**4. Nucleation and droplet size distribution**

According to the classic nucleation theory, a substrate with low contact angles can decrease energy barrier and increase nucleation rate. The geometrical structure can also enable spatial distribution of nucleation sites on a micro/nanostructured surface. More interestingly, the structure feature plays an important role in determining the wetting mode of a nucleated droplet and subsequent droplet growth and dynamics. For example, a droplet in Cassie state has low adhesion and high mobility, which is favorable for dropwise condensation. Various theoretical and semiempirical models have been proposed to describe the relationship between surface structure and nucleation, for example, interfacial free energy analysis and cluster theory [16, 17, 19, 74]. To obtain more detailed information on the droplet nucleation in micro/nanostructures, many numerical modeling methods have been developed such as the Gibbs free energy analyses, finite element method, lattice Boltzmann method, lattice density functional method, and molecular dynamics (MD) simulations [13, 18, 75–80].

increase of contact angle with near-constant contact line, and finally growth with constant contact angle. The droplet growth rate can be improved by increasing the size of high-energy sites. Moreover, the microstructures (**Figure 6c**), for example, micro-pores and microgrooves, can promote droplet nucleation due to the lower energy barrier caused by the larger solid-liquid contact area when compared with a smooth surface. The preferred nucleation in the microstructures was also observed

*Effect of surface energy and structure feature on the nucleation. (a) Random nucleation occurring on a uniform smooth surface. (b) Initial nucleation occurs selectively in the high-energy sites on the hybrid surfaces with wetting contrast [13]. (c) Effect of structure features (micropores and microgrooves) on the initial nucleation. (d) ESME images showing droplets prefer to nucleate in the micro-defects on a nanostructured surface.*

After droplets nucleate on a solid surface, they can grow through two mechanisms: direct growth without coalescence for the small droplets by vapor condensing on the liquid-vapor interface, and coalescence-dominated growth for the large droplets by merging with neighboring droplets. As a result, condensed droplets with a wide range of sizes exist on a surface, forming a unique self-similarity. The transient characteristic of droplet size distribution has been widely studied to understand the heat transfer mechanism of dropwise condensation [14, 17, 32, 34, 51, 81, 82]. In the initial droplet growth stage after nucleation, all the droplet sizes are uniform and droplet size distribution has a peak value, showing the lognormal distribution (**Figure 7a**). With droplets growing up, the average droplet size increases. Then, the coalescences among droplets lead to a wider span of droplet size and decline of the right peak. With the increasing space among the first generation droplets, new droplets nucleate and grow at the exposed substrate, forming a bimodal distribution. With further coalescence and nucleation, the

on the nanostructured surfaces with micro-defects (**Figure 6d**).

*Advances in Dropwise Condensation: Dancing Droplets DOI: http://dx.doi.org/10.5772/intechopen.92689*

**Figure 6.**

**237**

Nucleation of water droplet on the surfaces with different wettability and structural features has been widely investigated by MD simulations. The nucleation process and wetting state of nucleated droplets were obtained by manipulating the interaction potentials between surface atoms and vapor molecules [13]. The nearconstant contact angles can be established for nanoscale nucleated droplets on a surface, decreasing as the solid-liquid interaction intensities linearly. On an uniform high-energy surface (**Figure 6a**), the interactions between surface atoms and water molecules are strong to capture water molecules rapidly once they get close to the surface. A large number of water molecules are absorbed on the surface and the nucleation occurs almost instantaneously. On a hybrid surface with different surface energies (**Figure 6b**), where high-energy sites distributed on a low-surface energy surface, the nucleation preferably initiates on the high-energy sites. The clusters are trapped on the high-energy sites instead of migrating around observed on an uniform surface (**Figure 6a**). The growth of nucleated droplets on highenergy particles can be divided into three stages: the formation of a wet-spot,

*Advances in Dropwise Condensation: Dancing Droplets DOI: http://dx.doi.org/10.5772/intechopen.92689*

#### **Figure 6.**

methods for metal micro/nanostructures, the electrochemical deposition (or electroplating) method with the assistance of templates is considered to be a convenient and versatile approach [21, 65, 70, 71, 73]. **Figure 5d** shows the gold nanowires fabricated to promote droplet coalescence without direct coalescence [70]. Using a two-step template-assisted electrochemical deposition method, closely spaced copper nanowires were fabricated to promote the formation of mobile droplets in suspended wetting states (**Figure 5e**) [65]. To prevent the formation of microdefects between agglomerated nanowire arrays, the 3D copper nanowire networks consisting of interconnected nanowires with nanoscale bumps were fabricated using 3D porous anodic alumina oxide templates (**Figure 5f**). These nanobumps on the nanowire walls serve as anchors to fix adjacent nanowires [71]. **Figure 5g** shows a hierarchical surface with micro-patterned copper nanowire arrays [21]. In addition, the commercial metal materials, such as copper mesh, foam, and particles, have been applied to develop low-cost superhydrophobic surfaces and to provide a solution for large-scale deployment of enhanced heat transfer surfaces in a diverse range of technologies (**Figure 5h**) [72]. Despite that diverse micro/nanostructured surfaces have been fabricated to meet the multiple length scale need in manipulating droplet growth as mentioned above, there remains a need for advanced surface features to optimize droplet behaviors for future vapor

According to the classic nucleation theory, a substrate with low contact angles can decrease energy barrier and increase nucleation rate. The geometrical structure can also enable spatial distribution of nucleation sites on a micro/nanostructured surface. More interestingly, the structure feature plays an important role in determining the wetting mode of a nucleated droplet and subsequent droplet growth and dynamics. For example, a droplet in Cassie state has low adhesion and high mobility, which is favorable for dropwise condensation. Various theoretical and semiempirical models have been proposed to describe the relationship between surface structure and nucleation, for example, interfacial free energy analysis and cluster theory [16, 17, 19, 74]. To obtain more detailed information on the droplet nucleation in micro/nanostructures, many numerical modeling methods have been developed such as the Gibbs free energy analyses, finite element method, lattice Boltzmann method, lattice density functional method, and molecular dynamics

Nucleation of water droplet on the surfaces with different wettability and struc-

tural features has been widely investigated by MD simulations. The nucleation process and wetting state of nucleated droplets were obtained by manipulating the interaction potentials between surface atoms and vapor molecules [13]. The nearconstant contact angles can be established for nanoscale nucleated droplets on a surface, decreasing as the solid-liquid interaction intensities linearly. On an uniform high-energy surface (**Figure 6a**), the interactions between surface atoms and water molecules are strong to capture water molecules rapidly once they get close to the surface. A large number of water molecules are absorbed on the surface and the nucleation occurs almost instantaneously. On a hybrid surface with different surface energies (**Figure 6b**), where high-energy sites distributed on a low-surface energy surface, the nucleation preferably initiates on the high-energy sites. The clusters are trapped on the high-energy sites instead of migrating around observed on an uniform surface (**Figure 6a**). The growth of nucleated droplets on highenergy particles can be divided into three stages: the formation of a wet-spot,

condensation enhancement.

*21st Century Surface Science - a Handbook*

(MD) simulations [13, 18, 75–80].

**236**

**4. Nucleation and droplet size distribution**

*Effect of surface energy and structure feature on the nucleation. (a) Random nucleation occurring on a uniform smooth surface. (b) Initial nucleation occurs selectively in the high-energy sites on the hybrid surfaces with wetting contrast [13]. (c) Effect of structure features (micropores and microgrooves) on the initial nucleation. (d) ESME images showing droplets prefer to nucleate in the micro-defects on a nanostructured surface.*

increase of contact angle with near-constant contact line, and finally growth with constant contact angle. The droplet growth rate can be improved by increasing the size of high-energy sites. Moreover, the microstructures (**Figure 6c**), for example, micro-pores and microgrooves, can promote droplet nucleation due to the lower energy barrier caused by the larger solid-liquid contact area when compared with a smooth surface. The preferred nucleation in the microstructures was also observed on the nanostructured surfaces with micro-defects (**Figure 6d**).

After droplets nucleate on a solid surface, they can grow through two mechanisms: direct growth without coalescence for the small droplets by vapor condensing on the liquid-vapor interface, and coalescence-dominated growth for the large droplets by merging with neighboring droplets. As a result, condensed droplets with a wide range of sizes exist on a surface, forming a unique self-similarity. The transient characteristic of droplet size distribution has been widely studied to understand the heat transfer mechanism of dropwise condensation [14, 17, 32, 34, 51, 81, 82]. In the initial droplet growth stage after nucleation, all the droplet sizes are uniform and droplet size distribution has a peak value, showing the lognormal distribution (**Figure 7a**). With droplets growing up, the average droplet size increases. Then, the coalescences among droplets lead to a wider span of droplet size and decline of the right peak. With the increasing space among the first generation droplets, new droplets nucleate and grow at the exposed substrate, forming a bimodal distribution. With further coalescence and nucleation, the

the peak value decreases as the size distribution curve is close to a lognormal distribution with time, which is distinctly different from the homogeneous equilib-

Surface modification plays a crucial role in manipulating droplet wetting and dynamics. When micro/nanostructures are covered with hydrophobic coatings, the resulted superhydrophobic surfaces can promote the formation of highly mobile droplets in the Cassie states. More interestingly, it has been demonstrated that on such micro/nanostructured superhydrophobic surfaces small microdroplets can undergo coalescence-induced droplet jumping due to the release of excess surface energy (**Figure 8a**), which is independent of gravity [75, 76, 83, 84]. Jumping

*Wetting and dynamics of the droplet on the superhydrophobic surfaces. (a) Droplet jumping due to the merge of two small droplets [83]. (b) Various wetting modes of condensed droplets on a nanostructured surface [85]. (c–i) Wetting transition of condensed droplets with the increase of surface subcooling: (c) schematic illustrating suspended droplets under a small subcooling, (d) time-lapse images of jumping droplets, (e) schematic illustrating immersed droplets under a large subcooling, (f) time-lapse images of flooding phenomenon, (g) schematic illustrating droplet nucleation in the nanostructure as a function of subcooling. ESEM images of a suspended (h) and immersed (i) droplet in nanostructures at a small and a large surface subcooling,*

**5. Condensation on superhydrophobic surfaces**

*Advances in Dropwise Condensation: Dancing Droplets DOI: http://dx.doi.org/10.5772/intechopen.92689*

rium distribution (**Figure 7d**).

**Figure 8.**

*respectively [24].*

**239**

**Figure 7.**

*Evolution of droplet size distribution on a plain hydrophobic surface. (a) Experimental results of the transient droplet size distribution in condensation [14]. (b–d) Evolution of cluster size distribution by MD simulation: (b) cluster size distribution as a function of time, (c) cluster size distribution as a function of time by MD simulation and (d) evolution of cluster radius distribution [17].*

number of larger droplets decreases while the number of smaller droplets increases, and droplet size range becomes wider. Finally, the right peak vanishes and the distribution becomes unimodal. Due to the self-similarity feature of droplet size distribution, a representative large enough surface area contains various growth stages, resulting in a steady-state droplet size distribution [14, 82].

To further understand the evolution of droplet size distribution, a transient cluster size distribution model was introduced to investigate the kinetics of the initial condensation stage by MD simulation method [17]. It is well known that the growth/decay of clusters is significantly affected by the cluster size and contact angles of the condensing surface. With the increase of cluster sizes, more surface area of the cluster is exposed to the vapor, as well as the increased attachment/ detachment frequencies. When the contact angle decreased to a certain value, the attachment frequency becomes larger than detachment frequency, resulting in the continuous growth of the clusters and subsequent nucleation. The results of the evolution of cluster/droplet size distribution indicate that the transient cluster size distribution translates from a monotonic decreasing distribution to a unimodal distribution with time (**Figure 7b,c**). The cluster radius at the peak of cluster/ droplet size distribution curve shifts to the larger cluster sizes with time. However, *Advances in Dropwise Condensation: Dancing Droplets DOI: http://dx.doi.org/10.5772/intechopen.92689*

the peak value decreases as the size distribution curve is close to a lognormal distribution with time, which is distinctly different from the homogeneous equilibrium distribution (**Figure 7d**).
