Open Access Article

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DOI: 10.1039/C8NA00237A
(Paper)
Nanoscale Adv., 2019, Advance Article

Aref Vandadi‡
^{a},
Lei Zhao‡^{b} and
Jiangtao Cheng*^{ab}
^{a}Department of Mechanical and Energy Engineering, University of North Texas, Denton, TX 76207, USA.
^{b}Department of Mechanical Engineering, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA. E-mail: chengjt@vt.edu

Received
23rd September 2018
, Accepted 20th December 2018

First published on 20th December 2018

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.

Achieving sustained dropwise condensation on micro/nano-engineered surfaces places stringent requirements on the proper design of the configuration and length scale of surface structures and necessitates meticulous control of nucleation density and droplet morphology, which entail deep understanding of the underlying mechanisms governing condensate growth dynamics. Recently droplet jumping condensation on a nanostructured superhydrophobic CuO surface was demonstrated to enhance heat transfer efficiency by 30%.^{8} However, these condensation experiments were conducted in carefully-tailored and mild thermal conditions with a low supersaturation (<1.12) and a relatively low critical heat flux (<8 W cm^{−2}). Otherwise, condensate flooding on the condenser surface would occur and give rise to the unacceptable rise of thermal resistance thereon. Based on interfacial energy analysis, Rykaczewski et al. reported that the growth mechanism of individual condensate microdroplets on nanotextured surfaces is universal and independent of the surface architecture.^{9} The key role of the nanoscale topography is to confine the base area (i.e., wet spot) of forming embryos, which allows droplets to grow only through contact angle increase. Through observing condensation on one tier of nanowires in ESEM, they claimed that the formation of condensate nuclei could be controlled close to the top of the nanostructures.^{9} But how to form discrete condensate distribution on the surface of nanowires postponing flooding as well as validation of condensation heat transfer enhancement were not systematically studied in this work. Up to date, the rationale for designing biomimetic surfaces with optimum configuration of surface roughness in real thermal conditions are still lacking. Most of such reported condenser surfaces^{3–11} are characterized by fortuitous design and configuration, rather than driven by the fundamental principles of thermal and physical processes. In order to achieve sustained dropwise condensation on an engineered surface, it is imperative to understand the formation of condensate embryos, nucleation site and condensate density distribution, the dynamic growth of condensate droplets and evolution of droplet morphology.^{3}

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.

Our study of surface roughness optimization starts with the two-tier hierarchical structures as introduced above. Fig. 2 shows a close inspection of condensation process in ESEM. The formation of nucleation embryos in nanostructure cavities and in the first tier pillar cavities is ubiquitous since the critical nucleation radius of water is around tens of nm. The free energy barrier to condensate formation in the roughness cavity, i.e., the change in the free energy associated with embryo formation, could be calculated as:^{17,18}

(1) |

Furthermore, we compared condensation process of water vapor on the two-tier superhydrophobic surface in ESEM^{12} 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. |

Adhesion energy is one of the main barriers impeding the PW-Cassie transition as discussed in previous studies.^{20} The crucial step to form a mobile Cassie droplet is to first detach the condensate tail from the cavity and subsequently to expel the tail to the top of the pillars (PW-Cassie transition). The surface free energy change during the detachment of the tail per unit surface area would be W_{adh} = σ(1 + cosθ),^{21} where θ ≈ 115° is the Young's contact angle on the fluoropolymer-coated (FluoroPel™ PFC1601V, Cytonix or Teflon®)^{12} smooth surface. The contact area of the tail with the nanostructures on the cavity base is f_{n}πr_{b}^{2}, where is the radius of the tail base, W = 2(l − b), and θ_{n} ≈ 150° is the contact angle on the second tier nanostructures (Fig. 1(d)). The contact area of the tail along the pillar sidewalls is ∼L_{cl}h, where L_{cl} is the length of the three-phase contact line as illustrated by the pink segments in Fig. 1(a) and the derivation of L_{cl} can be found in the ESI.† Then the work required to detach the condensate tail from the cavity is

E_{adh} = W_{adh} × A_{adh} = σ(1 + cosθ)f_{n}(πr_{b}^{2} + L_{cl}h)
| (2) |

Consequently, actual contact area A_{adh} is scaled down by a factor of f_{n} due to the secondary nano-roughness.

The viscous dissipation incurred by the internal convection of a PW droplet makes another part of the resistant energy. As evidenced by the nonuniform heat transfer on the base of a condensate droplet, condensate droplets are usually not in a thermodynamically equilibrium state.^{22} The gradually expanding upper portion of a PW condensate leads to a decreasing capillary pressure therein whereas its tail portion remains under high capillary pressure. Thus-generated internal pressure gradient^{23,24} would pull the condensate tail upward, and at a specific size the droplet base would detach from the first tier valley, leading to self-pulling (or self-jumping) of the stretched PW droplet without coalescence. The self-pulling mode has been recently reported by Aili et al.^{6} in their study of condensation on nanostructured microporous surfaces. To analyze self-pulling of a condensate tail in a first tier cavity, we assumed Laplace pressure force F as the intrinsic mechanism in this study and the magnitude of viscous dissipation during the tail elevation is estimated as^{25}

(3) |

(4) |

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 effect^{19} 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 .

Since the size of the entrapped tail is much smaller than the capillary length of 2.7 mm of water, the bottom portion of the tail is assumed to be a spherical cap as shown in Fig. 1(d). So the curvature of the bottom tail can be estimated as r_{s} = W/2cos(π − θ_{n}).^{21} Given the larger radius of the condensate droplet on top (Fig. 1(c)), the pressure in the top portion of the tail should be much smaller than the bottom pressure. Consequently the pressure difference in the tail can be estimated as and the vertical driving force F is scaled as

(5) |

As such, depinning of the solid–liquid–vapor contact line in the cavity occurs when a certain value of critical pressure inside a droplet is surmounted.^{26} With the tail mass m ≈ ρπW^{2}h/4, we have and the energy loss due to viscous dissipation in the self-pulling process could be estimated as

(6) |

Our vapor condensation study in ESEM^{12} and other moisture condensation studies^{11} have observed extensive immobile coalescence on superhydrophobic surfaces, which can be partially ascribed to contact line pinning and hence contact line dissipation.^{27,28} Therefore, contact line dissipation should be given a careful evaluation in the energy analysis. From the point of view of molecular kinetic theory (MKT),^{29–33} the displacement of the three-phase contact line is determined by the forward and backward jumping frequencies, K^{+} and K^{−}, of liquid molecules in the contact zone. At equilibrium, K^{+} = K^{−} = K_{0}, where is the equilibrium displacement frequency, k_{B} is the Boltzmann constant, T is the absolute temperature, and v_{L} is the unit volume of flow of the liquid at the contact line. The activation free energy ΔG arises from the solid–liquid interaction, which can be taken as the work of adhesion, and the liquid–liquid viscous effect.^{33} Disturbed by an external driving work (shear stress, capillary force, hydration force, van der Waals interactions, etc.), K^{+} and K^{−} become unbalanced and the contact line will start moving. For the condensate tail in the cavity, the out-of-balance surface tension force is given by σ_{sv} − σ_{sl} − σcosθ_{d}, where σ_{sv} is the solid-vapor interfacial tension, σ_{sl} is the solid–liquid interfacial tension, θ_{d} is the dynamic contact angle (i.e., contact angle associated with the moving contact line) of condensate on the nanostructures inside the cavity. When the difference between the static contact angle θ_{0} and the dynamic contact angle θ_{d} is not significant, the contact line velocity u_{c} can be linearized as (see ESI† for detailed derivation)

(7) |

Instead of polyethylene terephthalate,^{30} fluoropolymer-based polytetrafluoroethylene (PTFE) was used as the water repellent coating material on the engineered structures in this work. Little work has been done on contact line friction of water on smooth or rough surfaces coated by fluoropolymer. Therefore we used molecular dynamics (MD) simulation to study the wetting behaviour of water on a PTFE surface in the framework of MKT.^{35–38} For water droplets of tens of nanometers in diameter on a smooth PTFE surface, our MD simulation shows CLFC on the smooth PTFE surface ∼ 1.2 × 10^{−3} Pa s at 5 °C, which is on the same order of magnitude as the dynamic viscosity of water. For condensate tails entrapped in the first tier cavities, the body of each condensate tail is actually in direct contact with the secondary CNT structures, which is conformally coated by PTFE. Since the condensate tail stays at the Cassie-state with regard to the second-tier nanoroughness, the contact line dissipation is scaled down by a factor of f_{n}. From Rayleigh dissipation function, the total energy loss due to contact line dissipation in the first tier cavity can be evaluated as:

(8) |

Combining eqn (2), (6) and (8) yields the self-pulling energy barrier of the PW-Cassie transition including adhesion energy E_{adh}, viscous dissipation E_{vis} and contact line dissipation E_{cl} in a first tier cavity:

(9) |

Using pillar height h as the characteristic dimension, the nondimensionalized form of eqn (9) becomes:

(10) |

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.

On the other hand, an ideally-structured condenser surface should be able to promote out-of-plane (vertical) growth of condensate droplets rather than in-plane (lateral) spreading.^{10} We compared surface energy change ΔE_{vertical} due to an infinitesimal vertical growth dz with surface energy change ΔE_{lateral} induced by an infinitesimal lateral growth dr. The energy ratio as a function of cavity width s* and first tier solid fraction f_{f} is plotted in Fig. 5 (see ESI† for detailed derivation). For < 1, the first tier pillars are able to block the lateral wetting of condensates towards the neighbouring cavities, indicative of a thermodynamically favorable configuration of surface structures. The critical cavity width with = 1 and the optimum cavity width with minimum resistant energy are shown in Fig. 4 for various Oh (i.e., pillar height h) values. It can be seen that for each cavity width larger than the critical width there does exist an optimum cavity size , which not only minimizes resistant energy but also favors vertical growth of condensates. Importantly, by minimizing the resistant energy, the condensate tails formed in the cavities are more apt to the PW-Cassie transition via self-pulling mode. It needs to be mentioned that pillar height h cannot be arbitrarily short in engineered surface design not only for maintaining proper surface roughness but also for preventing potential sagging of condensate droplets in to the structure cavities even after the PW-Cassie transition.^{39}

From Fig. 4, it is clear that for each Oh (hence the height h of the pillars) there exists an optimum cavity width leading to the lowest resistant energy in a unit cell. Actually, the value of optimum is constrained by the vertical growth preference factor and sagging phenomenon^{40} of condensate droplets respectively. The critical cavity width , which is set by = 1, defines the lower bound for . The upper bound for can be determined by preventing the occurrence of complete sagging, i.e., Cassie to Wenzel transition, of the condensate droplet after the PW-Cassie transition. The critical pillar height h_{sag} for the occurrence of complete or full sagging of a droplet sitting on top of pillars is (see ESI†)

(11) |

(12) |

Assuming θ_{a} ≅ θ_{n} ≅ 150° on the CNTs with f_{n} = 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 s_{m}, 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 s_{m}, 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 s_{m}, b and h is displayed to demarcate the micron and submicron regimes. As Oh increases, the values of s_{m}, b and h decrease as shown in Fig. 6 as well. For Oh > 0.2, the values of s_{m}, 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. 7 The nondimensional optimum cavity width , pillar height h, pillar width b and optimum cavity width s_{m} 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 σh^{2}. 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). |

Discrete dropwise condensates should be methodically maintained while the uncontrolled lateral spreading should be avoided or at least delayed during condensate growth in surface cavities, otherwise dropwise condensation may not continue and the surface would eventually get flooded. To satisfy this criterion the spacing between the condensate sites L should be at least 2 times the roughness period or spacing l, i.e., .^{41} The spacing between condensate sites could be given by , where N_{s} denotes the density of condensate droplets. Then the above site criterion can be rewritten as

(13) |

Fig. 9 shows the critical values of condensate sites versus roughness spacing l (eqn (13)). The region beneath the critical curve (green area) represents the condensate site densities satisfying dropwise condensation criterion mentioned above. For N_{s} values above the critical curve (red area) multiple condensates may form around a unit cavity. As these condensates grow, they would merge with those in neighbouring cavities leading to film condensation and may eventually flood the surface. Regarding one of our two-tier superhydrophobic surfaces with l_{0} = 9 μm, the benchmark site criterion is . In a general case with varying roughness spacing l, the critical condensate site density N_{s,critical} can be related to N_{s0} as

(14) |

To guarantee N_{s} falls beneath the critical curve in Fig. 9, the following criterion regarding the power n of must be met

(15) |

As shown in Fig. 10, the power n of was chosen to be 1, 1.3, 1.6, and 1.8 respectively to make N_{s} 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.

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 surfaces^{42} or chemical micropatterns^{43} 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. |

On the other hand, when the surface roughness approaches nanoscale, the intrinsic line tension effect^{47} may play a certain role in the PW-Cassie transition. For a PW droplet anchored in the surface cavities, the line tension occurs at the three-phase contact line and its distribution is illustrated in Fig. 12(a). The resistant energy E_{lt} caused by line tension can be calculated by:

E_{lt} = σ_{κ}(L_{cl} + L_{base} + L_{side}) ≈ σ_{κ}(L_{cl} + 2πr_{b} + 8 h)
| (16) |

σ_{κ} = (σ_{sv} − σ_{sl} − σcosθ_{0})κ
| (17) |

We also studied the relative strength e_{E} of line tension in the resistant energy associated with the PW-Cassie transition, which is defined as

(18) |

And the values of e_{E} for different Ohnesorge numbers are shown in Table 1.

Oh | 0.10 | 0.12 | 0.14 | 0.16 |
---|---|---|---|---|

e_{E} |
1.16% | 1.36% | 1.53% | 1.68% |

Therefore, according to our analysis, line tension is not a dominant factor in resisting PW-Cassie transition at least in nanoscale roughness.

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## Footnotes |

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8na00237a |

‡ These two authors contributed equally to this work. |

This journal is © The Royal Society of Chemistry 2019 |