Source: https://pubs.rsc.org/en/content/articlehtml/2019/na/c8na00237a
Timestamp: 2019-04-20 00:17:26+00:00

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aDepartment of Mechanical and Energy Engineering, University of North Texas, Denton, TX 76207, USA.
Recently the development of superhydrophobic surfaces with one-tier or hierarchical textures has drawn increasing attention because enhanced condensation heat transfer has been observed on such biomimetic surfaces in well-tailored supersaturation or subcooling conditions. However, the physical mechanisms underlying condensation enhancement are still less understood. Here we report an energy-based analysis on the formation and growth of condensate droplets on two-tier superhydrophobic surfaces, which are fabricated by decorating carbon nanotubes (CNTs) onto microscale fluorinated pillars. Thus-formed hierarchical surfaces with two tier micro/nanoscale roughness are proved to be superior to smooth surfaces in the spatial control of condensate droplets. In particular, we focus on the self-pulling process of condensates in the partially wetting morphology (PW) from surface cavities due to intrinsic Laplace pressure gradient. In this analysis, the self-pulling process of condensate tails is resisted by adhesion energy, viscous dissipation, contact line dissipation and line tension in a combined manner. This process can be facilitated by adjusting the configuration and length scale of the first-tier texture. The optimum design can not only lower the total resistant energy but also favor the out-of-plane motion of condensate droplets anchored in the first-tier cavity. It is also shown that engineered surface with hierarchical roughness is beneficial to remarkably mitigating contact line dissipation from the perspective of molecular kinetic theory (MKT). Our study suggests that scaling down surface roughness to submicron scale can facilitate the self-propelled removal of condensate droplets.
In this paper, we report our energy-based analysis of growth dynamics of dropwise condensates on biomimetic surfaces with two-tier hierarchical textures and their structural optimization. In particular, we focus on the intrinsic energetics associated with the transition of a condensate tail, which is entrapped in the first tier cavity but exhibits Cassie state relative to the second-tier nanoscale roughness (exclusively referred to as the PW condensates in this work), to the Cassie state through the self-pulling mode without coalescence with neighbouring droplets. In this analysis, the resistant energy associated with the expulsion of a condensate tail in the PW morphology is divided into three main parts. The first part is the adhesion energy due to the adhesion of the tail to the nanotextures in the first tier cavity. The second part is the viscous dissipation during the upflow of the condensate tail from the first tier cavity to the bulky portion of the PW droplet. The third part is related to contact line dissipation that occurs within three phase (liquid/vapor/solid) contact zone of the condensate tail. Besides, the effect of line tension on condensate state transition is also discussed. By minimizing the energetic resistance, the first tier roughness is optimized to favour the PW-Cassie transition of a condensate tail in the cavity. This work can advance our understanding on the growth dynamics of PW condensate droplets and aid the design of micro/nano-engineered lotus-leaf-like condenser surfaces promoting continuous dropwise condensation.
In our dropwise condensation studies we fabricated biomimetic lotus-leaf-like surfaces with two-tier hierarchical structures as shown in Fig. 1(a) and (b).12 Regularly-positioned silicon micropillars12–16 were adopted as the first-tier texture, i.e., the primary roughness, and carbon nanotubes (CNTs) were decorated on the micropillars as the second-tier texture, i.e., the secondary roughness. The width, height and center-to-center pitch (period) of the first tier pillars are denoted by b, h and l, respectively, while rp stands for the corner radius of the first tier pillars. In this work, the first tier has a solid fraction of ff = b2/l2 ≈ 0.32 and the second tier nanostructures has a solid fraction of fn ≈ 0.25, so the solid fraction of the two-tier textures is f = fffn = 0.08, which is much smaller than 1.11,12 As-formed micro/nanostructured surface was conformally coated by a thin layer of fluoropolymer (FluoroPel™ PFC1601V, Cytonix or Teflon®) to strengthen its hydrophobicity to superhydrophobicity. This type of engineered surfaces with regular geometries provides an ideal platform to study the dynamics of droplet growth and expulsion of the tail of a PW condensate droplet in a unit cell, i.e., the cavity volume surrounded by four micropillars as shown in Fig. 1(b), as well as the spatial distribution of condensate droplets.
Fig. 1 (a) Schematic (top view) of the first tier pillar array on a two-tier textured surface with the second tier nanotextures shown in red bars. The four short pink segments indicate that the condensate droplet formed in the cavity anchors on the vertical corner edges of the four surrounding pillars. (b) SEM image (plan view) of a cavity cell confined by the surrounding first tier pillars, which are decorated by CNTs. (c) Side view of a condensate droplet in the PW morphology. The tail can enhance heat conduction by bridging the gap between the condensate bulk and the cavity base. For simplicity, only one tail out of multiple tails is shown under the droplet. In the self-pulling mode, out-of-plane expulsion of the condensate tail is intrinsically driven by Laplace pressure gradient. (d) The base of the condensate tail is adhered to the nanotextures in the cavity valley (surface contact) while the tail body is in contact with the nanotextures along the first tier vertical edges. Droplet portion above pillars is not shown for the purpose of simplicity. It is assumed that nucleation embryos start growing close to the top portion of the nano-structures forming a wet spot and therefore as-grown condensate microdroplet is at the Cassie state relative to the nanoscale roughness.
where re = 2νlσTw/hlv(Tsat(Pv) − Tw) is the equilibrium radius and σ, rc, θ, νl, hlv, Tw, Tsat, Pv are the liquid–vapor surface tension, the characteristic size of formed embryos, the Young's contact angle, liquid specific volume, latent heat, wall temperature, saturation temperature, and vapor pressure, respectively. Assuming θ ≈ 115° on fluoropolymer-coated surface, hlv ≈ 2500 kJ kg−1, Tsat = 278 K, Tw = 273.15 K, σ ≈ 0.075 N m−1, and νl ≈ 0.001 m3 kg−1, the equilibrium radius re is ∼3 nm in our ESEM experiment. The free energy barrier to form condensate nanodroplets within the CNT interstices of ∼100 nm is about −6.8 × 10−15 J. By analogy, the free energy barrier to form condensate microdroplets in the first tier cavities of 4–5 μm wide is about −1.15 × 10−11 J. Therefore, it is energetically favourable to form condensate droplets in the first tier cavities while being fed by nucleation embryos formed in the interstices of CNTs. As such, condensate droplets firstly emerged at the bottom edges in the first tier cavities as indicated by the white dotted circle in the 2′′ time frame of Fig. 2 in order to minimize the area of liquid–vapor interface.4 The formation of the primitive microdroplets, which seem to root in and sprout from the first tier cavities in spite of ubiquitousness of nucleation embryos on nano-roughness, has been confirmed both in ESEM12 and under optical microscope.11 Subsequently morphing-induced Laplace pressure gradient worked as the driving force to laterally (in plane) propel the condensate body among the pillars (frame 5′′ to frame 17′′), during which the continuously growing condensates eventually took the shape of sphere (droplet) to maintain a relatively lower surface energy.4 During this growing process, the forming condensate microdroplets in the first tier cavities swept and swallowed the surrounding nucleation embryos. Droplet in-plane movement continued until it reached the cavity centre and got anchored by the surrounding pillars (frame 19′′). Then it started to grow upward in the out-of-plane direction (z direction) with constrained in-plane lateral spreading (frames 22′′ to 29′′). Eventually it reached the pillar top and evolved over them. Thereby a stretched condensate droplet formed in the PW state with its tail entrapped in the cavity (frame 33′′). Such PW droplets can potentially give rise to enhanced heat transfer3 but still need to be expulsed to the Cassie state for agile removal.
Fig. 2 ESEM snapshots (in seconds) of dynamic growth of condensate droplets in the first tier pillar cavities (b ≈ 5 μm, h ≈ 6 μm, l ≈ 9 μm) on the hierarchical superhydrophobic surface. The video was captured at 12.5 kV and a beam current of 0.2 nA. Vapor pressure was ∼5 torr in ESEM. Dotted line in the 33′′ snapshot indicates the tail of a condensate droplet formed in the PW morphology within four neighboring pillars. Hierarchical roughness can effectively control and confine spatial distribution of emerging condensate droplets out of first tier cavities.
Furthermore, we compared condensation process of water vapor on the two-tier superhydrophobic surface in ESEM12 with that on a smooth hydrophobic surface, which was formed by spin-coating a thin layer of fluoropolymer (FluoroPel™ PFC1601V, Cytonix or Teflon®) on a silicon wafer. The average sizes of condensate droplets during condensation on the two surfaces were calculated and plotted in Fig. 3(a). On the smooth surface, condensate droplets continuously coalesced with neighbouring condensates forming larger droplets, but the majority of the condensates still remained on the surface (see ESI†). Consequently, the average droplet diameter continuously grew as shown in Fig. 3(a). On the contrary, the average drop diameter on the two-tier surface started growing at the early stage of condensation but reached a saturated value of ∼15 μm during the subsequent stage. This could be explained by the self-cleaning ability of the two-tier superhydrophobic surfaces, which removes the condensate droplets by coalescence-induced jumping.12 The surface coverages of condensates on both the smooth and two-tier surfaces are plotted versus time in Fig. 3(b). Initially surface coverages on both surfaces increased along with condensation and got flattened out respectively following the early stage. For smooth hydrophobic surface the surface coverage levelled out at 0.58 while the superhydrophobic surface coverage got stabilized at around 0.28, which is almost half of the smooth surface coverage. Maintaining a higher percentage of unwetted bare surface could lead to less thermal resistance towards condensation, which can potentially enhance the rate of condensation and subsequently increase heat transfer on the two-tier structured surface.
Fig. 3 (a) Average droplet diameter and (b) surface coverage in the condensation process on smooth and two-tier surfaces.
Regarding a forming PW condensate, its tail is entrapped in the cavity of primary roughness but could be expelled either by self-pulling6 because of morphing-induced Laplace pressure gradient (intrinsic mechanism), or jump off via coalescence with neighbouring droplets (extrinsic mechanism). Nevertheless, immobile coalescence13 has been observed in the event that the released surface energy in coalescence is smaller compared to adhesion work, viscous dissipation consumption and contact line dissipation/pinning.19 For the purpose of structural optimization of surface roughness, we are not concerned with interactions (e.g., coalescence) between condensate droplets already sitting on top of surface roughness. Alternatively, we focus on the resistant energy preventing a single condensate tail from transitioning to the Cassie state, which would be the favourable state for agile condensate removal. In our analysis, we divide the total resistance energy into three major parts, i.e., adhesion energy, viscous dissipation and contact line dissipation.
Consequently, actual contact area Aadh is scaled down by a factor of fn due to the secondary nano-roughness.
where μ is the viscosity of the condensate (water), R the radial coordinate (in plane), ϑ the azimuthal coordinate, uz the out-of-plane velocity, and U the eventual vertical velocity of the expulsion process.
Regarding the dynamic growing process of a droplet on the pillar-arrayed surface, the surface tension σ, the viscosity μ and the inertia govern the liquid motions. For liquid, the ratio of viscous dissipation to surface tension and inertia is characterized by the Ohnesorge number Oh = μ/ρσL, where ρ and L are the fluid density and the characteristic length scale (L = pillar height h in this work, i.e., the height of the condensate tail in the cavity), respectively. In typical vapor condensation experiments, Oh ∼0.1 (ρ = 998 kg m−3, μ = 0.001 Pa s, σ ≈ 0.075 N m−1 for water at 5 °C and 1 bar, and L is on the order of μm) indicating viscous dissipation starts to dominate the surface energy effect19 and the self-pulling elevation of the tail is a capillary-inertial process (Oh < 1) in the cavity. Thus the average velocity could be assumed to be U/2 and the upward displacement of the mass center of the tail is h/2, then the time scale for the PW-Cassie transition can be estimated as , which is actually scaled as the capillary time (see ESI† for detailed derivation). By the momentum law the velocity of the droplet tail could be scaled as .
where is the contact line friction coefficient (CLFC) that determines the dissipation rate ξuc2 within the three-phase contact zone,34 λ is the jumping step length of liquid molecules and n is the number density of the adsorption sites on the surface.
where LT = Lcl + Lside is the total length of receding contact line of the tail in the PW-Cassie transition, the lateral contact line length Lcl ∼3.44rp (see ESI† for detailed derivation), the vertical contact line length Lside ∼8 h, i.e., along the surrounding pillar side walls, and uc ∼ U.
The dimensionless variables are represented with an asterisk, e.g., . It is noteworthy that the Ohnesorge number Oh is defined as Oh = μ/ρσh in this analysis. This energy barrier must be overcome by a condensate tail in order to accomplish the PW-Cassie transition via self-pulling mode.
We conducted parametric studies to characterize the influence of the first tier pillar geometry on the energy barriers. Fig. 4 shows the variation of resistant energy versus nondimensional cavity width s* (i.e., s* = s/h and s = l − b) with the solid fraction f = 0.08. It can be seen that for each Oh number there does exist an optimum (and hence optimum pillar width ) leading to the minimum resistance.
Fig. 4 Resistant energy associated with a condensate tail in a first tier cavity versus nondimensional cavity width s* with f = 0.08 and Oh be defined as . The critical cavity width is 0.49, which is determined by = 1. Dot on each curve stands for the optimum cavity width in each configuration. Square on each curve indicates the primary cavity size where adhesion work equals to viscous dissipation.
Fig. 5 Surface energy change ratio with respect to cavity width s* and first tier solid fraction ff (with , Oh = 0.10). Area below = 1 implies vertical growth of condensate droplets becomes energetically favorable.
Assuming θa ≅ θn ≅ 150° on the CNTs with fn = 0.25, the upper limit of cavity width bounded by sagging occurrence on the textured surface is .
Fig. 6 shows the effects of Oh on the nondimensional optimum cavity width , dimensional optimum cavity width sm, the pillar height h and pillar width b of the primary roughness. As Oh approaches to 0.035, indicative of a pillar height of ∼15 μm, the optimum reaches the critical lower bound , beyond which the surface structures cannot effectively prevent the undesired lateral spreading. As Oh increases, the nondimensional optimum increases whereas the optimum cavity width sm, pillar height h and pillar width b eventually decrease to the submicron levels, respectively. When Oh approaches to 2.7, indicative of the pillar height as low as tens of nanometers, the optimum gets close to the upper bound . But between the lower and the upper limits for , the shorter the height of the pillar (and hence the narrower the width of the cavity), the sooner the cavity is filled up with condensate and the sooner the self-pulling stage can be achieved. Fig. 7 shows a portion of the Fig. 6 for Oh in the range of 0.1–0.3. The 1 μm line for sm, b and h is displayed to demarcate the micron and submicron regimes. As Oh increases, the values of sm, b and h decrease as shown in Fig. 6 as well. For Oh > 0.2, the values of sm, b and h enter the submicron regime. Therefore, having both tiers of roughness in the submicron scale and also designing the first tier structure to match the optimum value shown in Fig. 4 (there exists an optimum value for each Oh, and hence for each pillar height h) can facilitate the PW condensate removal as a result of the remarkably alleviated resistant energy. On the other hand, the design of such structured surfaces with both the two tier roughnesses in submicron scale also imposes a limitation on how small the first tier can be, which is beyond the scope of this work.
Fig. 6 The nondimensional optimum cavity width , pillar height h, pillar width b and optimum cavity width sm with respect to . The geometries and configuration of primary roughness in the green regime can be adopted in condenser design for facilitating continuous dropwise condensation.
Fig. 7 The nondimensional optimum cavity width , pillar height h, pillar width b and optimum cavity width sm with respect to the Oh number in the range of 0.1–0.3.
We further compared in Fig. 8 the evolution of viscous dissipation , adhesion work , contact line dissipation and resistant energy with pillar gap s* while f = 0.08 and Oh = 0.1. Same as resistant energy , the above energy factors are nondimensionalized by σh2. Increasing pillar gap s* has a prominent mitigating effect on viscous dissipation and contact line dissipation as opposed to an intensified effect on adhesion. As discussed above, there exists an optimum cavity size giving rise to the minimum resistant energy . It is noteworthy that for the work of adhesion, viscous dissipation and contact line dissipation are all prominent, but for the resistance is dominated by viscous dissipation and contact line dissipation.
Fig. 8 Variation of the nondimensionalized viscous dissipation , adhesion work , contact line dissipation and resistant energy in a cavity with roughness spacing s* (f = 0.08 and Oh = 0.10).
The topography of a condenser surface has significant influence on nucleation site density as nucleation sites generally increase with the decrease of the characteristic size of surface roughness. Nevertheless, the importance of controlling nucleation density in pursuit of CDC could be evidently illustrated by selecting different hypothetical condensate spatial densities, indicative of the influence of surface structure density on resistant energy. In this analysis, condensate site on a two tier rough surface refers to a first tier cavity with a forming PW condensate entrapped therein after the initial nourishing process of a nucleate embryo.
Fig. 9 Critical condensate site density Ns,critical for different roughness period l.
As shown in Fig. 10, the power n of was chosen to be 1, 1.3, 1.6, and 1.8 respectively to make Ns remain below the critical curve, i.e., in the CDC region. These condensate site density curves are hypothetically chosen in order to provide an insight into the effect of condensate site density on the overall resistant energy of surfaces of different roughness characteristics.
Fig. 10 Different hypothetical condensate site density Nsversus roughness spacing l.
It can be seen in Fig. 11 that for different condensate site densities the overall resistant energy of the surface displays various behaviours as the primary cavity size s decreases. When the condensate site density is maintained at a moderate level (n = 1 or 1.3), the resistant energy of the surface is approximately in proportion to s. In contrast, as the condensate site density rises to even higher levels (n = 1.6 or 1.8), the overall resistant energy of the surface could increase especially for smaller roughness sizes, despite the resistant energy of a unit cell is decreasing (also see ESI†). In other words, the significant increase in the number of condensate sites leads to more condensate tails formed in the cavities of first-tier structures. Therefore, the overall resistant energy rises as a result of the increase of condensate sites. From the experimental point of view, to maintain condensate sites within a proper range (n < 1.6), superbiphilic surfaces formed by lithographically patterning superhydrophilic islands on superhydrophobic surfaces42 or chemical micropatterns43 can be employed for spatial control of at least microscale droplets during condensation, which is a new research theme in dropwise condensation on engineered surfaces.
Fig. 11 Resistant energy per unit area versus primary cavity size s for different hypothetical condensate site densities.
Even though it has been a challenge to realize sustained CDC on engineered surfaces in real applications, there have recently been reported experimental works demonstrating jumping-droplet-enhanced condensation on nanostructured surfaces in carefully-designed vapor chambers.8,24,44–46 Instead of using conventional wicks, Boreyko and Chen developed a novel vapor chamber with jumping-drop liquid return,44 in which superhydrophobic condenser is composed of 150–300 nm clusters of 50–100 nm silver nanoparticles on copper, and the overall lumped heat transfer coefficient has reached as high as 55 kW m−2 K−1. Both tiers of silver nanoparticles structures on the condenser are in nanoscale, which is in excellent agreement with our analysis of structural optimization. Two other recent works, which used superhydrophobic Si nanowires45 and hydrophobic Cu nanowires46 respectively on their condensers, have confirmed enhanced condensation heat transfer due to droplet jumping behaviour. It is noteworthy that these nominal single layer nanostructures actually consist of bundles of nanowires with microscale gaps, indicative of essentially two-tier hierarchical structures. Even though our resistant energy analysis starts with two-tier structures, it can be easily extended to surfaces with nominal one-tier nanotextures,8 which may be essentially composed of dual-scale roughness as more obvious in the clustered ribbed-nanoneedle structures.24 Due to the irregular and nonuniform geometries of surface roughness present in these works, we could only conduct a rough comparison of these experimental studies with our theoretical analysis. Based on the major features or characteristic lengths extracted from the surface structures of the abovementioned works, the structural comparison in the form of scattered points is shown in Fig. S8.† It can be seen that most of these surface configurations satisfy the continuous dropwise condensation criteria, which is illustrated by the central green region, proposed by us. In the condensation experiment by Yang et al.,46 the nanofibers are ∼20–30 μm tall with a high aspect ratio so that condensate droplets may be initially formed in the Cassie state instead of in the partial wetting (PW) morphology, which is not in accord with the situation of the PW-Cassie transition as discussed in our analysis.
where κ is the surface curvature of liquid, and here θ0 refers to the static contact angle of nanodroplets obtained by MD simulations. The positive value of σκ enhances surface hydrophobicity, whereas the negative value promotes surface hydrophilicity. The calculation of each component of the interfacial tensions followed the standard procedures suggested by a previous study.48 And the line tension coefficient σκ for water spreading on fluorinated surfaces was calculated to be ∼2 × 10−10 J m−1. As shown in Fig. 12(b), the difference between the total resistant energy, which takes the line tension into consideration, and the resistant energy excluding the line tension effect is negligible for Oh = 0.10.
Fig. 12 (a) Top view of line tension distribution of a PW droplet. The red dots indicate the eight vertical segments along the first tier pillars. (b) Effect of line tension on the resistant energy for Oh = 0.10.
And the values of eE for different Ohnesorge numbers are shown in Table 1.
Therefore, according to our analysis, line tension is not a dominant factor in resisting PW-Cassie transition at least in nanoscale roughness.
Continuous and sustained dropwise condensation on engineered surfaces places stringent requirements on careful design of surface structure length scale and geometry, as well as meticulous control of condensate morphology and spatial distribution.49–51 We carried out optimization of two-tier structured surfaces by minimizing the resistant energy that impedes the PW-Cassie transition. Our analysis of condensate growth in a first tier cavity shows that there does exist an optimum first tier geometry that can tremendously mitigate the resistant energy while allowing condensates to preferentially grow in the out-of-plane direction. We further showed that reducing surface roughness to submicron scale could be promising in achieving sustainable dropwise condensation due to the even lower resistance to the PW-Cassie transition. In addition to playing an important role in confining and controlling embryo formation, the second-tier nanotextures can effectively mitigate contact line dissipation and contact line pinning of the condensates. On the overall condenser surface, discrete condensate distribution is desired for delaying condensate flooding. Due to the relatively smaller energy barrier (Gibbs free energy) in the cavities of first tier structures, most of the condensate droplets in the PW state can be discretely confined therein. In this respect, superhydrophobic condenser surfaces with multiscale structures are superior to flat or solely nanotextured surfaces in controlling condensate spatial distribution.52,53 In order to further meticulously and precisely control condensate droplet density at elevated supersaturation, superbiphilic surfaces formed by lithographically registering superhydrophilic spots on superhydrophobic surfaces can be developed as novel condensers in order to achieve CDC. This energy-based analysis can help us design engineered surfaces that can sustain CDC in strong condensation with elevated supersaturation and consequently give rise to enhanced condensation heat transfer.
J. C. and A. V. conceived the concepts of this research. L. Z. obtained contact line friction coefficient by MD simulation. All authors contributed to finalizing this manuscript.
This work was supported by National Science Foundation under the Grant Numbers CBET 1550299, ECCS 1808931 and ECCS 1550749, and VT Institute for Critical Technology and Applied Science (ICTAS) funds. The authors also acknowledge Advanced Research Computing at Virginia Tech (http://www.arc.vt.edu) for providing computational resources and technical support that have contributed to the results reported within this paper.
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‡ These two authors contributed equally to this work.

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