Topography Optimization for Sustainable Dropwise Condensation: The Critical Role of Correlation Length

An effective pathway to enhance the heat transfer process is to induce the formation of highly mobile condensate droplets, employing micro‐nanoengineered superhydrophobic surfaces. However, the design of the topography of these surfaces for sustained high performance constitutes a significant scientific and technological challenge. Herein, the critical role of the correlation length of topography is demonstrated as an important factor when designing superhydrophobic surfaces for heat transfer applications. Specifically, it is shown that a) a high correlation length value corresponds to increased space between surface structures and higher lateral distances between nucleating droplets, which results in lower droplet departure diameter and significantly delayed flooding of the surface and b) correlation length has to surpass a critical value for dropwise condensation (DWC) to be sustained in hierarchical structured surfaces, when the droplets are growing in a partial Cassie state. Following this rationale, droplets are categorized in three different energy and wetting states (Wenzel droplets, Cassie droplets of low kinetic energy and high energy jumping droplets), depending on the correlation length of the topography. Heat transfer experiments demonstrate an increase of 126% in the heat transfer coefficient (HTC) of surfaces exhibiting the maximum correlation length when compared to the flat hydrophobic surface.


Introduction
Condensation is essential for many applications which involve phase change heat transfer, such as steam cycle power generation, [1,2] water desalination, [3] and air conditioning systems. [4]Control of condensation can be achieved through DOI: 10.1002/adfm.202306756altering the wetting characteristics of a surface by manipulating its morphology and/or its surface energy.Heat transfer enhancement can be accomplished by forcing the steam to condense as distinct droplets with high mobility in order to be shed easily from the surface and allow for a new condensation cycle. [5]The main problem with the enhancement of phase change heat transfer is the balance between number of nucleation sites and departure rate of the condensed water of the surface.Heat transfer is enhanced by increasing the number of nucleation sites and reducing the droplet departure diameter. [6]11][12][13][14][15][16][17] Except from the typical superhydrophobic surfaces for dropwise condensation, many researchers have used special designs for enhancing this phenomenon.For example, Peng et al. [18] fabricated hierarchical microgrooved superhydrophobic surfaces by mechanical broaching and chemical etching, achieving an increase in heat flux of 90% compared to the flat surface for ΔΤ < 5 Κ, by inducing jumping droplets in two different length scales. [18]In the same path, Kumar et al. [19] investigated the combined effect of squared-grooved surfaces with a CNT coating in heat transfer coefficient.They measured heat transfer in grooved surfaces, with and without the CNT coating, and showed that the coated surfaces achieved a heat transfer coefficient of 61.11 kW m −2 K −1 for subcooling of 1 K.They also found that as the aspect ratio of the grooves increases, the higher the heat transfer coefficient is, due to the increased surface area of condensation. [19]Chen et al. [20] fabricated nano-textured micropyramidal superhydrophobic surfaces that combine high nucleation and high droplet removal rates.They achieved an ≈65% increase to the number of nucleated drops and an ≈450% increase in droplets removal rate compared to the just nanograssed superhydrophobic surface.Yao et al. [21] followed a similar direction by fabricating superhydrophobic cylindric and cubic micro structured silicon surfaces that achieved high heat transfer coefficients for high subcooling values. [21]any studies report the flooding of structures with the increase of surface subcooling.][24][25][26] Recent reports have shown that the nucleation site density not only depends on the surface subcooling and fluid properties, but it is also strongly influenced by the surface topography.Mu et al. [27] showed that nucleation site density increases with the surface fractal dimension: as the fractal dimension increases, there are more geometrical features on which condensation can take place such as sidelines, angles and edges.Ucar et al. [28] showed that at the initial stage of condensation, the nucleation density and the surface coverage from condensed droplets increases with the roughness of the surface.Enright and co-workers [29] have proposed two criteria for achieving superhydrophobic condensation.The first concerns the growth of individual droplets and determines whether the droplet contact line remains pinned, resulting to a partial Cassie state, or is de-pinned and the droplet wets the structures, leading to a Wenzel state.The second criterion highlights the effect of nucleation density, which determines the mean distance between nucleation sites 〈L〉.This criterion requires this distance 〈L〉, to be at least two times higher than the characteristic spacing l of the roughness, or equivalently, the nucleation density to be lower than the density of surface structures, N < l −2 .This second criterion is derived from the mechanism that leads the droplets to a non-jumping, pinning mode.In order to achieve superhydrophobic condensation, the excess surface energy released after the coalescence of two droplets, must surpass the adhesion work of the droplets with the surface.This energy is highly influenced by the distance of nucleation sites, i.e., the nucleation density, and decreases as the distance of nucleation sites approaches the length scale of the structures.When the droplets are in a partial wetting state, with pinned contact line, the smaller distances between nucleation sites, force the formed droplets to coalesce in lower radius, resulting in released surface energy that do not surpass the adhesion work, and formation of Wenzel droplets. [29,30]These studies show a clear dependence of the nucleation density with the surface topography and as key feature to enhance phase change heat transfer and prevent flooding, the balance between nucle-ation rate and departure rate from the surface. [7,31]However, no specific design factor have been used as guideline for the design of superhydrophobic surfaces intended for heat transfer applications.
Herein, we show that the condensation performance of micronanostructured, superhydrophobic surfaces can be controlled using correlation length as a design factor.The critical role of correlation length is demonstrated by experimental measurements of the topography changes of aluminum surfaces during their etching with HCl, along with their condensation performance.The metrological analysis of aluminum surface topography is coupled with a simulation of the etching process to show that the correlation length does not vary monotonically with etching duration, but exhibits a maximum value at an intermediate etching time.Higher correlation length values lead to increased distance between nucleating droplets and the formed droplets can be de-pinned from the structures and move freely, while for even higher values, jumping droplets are formed.Wider structures also enable the ejection of jumping droplets outside the topography since there is more space between surface structures (i.e., wider solid angle) allowing them to escape the surface more easily.In order to provide a quantified metric for the design of superhydrophobic surfaces which enable sustainable dropwise condensation, a critical ratio between the distances of micro and nanotopography is established for hierarchical structured surfaces during condensation, when the partial Cassie state is favored.The presented calculations clearly show that for correlation length below this critical value, the surface energy surplus created from the coalescence of two droplets, cannot surpass the adhesion of the droplets with the surface, and Wenzel droplets are formed after the coalesce.Intermediate correlation length structures lead to the formation of de-pinned Cassie droplets of low energy, while higher correlation length values lead to high energy jumping droplets.Heat transfer measurements in the surfaces which are following the extracted design guidelines, exhibit enhanced HTC of ≈160 kW m −2 K −1 for low subcooling, which is 126% better compared to flat hydrophobic surfaces.

Superhydrophobic Surfaces Topography
Microtopography is first fabricated on aluminum surfaces using dislocation-selective etching by immersion into an aqueous solution of hydrochloric acid for different durations (7, 8, 10, 12, and 15 min).To achieve a superhydrophobic state, 7 min of wet etching in HCl is required.Shorter durations resulted in nonuniform structures and isolated pits on the surface (poor superhydrophobicity), while durations longer than 15 min led to reduction of sample thickness and created large topography structures which are not suitable for condensation applications, since spacing and height of the topography become much larger than optimal (see Figure S1, Supporting Information).
The analysis of the autocorrelation function C(r) [32] of SEM images at different magnifications for the aforementioned etching durations shows that the correlation length does not have a monotonic behavior but it maximizes at an intermediate etching time (10 min) (Figure 2a).This maximization of correlation length, evident in the analysis of images of different magnification, indicates that this trend is not only local, but extends to all topography scales (Figure 2a), which reinforces the finding importance.Moreover, using the measured correlation length of different magnifications, we can extract data for the surface structure distances of different scales.For example, from low magnification images (×100, ×500) we extract the average distances between the formed cavities on micro-scale, while the correlation length derived from higher magnifications (×5000, ×10 000) quantifies the mean distances of structures inside cavities (Figure 2a, inset).
These findings are in full agreement with the analysis of SEM-like images, derived from the simulation of the etching process (Figure 2b and Figure S9, Supporting Information).The maximization of correlation length in both Figure 2. a) Correlation length obtained from the metrological analysis of SEM images for magnifications ×100, ×500, ×5000, and ×10 000.In all magnifications a maximum value in the correlation length is observed for the 10 min process.Inset: Schematic explanation of correlation length in the images with different magnification.b) Correlation length versus computational etching time resulting from the analysis of the SEM-like images of the simulated topography (black line).The computational etching time refers to the number of repetitions of the simulation.The pixel size is selected appropriately (188 nm) to match with the values from the analysis of real SEM images at magnification ×500 (blue line).c) Surface area and rms roughness related to etching duration as predicted by simulation.Both surface area and rms roughness increase exponentially with etching duration.d) Schematic representation of the topography after dislocation selective etching.In the intermediate time, the final pits have bigger distances between their edges, compared to the structure's distances resulted from higher and lower etching durations.
simulation and experimental results gives us the capability to calibrate the computational etching time (number of repetitions) with the actual etching duration in hydrochloric acid.The simulation of the etching process also reveals an exponential dependence of rms roughness and surface area on etching time (Figure 2c).
This peak in the correlation length indicates that an intermediate etching duration results in wider structures with higher distances between them, which can be explained as follows: when the aluminum surface is immersed in HCl, the dislocation defects are dissolved from the etchant forming small pits.These pits increase the etchable area and reveal more defects.The defects increase the etching rate (defined as subtracted material per unit time) leading to a subsequent increase of exposed area and an exponential dependence of the surface area with the etching time.Thus, the etching rate is lower in the upper tiers of the cavity and increases with the depth of the cavity.In addition to the surface area, the total rms roughness also increases exponentially with etching duration (see Figure 2c,d).As the etchant fills the cavity, etching occurs simultaneously throughout the material.If the etchant was in contact only with the bottom of the cavity, the lateral distances, at the same depth of the cavity, would increase monotonically with the etching duration.The etching process creates pits with the form of reversed pyramid, meaning that the lateral distances inside a pit are smaller in the deeper regions of the pit.When all the cavity is covered by the etchant, the upper tiers are also etched.Therefore, at some point, the monotonic increase in lateral distances with etching duration will stop due to the removal of the higher lateral distance tiers in the cavities, leaving only the deeper tiers with smaller lateral distances, as shown in Figure 2d.This means that at some intermediate point during etching, a peak in the lateral distances will appear.

Effect of Topography on Droplet Departure Size
In order to examine the effect of correlation length on condensation, we tested surfaces with high differences in the values of correlation length.The condensation performance was evaluated by measuring the departure diameters in the setup described in Section 4 (Condensation Evaluation).The comparison of departure diameters was conducted amongst the surfaces shown in Table 1, fabricated as will be discussed in Section 4 (Surface Micro-Nanotexturing and Hydrophobization).All three hierarchical surfaces are superhydrophobic with static contact angle (SCA) higher than 170°and hysteresis (Advancing Contact ngle (ACA)-Receding Contact Angle (RCA)) below 2°(Table 1).The nanostructured superhydrophobic surface also has high contact angle 170°but its hysteresis is higher compared to the hierarchical surfaces (Table 1).The hydrophobic surface has SCA 110°a nd hysteresis >20°.The roll-off angle is <1°in all hierarchical superhydrophobic surfaces and 3°± 0.5°for the case of nanostructured superhydrophobic surfaces.
The condensation evaluation was performed by measuring the maximum departure diameter on each surface, during condensation of saturated steam of 100 °C and subcooling of ≈1 °C.We consider that the measurement of droplet departure diameter is a useful method for the evaluation of condensation because it reflects the ability of the surface to get cleared from condensate, as the maximum departure diameter d max mainly depends on the surface hysteresis. [33]The increase of departure diameter during condensation corresponds to an increase of apparent hysteresis due to possible flooding of the structure and/or pinning effects caused by this flooding (see Figure S3, Supporting Information).
Figure 3a shows the average value of the diameter of the three largest droplets on: a) the flat hydrophobic, b) the nanostructured superhydrophobic, and c) the 7, 10, and 15 min hierarchical superhydrophobic surfaces.The flat hydrophobic surface exhibits a quite large drop departure diameter already early during condensation and transitions to filmwise condensation soon thereafter (within the first 20 min of the experiment).The nanostructured superhydrophobic surfaces exhibit smaller droplets, but the departure diameter increases rapidly to over 1 mm after 20 min, indicating that transition to filmwise condensation is taking place (see Figure S4, Supporting Information).Hysteresis of hydrophobic and nanostructured surfaces was slightly increased after condensation, while in the case of hierarchical surfaces, hysteresis remained unchanged, sustaining dropwise condensation for at least 1 h.Thus, flooding of a nanostructured surface occurs much faster compared to hierarchical surfaces.This occurs due to the following reasons: a) The increased surface area of hierarchical surfaces enables the formation of more nucleation sites and the three-dimensional coalesce of droplets inside the cavities, which increases the departure rate of condensate, comparing with the two-dimensional coalesce of the nanostructured surfaces.b) The existence of microstructures enables also the departure and self-navigation of bigger droplets through Laplace pressure instabilities, [34] a mechanism which is not applicable on the nanostructured surfaces.c) Nanostructured surfaces exhibit higher hysteresis angle, which also contributes to bigger droplets and faster flooding, compared to hierarchical surfaces.On the contrary, the droplet departure diameter on the surface etched for 10 min does not exceed 0.6-0.8mm even after 2 h of condensation, while the departure diameter of the 7 min-etched surface again exceeds 1 mm after 80 min.Similar to the 7 min etching, the departure diameter of the 15 min etched surface reaches 1 mm after 60 min.Another important finding is that the droplet departure diameter of superhydrophobic surfaces is ≈4 times smaller compared to the flat hydrophobic surfaces.
When a condensation transition from dropwise to filmwise is observed, droplets with larger diameters appear and the standard deviation increases.Thus, the increase of error bars versus time in Figure 3a shows the increased standard deviation of the diameters of the three largest droplets.However, this evaluation technique using the sizes of the 3 largest droplets has a significant drawback, which is that it cannot quantify the condensation behavior of a surface after the transition to filmwise.This can be achieved by monitoring the projected area of the surface that is covered by condensed water (droplets with diameters larger than 50 μm or film), as it is shown in Figure 3b. Figure 3b clearly shows that the 10 min-etched surface is covered with condensed water at a fraction under 20% during the entire duration of the condensation experiment, whereas in the other two surfaces this fraction increases significantly after 100 min, indicating that large areas of the surface are covered with water film (see Figure S5, Supporting Information).The 10 min surface also exhibits a high number of jumping droplets (see Figure S6, Supporting Information), which prevents the surface from flooding and shifts the droplet distribution systematically to smaller values (Figure S7, Supporting Information).In Figure 3c we demonstrate that the 10 min surface is durable and maintains dropwise condensation for at least 240 min (see Figure S3, Supporting Information) with droplet diameter smaller than 0.8-1 mm (Figure 3a,c).

Effect of Topography on the Transition from Dropwise to Filmwise Condensation
The effect of topography can be explained by understanding the condensation dynamics inside the cavities.Starting from nucleation events, nucleation mainly occurs in the space between the nanostructures.Increase of the surface area by chemical etching will lead to an increase in the number of nucleation sites and thus, the surfaces etched for 7, 10, and 15 min will have ≈12×, 30×, and 216× more nucleation sites compared to the nanostructured surface, respectively (Figure 2c).Although the increase in number of nucleation sites increases the heat flux during the phase change phenomena, the increased number of closely spaced nuclei may cause flooding and transition to filmwise condensation, depending on the width of the microtopography, which is described by the correlation length.Thus, higher values of correlation length can lead to a delayed transition to filmwise condensation via two different mechanisms, acting at different scales: Micro-nanoscale mechanism: The higher distances between the nucleated droplets enable formation of larger drops which can overcome the adhesion force with the surface when merged. [30]This mechanism is mainly described from the correlation length of magnifications ×5000 and ×10 000 which correspond to the edge distances of the structures inside and near the bottom of the cavities.
Microscale mechanism: cavities with wider structures allow the jumping droplets to be ejected over a wider solid angle.This mechanism is mainly described from correlation length of magnifications lower than ×5000, which describe upper parts of the cavities.
During condensation, if the criterion E * = −1 k⋅cos , introduced by Enright et al., [29,30] is < 1, droplets are growing in an energy favorable partial-Cassie state, meaning that their contact area is pinned and constant, while they are growing in a balloon-like shape.In this equation, k refers to the ratio of the actual surface to the projected surface.Here, k was calculated by recreating the nanotopography of SEM images, by using the software nanoTOPO-SEM.The constant contact of the droplets entails also a constant work of adhesion W between the droplets and the surface.As such neighboring droplets grow and start to merge into even larger drops, energy ΔE is released due to the droplet surface change.The excess energy, ΔE, may or may not exceed the adhesion work W of the droplets on the surface and this energy difference determines if the merged droplet is pinned, mobile or able to jump.For a droplet to de-pin from the structures after coalescence, ΔE has to overcome the work of adhesion W of each droplet existing before the coalescence.As condensation proceeds, the increasing size of the droplets will eventually lead to a drop with energy surpassing this adhesion work.
For two droplets of equal size having a fixed contact area with radius r c , the critical minimum radius of the droplet r, at which the surface energy surplus exceeds adhesion work, will be (see also Supporting Information for more details): where  the microscopic contact angle.The radius of contact area r c is half the distance between the nucleated droplets l NS and thus If the average distance between the nucleation sites l NS is equal to the distance of nanostructures l nano , then Equation (2) becomes: This means that, in order a de-pinned Cassie droplet to be formed, the two coalescing droplets has to be within microstructures with distances l micro (equal to correlation length CL) higher than 4 • r (see index of Figure 4), and thus If l micro is smaller than the quantity of inequation ( 4), the coalescing droplets will have radius below the critical value of Equation (3) and the formed droplet will be in Wenzel state.
From (4) one can derive a critical ratio between the distances of micro and nanostructures for sustaining DWC when the partial Cassie state is favored: The above ratio indicates the relative lateral distance between the micro and nanotopography for achieving sustainable dropwise condensation and delaying flooding of the surface, when the drops are growing in a partial Cassie state.For hydrophobic coatings with typical values of contact angles, ranging from slightly higher than 90°up to 110°, the above ratio will be between 3.1 to ≈2, accordingly.
In our case, considering a fixed contact radius of the coalescing droplets, of 135 nm (which is the half of the approximate spacing between the nanostructures l nano ) as E * < 1, ΔE∕W becomes >1 when each droplets radius grows above 140 nm.This value corresponds to a minimum spacing between microstructures of at least ≈560 nm, as CL minimum ≅ 4 • r.
Figure 4 shows that there are three types of droplets inside the micro-nanotopography.When ΔE W < 1 adhesion work is higher from the interfacial energy surplus after coalescence, and the formed droplet remains in a pinned Wenzel state (Region I, Figure 4).The increase of droplet radius to 140 nm offers enough surface energy that surpass the adhesion work enabling the formed droplet to be de-pinned from the nanostructures and transit to a Cassie state.In order to achieve such energy surplus, the distance between the microstructures, i.e., the correlation length, has to be at least 560 nm (Region II, Figure 4).In this region the formed droplets are moving freely between the micro-nanotopography but do not have enough kinetic energy for jumping.Further increase of droplets radius results to enough kinetic energy that allows droplets to jump from the structures, as the dynamic energy needed to escape from the deeper cavity (rms roughness of 250 μm) is in the order of 10 −19 to 10 −20 J, which is 5 to 6 times order of magnitude lower than the kinetic energy of the formed droplets (Region III, Figure 4).As indicated in Figure 4, correlation length should be at least 1000 nm in order for the droplets to jump.The kinetic energy of droplets right after coalescing is where E viscous is the viscous energy dissipation after coalescing and is calculated as: [35,36] (2r) 1.5 (   ) 0.5 (7)   where  and  are the water viscosity and density respectively.The metrological analysis of ×10 000 magnifications images shows that the structures in those length scales (bottom of microstructures) are below the limit of 560 nm, also for the case of the 10-min structured surface.This means that in these structures, eventually Wenzel droplets will form and the flooding will begin from there, even for the 10 min-etched surface.The difference with the other two topographies is that, in the 7 and 15-min etched surfaces, flooding will start in more tiers, compared to the 10 min etched surface, and this is the reason of their quicker transition to filmwise.
The second mechanism is related to the solid angle of droplet ejection, for droplets formed in structures of higher correlation lengths.Cavities with larger cavity openings allow the jumping droplets to be ejected at higher solid angle.Thus, the 10 min etched surfaces exhibit much larger jumping solid angle (compared to 7-and 15-min etched surfaces).
In conclusion, the phenomenon can be described in Figure 5. Topographies with higher correlation length allow the formation of Cassie droplets that self-navigate between structures and escape the surface, while topographies with correlation length below the critical minimum, result to formation of pinned droplets that eventually flood the structures (Figure 5a).Flooding starts from the bottom of microstructures due to the first mechanism of the formation of Wenzel droplets, while the second mechanism of the entrapping of jumping droplets, is the driving mechanism during flooding at higher length scales.As a result, when the water film reaches the outer droplets, pinning takes place and the apparent hysteresis of the droplets increases, resulting in larger drop departure diameter (Figure 5b).

Heat Transfer Measurements
Figure 6a,b shows the heat transfer coefficient (HTC) and heat flux (Q) as function of surface subcooling.Heat transfer measurements have been performed at 30 mbar in an industrial-like setup (see subsection Heat Transfer Evaluation Setup in Experimental Section), different from the setup of preliminary condensation experiments at atmospheric pressure.The hydrophobic flat surface exhibited a mean HTC of ≈50 kW m −2 K −1 for subcooling ranging from 0 to 5 K while superhydrophobic nanostructured surface showed a maximum value of 76 kW m −2 K −1 for subcooling around 0.4 K.This value of subcooling corresponds to supersaturation of 1.02 in which jumping effects are dominant in the departure behavior.Increasing of supersaturation to 1.15 for subcooling 2.37 K leads to flooding of the nanostructures and HTC becomes similar to flat hydrophobic.Hydrophobic surface sustains dropwise condensation in the entire duration of the condensation experiment and a partial transition to filmwise is observed only in coating defects (see Video S1, Supporting Information).On the contrary, the nanostructured surface transitions to filmwise for subcooling higher than 3 K, and heat flux and HTC = Q ΔT are reduced lower to that of flat hydrophobic (see Video S2, Supporting Information).This occurs due to the flooding of the nanostructures and the increase of apparent hysteresis which leads to inability of shedding the condensed water that has trapped between the structures.In the absence of structures, the wettability of the flat hydrophobic surfaces depends only on the coating durability.Hysteresis of nanostructured and flat hydrophobic surfaces was increased after condensation experiment, indicating the degradation of the hydrophobic coating (see Video S5, Supporting Information).The hierarchical surfaces maintained similar performance for the entirety of condensation experiment and hysteresis did not increase after the experiment (see also Video S4, Supporting Information).
Hierarchical superhydrophobic surfaces reach an HTC value of ≈160 kW m −2 K −1 for surface subcooling up to 1 K (3 times higher compared to hydrophobic) (see also Video S3, Supporting Information).This value is directly comparable or even higher from other reports in the literature [10,15,22,26] and it can be attributed to the special topography feature characteristics that the 10 min hierarchical superhydrophobic surface exhibits.Heat transfer coefficient for higher subcooling declines down to 68 kW m −2 K −1 for supersaturation higher than 1.15.This effect is attributed mainly to the edge effect which initiates the formation of large droplets. [37]These large droplets enhance flooding, starting from the edges, which gradually grow to other areas.

Conclusion
In this work a clear link between correlation length and condensation behavior was demonstrated.Metrological analysis of SEM images from surfaces etched through dislocation selective etching with hydrochloric acid showed that an intermediate etching duration results in higher correlation length and proves the nonmonotonic dependence of surface structure distances on etching time.These results are also valid at all length scales and have been confirmed by a developed kinetic Monte-Carlo model of the etching process.Monitoring of droplet departure diameters during condensation on hierarchical superhydrophobic surfaces showed that high values of correlation length result in significantly delayed transition to filmwise and smaller droplet departure diameters.This can be explained by the higher lateral distances of nucleation sites, when the correlation length is higher, as the formed droplets can grow independently and coalesce in bigger diameters.Specifically, when correlation length surpasses a critical value, the surface energy released from the coalescence, surpasses the work of adhesion allowing the merged droplet to depin from the nanostructures and move freely upward in micronanotopography.Coalescence in even higher lateral dimensions results in merging of droplets with bigger radius and formation of droplets with high kinetic energy that jump outside the cavities.Also, the wider the structures are, the easier it is for jumping droplets to be ejected, as the angle field is increased, and the probability of droplets are entrapped from structures af-ter jumping is lower.Thus, a critical ratio between microstructures distances (correlation length) and distance of nucleation sites has been established for condensation of droplets in a partial Cassie state in hierarchical surfaces for sustainable dropwise condensation.This ratio is equal to a ratio between the distances of micro and nanotopography when the distance of nucleation sites is same as the size of nanostructures.Therefore, surfaces used for condensation should satisfy this lower limit of correlation length in all tiers of their structures in which nucleation takes place, in order to sustain dropwise condensation for longer durations.Heat transfer measurements showed that the most durable, in terms of DWC, hierarchical surface, exhibited heat transfer coefficients of ≈160 kW m −2 K −1 at 0.7 K subcooling, which is translated to an improvement in HTC of 126% compared to the flat hydrophobic surface.Correlation length can be considered as a critical surface parameter for adjusting nucleation site density during condensation on randomly structured surfaces.

Experimental Section
Materials: Pristine 1.5 mm thick aluminum substrate (99.5%) was cut into 20 × 50 mm samples.The samples were cleaned using isopropyl alcohol (IPA), acetone and deionized (DI) water.Samples were then immersed into 0.1 m aqueous solution of NaOH for 20 min in order to remove the native aluminum oxide.Hydrochloric acid (37%) used for Al etching was purchased from Sigma-Aldrich.
Surface Micro-Nanotexturing and Hydrophobization: Microtexturing: Aluminum samples were immersed into 9.25% v/v aqueous solution of hydrochloric acid for 7, 10, and 15 min for the fabrication of microscale features, using the dislocation-selective etching method. [38]anotexturing: A fast and environmentally friendly method for fabrication of nanostructures on aluminum surface is the boehmitage process. [39]After aluminum texturing with hydrochloric acid, Al surfaces are immersed into boiling DI water for 5 min in order to fabricate nanotexturing (hierarchical topography).
Hydrophobization: It has been conducted through C 4 F 8 plasma deposition in an Inductively Coupled Plasma (ICP) reactor (Micromachining Etching Tool, Alcatel).The deposition conditions were: Plasma power 900 W, pressure 40 mTorr, C 4 F 8 flow rate 25 sccm, 0 V bias and sample temperature 0 °C. [40]These conditions induce a deposition rate of a hydrophobic film ≈30 nm min −1 when measured on a flat Si sample.Coating thickness is expected to be lower in a rough surface compared to a flat one.The duration of deposition on flat and hierarchical surfaces were 1 min while on nanostructured surfaces were 30 s. Plasma deposited coatings are rich in CF 2 -and -CFx-content, which makes them hydrophobic. [41]etting and Condensation Evaluation: Wetting properties characterization: Static and dynamic contact angle measurements were conducted at KRÜSS DSA 30S contact angle instrument using 5 μL water droplets.Surface energy measurement was conducted in the same instrument by using droplets of diiodomethane and DI water of 5 μL.For the calculation of surface energy, the OWRK model was used.[42] Condensation evaluation: To evaluate the condensation on fabricated surfaces, an environmentally isolated setup was constructed including a DSLR camera, a mount for the sample with an adapted heat sink and a beaker with boiling water, as is shown in Figure 7a.Hot steam from the beaker moves parallel to the surface with an adjusted speed of 139 mm s −1 .Steam velocity was calculated after measuring the evaporation rate of the boiling water.Steam is canalized parallel to the surface with the use of mounted PTFE flaps.Surface temperature is measured by using a K-type thermocouple.The condensation process is recorded with the DSLR camera.After recording, the images are analyzed using MAT-LAB Image Processing Toolbox (Figure 7b).All measurements reported in this paper were performed in this setup, except from the heat transfer coefficient measurements.
Metrological analysis: The experimental SEM images of Al surfaces etched in hydrochloric acid for the tested durations are analyzed by using the software nanoTOPO-SEM provided by Nanometrisis p.c. (https: //www.nanometrisis.com).The main focus of the analysis has been to calculate the correlation length of images which is defined as the distance  at which the normalized autocorrelation function C(r) of SEM image just drops below 1/e. [32]Practically, in a rough surface, the correlation length quantifies the average width of topography features.High values of correlation length indicate that the surface structures have longer distances between them, while lower values imply the opposite.The correlation length along with the rms roughness are the more widely used descriptors of the topography features of a surface.
Topography simulation: Measurements of rms roughness and surface area of the topography created after the dislocation-selective etching with hydrochloric acid are challenging because of the complicated geometry of the deep cavities comprising the etched surface.In order to surpass this problem, a Monte-Carlo model of the etching process was developed to be used for the simulation of topography and the measurement of its characteristics.In addition, the comparison of topography features that can be extracted from both real and simulated surfaces (such as correlation length) can confirm the findings of metrological analysis and vice versa.
In the proposed simulation model, the aluminum substrate is represented with a square cuboid with depth equal to the initial thickness of the substrate.The etching process is modeled with the successive removal of much smaller cuboids from the large substrate cuboid.Assuming that the positions of dislocations in substrate are randomly distributed, a kinetic Monte-Carlo approach can be applied and the center of each removed cuboid is chosen randomly on the surface of substrate.The dependence of the size of removed cuboid (both laterally and vertically) on the height of etch position leads to the increase of the amount of removed material with etching duration since more and deeper cavities are created on surfaces in the course of etching time, thus increasing the enhanced factor (z max − z).An analogous behavior is observed in the physical dislocation selective etching.An already etched cavity is etched with an increased etching rate as more surface area of the material is in contact with the acid and more dislocation defects appear. [43]However, it should be emphasized that given the above-mentioned time dependences, attention should be paid so that the lateral size of removed cuboids during the whole simulation is kept sufficiently smaller than that of the substrate to minimize finite-size effects and justify the statistical reliability of the obtained results.
The thickness r of the removed cuboid at each step is chosen to depend linearly on the current height z of the substrate at the position that the etching takes place (center of removed cuboid), according to the formula: where z max and z min are the maximum and minimum height of the substrate surface accordingly, while r i and r f are the predetermined minimum and maximum thickness of the removed cuboid.The lateral size of the removed cuboid also depends on the height of the substrate at the center of cuboid as: where D x , D y are the x and y dimensions of the subtracted element (lateral size), D x0 , D y0 the initial x and y dimensions accordingly and c is an augmented factor which is used to match the dimensions of the removed cuboid with the cavities formed during the dislocation selective etching (Figure S9, Supporting Information).
In order to match the simulation parameters to the experimental results, the output simulated surfaces were converted into SEM-like images.This can be made in a straightforward manner, exploiting the specific morphology of the obtained topographies (flat and vertical sides with steep edges in between) and the edge phenomenon in SEM image acquisition according to which the obtained electron signal is maximized at the edges of topography due to geometrical reasons.The comparison of the experimental SEM images with the artificial SEM-like images from simulations assists the calibration of the modeling parameters in order to obtain simulated images very close to the real ones and estimate the topography characteristics that cannot be measured directly (see also Figure S10, Supporting Information).
Heat transfer evaluation setup: Condensation heat transfer measurements are conducted with saturated steam at 30 mbar.Similar procedures are followed to the recent work, where full details of the setup construction, experimental procedures, and uncertainty estimation can be found. [44]The tested sample is placed vertically on a copper cooler in a custom-built chamber.Saturated steam at 30 ± 0.5 mbar with a corresponding saturation temperature of 24 °C flows horizontally across the surface.The flow is driven by the pressure difference between an electric boiler maintained at a constant pressure and a vacuum pump, resulting in a mean speed of ≈4.6 m s −1 in a flow cavity of height 13 mm.At the same time, the back of the copper cooler, on which the sample is mounted, is cooled by a recirculating chiller.An array of 7 RTDs is positioned between the sample and the back of the cooler to measure the condensation heat flux by a linear fit of the temperatures.Surface and chamber steam temperatures are each measured by 2 RTDs.Chamber steam pressure is measured with a capacitance gauge.During experiment, the chiller temperature is varied across a fixed set of temperatures.As the system stabilizes and achieves steady state at each chiller temperature, the resulting subcooling, heat flux and heat transfer coefficient are then measured for 1 min and the mean value is reported.The sample area used for the heat transfer measurements was 20 × 20 mm 2 .

Figure 3 .
Figure 3. a) Maximum drop diameter (average of the 3 largest droplets) during condensation of saturated steam (steam velocity of 139 mm s −1 ) on hydrophobic, nanostructured superhydrophobic and 7, 10, and 15 min hierarchical superhydrophobic surfaces.Condensation was performed with saturated steam of 100 °C with subcooling of ≈1 °C.b) Fraction of surface that is covered with water droplets of diameters larger than 50 μm during condensation, c) Drop departure diameter of the 10 min etched surface during a long-lasting condensation experiment (>2 h).Dropwise condensation is maintained for at least 240 min.Representative images of the surfaces during condensation are provided in Figures S3-S6 (Supporting Information).

Figure 4 .
Figure 4. Excess energy ΔE during merging of two droplets of equal radius, compared to adhesion energy and kinetic energy, normalized by adhesion work, versus droplet radius and topography determined by etching time and correlation length.Black line represents the ΔE∕W, which shows the ability of the drop to detach from pinning, and red line is the kinetic energy of the final droplet after coalescence normalized by the work of adhesion E kinetic /W.When ΔE∕W > 1 droplet is depinned from the structures and when E kinetic /W positive the drop is expected to jump.Both ΔE∕W and E kinetic /W are plotted versus the droplet radius before coalescence.The blue color line is the correlation length of the structures versus the etching time of surface in hydrochloric acid.Three different regions are distinguished (I, II, III) depending on the energy and wetting state of the final droplet; Pinned Wenzel droplets that eventually will flood the structures (Region I), Cassie droplets of low kinetic energy that have been de-pinned and will self-navigate through further coalescences with other droplets (Region II) and high energy jumping droplets which are formed in the upper stages of cavities (Region III) where the correlation length allows it.

Figure 5 .
Figure 5. a) Droplet coverage of the topography resulting from 7, 10, and 15 min in hydrochloric acid and illustration of high and low lateral distances between nucleation sites/formed droplets (NS).Black dashed lines indicate the angle range of the ejected droplets.Dashed red lines indicate droplets after coalescing.b) Schematic representation of the flooding mechanism observed on the surfaces with low values of correlation length (7, 15 min) and the self-navigation of the Cassie droplet in the case of higher value of correlation length (10 min).The red contours indicate the growth stages of the outer droplets after the pinning caused by their bridging with the water film.This pinning results in higher droplet departure diameters in the cases of 7-and 15-min etched surfaces compared to the 10 min etched one.

Figure 6 .
Figure 6.Condensation on superhydrophobic hierarchical surfaces, superhydrophobic nanostructured and hydrophobic flat surfaces.a) Heat transfer coefficients for the tested surfaces, b) heat flux as a function of surface subcooling.

Figure 7 .
Figure 7. a) Illustration of the setup used for condensation evaluation and departure diameter measurement, b) condensation visualization and droplet diameter analysis.

Table 1 .
Wetting properties of the flat, nanostructured and the hierarchical surfaces after coating through C 4 F 8 plasma deposition.