the drop hysteresis caused by the advancing and receding contact angles and the contact line between the surface and the drop, or

_{h}*F*∼

_{h}*σD*(cos

*θ*− cos

_{R}*θ*). The viscous force scales with drop velocity, the thickness of the boundary layer between the drop and the surface,

_{A}*δ*, and the fluid viscosity, or

*F*∼ $\mu V\xaf/\delta $. Details regarding the time-changing hydrodynamic boundary layer are unknown from the experiments. However, the drop is accelerating from rest and speeds are low. Consequently, compared to the other two forces, the viscous drag force is much smaller. Neglecting the viscous drag force and simplifying Eq. (5) yields the following scaling for the average drop velocity in terms of inclination angle, drop diameter, fluid properties, and advancing and receding contact angles

_{v}*w*/

*h*, exert further influence on the results that are not captured by Eq. (7). This is likely due to their influence on the viscous drag at the interface between the drop and the surface. Analysis of the data shows that the influence of

*f*on $V\u0302$ is approximately a linear function. Thus, the scale analysis of Eq. (7) is modified to the form

_{s}The constants *k _{1}*, and

*k*were determined by minimizing the total mean squared error between Eq. (8) and the measured normalized average velocity. This was done separately for the smooth hydrophobic surface, post structured SH surfaces, and parallel and perpendicular oriented rib structured SH surfaces to give a different value of

_{2}*k*for each surface type, although

_{1}*k*remained constant at a value of 0.15. Table 3 provides the determined values of

_{2}*k*and

_{1}*k*that give the smallest mean squared error for smooth and post, parallel rib, and perpendicular rib surfaces. It is important to note that the first bracketed term in Eq. (8) is the most important and it captures more than 80% of the variability in $V\u0302$.

_{2}Structure type | $k1$ | $k2$ |
---|---|---|

Smooth | 0.95 | 0.15 |

Post | 0.57 | 0.15 |

Parallel rib | 0.28 | 0.15 |

Perpendicular rib | 0.51 | 0.35 |

Structure type | $k1$ | $k2$ |
---|---|---|

Smooth | 0.95 | 0.15 |

Post | 0.57 | 0.15 |

Parallel rib | 0.28 | 0.15 |

Perpendicular rib | 0.51 | 0.35 |

Shown in Fig. 12 is the measured value of $V\u0302$ plotted against the calculated value from Eq. (8). The data include all surface types, inclination angles, solid fractions, and wall temperatures, for smooth, post, and rib-structured SH surfaces. A total of 133 unique experimental conditions are included. Remarkably, nearly all of the data collapse to a single line with a slope close to 1, indicating that Eq. (8) incorporates all variables that exert influence on the average drop speed over the first 60 m traveled down an incline. The dashed line in the figure is a linear fit of the data. One of the most important outcomes of this work and Eq. (8) is that it allows prediction of the average speed of a drop rolling down smooth and SH surfaces for the entire parameter range that was explored in the present experiments.

### 3.2 Surface to Drop Heat Transfer Rates.

We now consider the thermal transport to a drop for drops on post patterned SH surfaces. The two most important variables that influence the heat transfer to the drop are the drop speed and the surface solid fraction. Shown in Fig. 13 is $d\epsilon /dt\u0302$ (panel a) and $\Delta \epsilon /\Delta t\u0302$ (panel b) as a function of $V\u0302$ for *f _{s}* = 1, 0.50, 0.27, and 0.09 and a drop volume of 20 $\mu $L. For all surfaces, the initial heating rate, $d\epsilon /dt\u0302$, is greater than the average rate, $\Delta \epsilon /\Delta t\u0302$. This is due to the fact that the slope of the

*ε*versus

*t*data is steepest at

*t*=

*0 and is constant through much of the transient process, but then begins to taper off at larger times (see Fig. 8). Experiments were also performed for stationary drops placed on surfaces with*

*f*$=1$ and 0.09 and these data points are shown at $V\u0302=0$. Trend lines are shown for each set of data and for all cases $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ appear to converge toward the stationary drop values as $V\u0302$ → 0.

_{s}The data of Fig. 13 show that both $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ increase with drop speed for all surfaces. This increase is much more pronounced for the surfaces with higher solid fractions. For the *f _{s}* = 1 surface, $d\epsilon /dt\u0302$ increased by approximately 500% and $\Delta \epsilon /\Delta t\u0302$ increased by approximately 250% as $V\u0302$ increased from 0 to 0.35. In contrast, for the

*f*= 0.09 surface, $d\epsilon /dt\u0302$ increased by only approximately 40% and $\Delta \epsilon /\Delta t\u0302$ increased by only approximately 20% for the same increase in drop velocity. A critical factor that affects the heat transfer is the contact area between the surface and the drop. This area is a direct function of the drop volume and the advancing and receding contact angles. For a classical surface, the product of this contact area, the local convection heat transfer coefficient, and the temperature difference between the drop and the wall represent the total instantaneous rate of thermal energy increase of the drop. This energy increase is equal to the product of the mass of the drop, the liquid specific heat, and the temporal gradient of the average drop temperature,

_{s}*dT*/

_{d}*dt*. Thus, the drop volume exerts significant influence on both $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$, since it affects both the contact area through which heat is transferred and the total thermal mass of the drop.

The solid fraction was also shown to significantly influence both the initial and average heating rates. Figure 14 shows $d\epsilon /dt\u0302$ (panel a) and $\Delta \epsilon /\Delta t\u0302$ (panel b) as a function of *f _{s}* for

*α =*10, 15, 20, and 25 deg, a drop volume of 20 $\mu $L, and

*T*= 50 °C. The red dashed line corresponds to the stationary drop case and is included for comparison. Data markers are not included for the stationary drop case because measurements were made only for the

_{w}*f*= 1 and

_{s}*f*= 0.09 surfaces. $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ increase with increasing

_{s}*f*for all inclination angles. The solid fraction strongly affects the total heat transfer to the drop and this is much more significant at higher values of

_{s}*α*and correspondingly larger drop speeds. As the inclination angle decreases toward a horizontal surface, the measured heat transfer approaches that for a static sessile drop. Further, at low values of

*f*, $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ approach the corresponding values for a static drop. The implication of this is that the speed of the drop is much less important with regard to heat transfer as the level of superhydrophobicity increases. This effect is demonstrated by the data being independent of the increase in droplet speed that prevails for the higher solid fraction surfaces.

_{s}The impact of variations in the drop volume is now discussed. Figure 15 shows $d\epsilon /dt\u0302$ (panel a) and $\Delta \epsilon /\Delta t\u0302$ (panel b) as functions of drop volume for varying solid fraction (*f _{s}* = 0.09, 0.27, 0.41, and 1),

*T*= 50

_{w}*°*C, and an inclination angle of

*α*= 20 deg. The data shows that rates of heat transfer increase as the volume of the drop increases. This is most significant for the smooth surface, where $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ increase by 50% and 25%, respectively as the drop volume increases from 12 to 40

*μ*L. This increase is likely due to the higher drop velocity and internal convection that prevails for the larger drop volumes. As

*f*decreases, however, the influence of drop volume is much less pronounced and this is evident in the data for all of the SH surface cases. Similar to the results for drop velocity, the wall temperature exerted no influence on the measured value of $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ for the temperature range considered here.

_{s}The type of SH surface exercises a small but not negligible influence on the initial heat transfer rate, with rib orientation modestly affecting the overall thermal transport. Shown in Fig. 16 is $d\epsilon /dt\u0302$ versus *f _{s}* for parallel and perpendicular oriented rib structured SH surfaces. The dashed lines shown in the figures represent curve fits of the data and illustrate overall trends. All data of Fig. 16 correspond to a drop volume of 20 $\mu $L. Panel (a) included results at an inclination angle of

*α*= 15 deg and panel (b) shows results for

*α*= 20 deg. Of course, for the

*f*= 1 (smooth) surface the results converge. At

_{s}*f*< 0.1 the $d\epsilon /dt\u0302$ values also appear to converge. Interestingly, at intermediate values of solid fraction, $d\epsilon /dt\u0302$ for the parallel rib orientation exceeds that for the perpendicular rib case and this behavior prevails at both

_{s}*α*= 15 deg and 20 deg. This behavior is similar to the average velocity data shown in Fig. 11. Recall that for drops moving on rib surfaces in the parallel rib configuration at intermediate values of

*f*the speed was faster than for the perpendicular rib configuration.

_{s}The data of Figs. 13, 14, and 16 demonstrate that *f _{s}* and $V\u0302$ exert a strong influence on heat transfer to the drop. To determine the level of dependency, the $d\epsilon /dt\u0302$ and $\Delta \epsilon /\Delta t\u0302$ data were evaluated as a function of the combined function $fsnV\u0302m$, where

*n*and

*m*are coefficients that were determined from the data to provide the best collapse of the data. This was done by adjusting

*n*and

*m*to minimize the mean squared error between the measured values of $d\epsilon /dt\u0302$ and the function $fsnV\u0302m$. The result of an optimization scheme showed that the best collapse of the data occurs when

*n*=

*1.2 and*

*m*=

*1.25. These coefficients suggest nearly first order dependency of $d\epsilon /dt\u0302$ on $V\u0302$ and $fs$. Recall from the discussion and data above that $V\u0302$ is also a strong function of*

*f*and this is evident in both the contact angle hysteresis term (that varies strongly with

_{s}*f*) and the [0.75 –

_{s}*k*] term of Eq. (8). Thus, the actual influence of

_{2}f_{s}*f*on the heat transfer is very significant. In particular, the solid fraction affects the heating rate by altering the (1) contact area between a drop and the surface and the (2) drop speed by increasing the slip length and associated reduced shear. Additionally,

_{s}*f*affects drop residence time on a surface, altering the time allotted for heat transfer to occur.

_{s}Shown in the left panel of Fig. 17 is $d\epsilon /dt\u0302$ as a function of $fs1.2V\u03021.25$ for every surface (smooth, post-SH, and rib SH), inclination angle, drop volume, and surface temperature considered in this study. The data show an excellent collapse with $fs1.2V\u03021.25$ and nearly all data points lie within 10% of the linear trend line which is also shown in the figure. An implication of this data collapse is that for a known average drop velocity and surface solid fraction, one can estimate the initial heating rate from the linear trend line shown in the right panel of Fig. 17. Recall that Eq. (8) allows prediction of $V\u0302$ for a given value of *f _{s}* and contact angle hysteresis and so using Fig. 17 and Eq. (8) together enables a priori estimation of $d\epsilon /dt\u0302$.

The right panel of Fig. 17 presents $\Delta \epsilon /\Delta t\u0302$ as a function of $fsV\u0302$ for all scenarios considered. Here the coefficients *n* and *m* are set to 1.0 since this provided the best collapse of the $\Delta \epsilon /\Delta t\u0302$ data. Recall from results presented previously (Figs. 13 and 14) the shape of the $\Delta \epsilon /\Delta t\u0302$ versus *f _{s}* or $V\u0302$ data differs from that of the $d\epsilon /dt\u0302$ data. $\Delta \epsilon /\Delta t\u0302$ is consistently lower than $d\epsilon /dt\u0302$ for higher solid fractions and higher drop velocity. This is caused by

*ε*plateauing at a later time and is a result of a decreased driving temperature difference between the drop and the surface. Consequently, greater variability in the $\Delta \epsilon /\Delta t\u0302$ data exists. However, as the data of Fig. 17(b) show, the $\Delta \epsilon /\Delta t\u0302$ data still show a very good clustering around a similar trend line.

## 4 Conclusions

This paper has experimentally explored the transient heat transfer to drops rolling on inclined heated smooth, post structured, and rib-structured SH surfaces. The transient mean drop temperature and displacement were measured using an infrared camera and from these measurements, the transient heat transfer and drop speed were determined. The influence of the SH surface solid fraction, inclination angle, drop volume, and wall temperature on the average drop rolling speed and the transient heating were explored. 133 unique experimental conditions were considered and analysis of the results leads to the following conclusions:

The speed of the drop is influenced strongly by the surface solid fraction and inclination angle. As the solid fraction is decreased the drop moves at higher speed and the behavior becomes similar to a block sliding down a frictionless inclined plane. The volume of the drop exerts a much smaller (but not negligible) influence on the drop speed.

The normalized average drop velocity was expressed as a function of the surface inclination angle, drop contact angle hysteresis, surface solid fraction, and surface structure pitch to height ratio. Coefficients were determined that allow drop speed prediction for smooth hydrophobic, post structured SH surfaces, and parallel and perpendicular oriented rib structured SH surfaces.

The transient heat transfer is primarily influenced by the solid fraction of the surface and drop velocity. As the solid fraction is decreased, the initial and average heat transfer rates are both reduced compared to the smooth surface. The decrease is as large as 80% for the initial heating rate, and as large as 74% for the average heating rate. An important implication is that SH surfaces with

*f*→ 0 offer a significant advantage for applications involving droplets on surfaces where low thermal transport is desired._{s}The data revealed that the heat transfer rates increase with increasing drop speed. This occurs for all surfaces, but the effect is much smaller for surfaces with very low values of

*f*. As the normalized drop speed decreased from 0.35 to 0, the initial heating rate showed a reduction of 80% for a smooth hydrophobic surface. In contrast, the reduction over this same change in drop speed was only 41% for a SH surface with a solid fraction of 0.09. Similarly, the average heating rate decreased by 58% for the smooth hydrophobic surface, and 23% for the SH surface with a solid fraction of 0.09 for the same reduction in drop speed._{s}The initial and average heat transfer rates behave similarly for low values of the solid fraction. However, at higher solid fraction, the average heating rate was consistently less than the initial heating rate. This occurs because the heating effectiveness eventually plateaus as the temperature gradient between the drop and the surface decreases.

For rib structured surfaces, both the average drop speed and the drop heat transfer were higher for drops moving parallel to the rib structures compared to the scenario where the drop moves perpendicular to the ribs.

## Funding Data

Directorate for Engineering, National Science Foundation (Grant No. CBET-1805805; Funder ID: 10.13039/100000084).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.