Numerical Simulation of Condensation on Structured Surfaces

Article pubs.acs.org/Langmuir

Numerical Simulation of Condensation on Structured Surfaces Xiaowu Fu, Zhaohui Yao,* and Pengfei Hao Department of Engineering Mechanics, Tsinghua University, Beijing 100084, People’s Republic of China ABSTRACT: Condensation of liquid droplets on solid surfaces happens widely in nature and industrial processes. This phase-change phenomenon has great effect on the performance of some microfluidic devices. On the basis of micro- and nanotechnology, superhydrophobic structured surfaces can be well-fabricated. In this work, the nucleating and growth of droplets on different structured surfaces are investigated numerically. The dynamic behavior of droplets during the condensation is simulated by the multiphase lattice Boltzmann method (LBM), which has the ability to incorporate the microscopic interactions, including fluid−fluid interaction and fluid−surface interaction. The results by the LBM show that, besides the chemical properties of surfaces, the topography of structures on solid surfaces influences the condensation process. For superhydrophobic surfaces, the spacing and height of microridges have significant influence on the nucleation sites. This mechanism provides an effective way for prevention of wetting on surfaces in engineering applications. Moreover, it suggests a way to prevent ice formation on surfaces caused by the condensation of subcooled water. For hydrophilic surfaces, however, microstructures may be submerged by the liquid films adhering to the surfaces. In this case, microstructures will fail to control the condensation process. Our research provides an optimized way for designing surfaces for condensation in engineering systems.

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INTRODUCTION

Surface roughness influences the wetting properties of a surface. For example, the well-known superhydrophobic lotus leaf surfaces are geometrically patterned. Some researchers are dedicated to the fabrication of special surfaces that are similar to the lotus leaf surfaces. Cao et al.1 designed and fabricated a kind of microtexture for inducing a superhydrophobic behavior on surfaces with an intrinsic water contact angle of ∼74°. Bellanger et al.2 reviewed some chemical and physical pathways to create superoleophobic surfaces. On the other hand, some researchers focused on elucidating the parameters that control the liquid state on a structured surface. Dorrer and Rühe3 studied the wetting behavior of microstructured post surfaces coated with a hydrophobic fluoropolymer. They pointed out that, under certain conditions, drops can transition from the Wenzel state to the Cassie state. Jung and Bhushan4 proposed a criterion to predict the transition from the Cassie and Baxter regime to the Wenzel regime. Forsberg et al.5 researched the wettability of surfaces microstructured with square pillars. Mishchenko et al.6 studied the behavior of dynamic droplets impacting supercooled nano- and microstructured surfaces. Rykaczewski7 investigated experimentally and theoretically the dynamics of droplet formation on nanostructured superhydrophobic surfaces and developed a quantitative model for droplet growth in the constant base area mode. Condensation of droplets on solid surfaces occurs in a variety of occasions and has a wide range of applications.8 Condensation in microfluidic devices will affect the performance of these devices. Understanding the mechanism of droplet condensation on surfaces will help us design suitable surfaces for microdevices. Many researchers have been dedicated to condensation on superhydrophobic surfaces with micro- or nanostructures over the past decade. Cheng et al.9 presented an in situ observation of © XXXX American Chemical Society

Figure 1. D2Q9 velocities.

Figure 2. Computational model.

water condensation and evaporation on lotus leaf surfaces inside an environmental scanning electron microscope. Chen et al.10 reported continuous dropwise condensation on a superhydrophobic surface with a two-tier texture. The two-tier texture was prepared by depositing carbon nanotubes on micromachined Received: September 1, 2014 Revised: October 17, 2014

A

dx.doi.org/10.1021/la503504r | Langmuir XXXX, XXX, XXX−XXX

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Figure 3. Contact angles are 180°, 90°, and 0°.

pillars. Narhe et al.11 investigated the growth dynamics of water drops in a condensation chamber on a hydrophobic surface with square pillars. Chen et al.12 found that the synergy of patterned active nucleation sites with pyramid-shaped hierarchical structures can lead to a continuous dropwise condensation. Rykaczewski et al.13 investigated the impact of microscale topography of hierarchical superhydrophobic surfaces on the droplet coalescence dynamics and wetting states during the condensation process. The dynamic behavior of droplets during the nucleation and growth is complicated. The process of condensation involves not only fluid−fluid interaction but also fluid−surface interaction. Solid surfaces can affect behaviors of droplets by the chemical

Figure 4. Superhydrophobic and hydrophilic surfaces: (a) superhydrophobic surface, with Gads = −100, and (b) hydrophilic surface, with Gads = −220.

Figure 5. Condensation process on superhydrophobic surfaces with microridges. The spacing, height, and width of ridges are 16, 10, and 2, respectively. B


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properties and micro- or nanostructures. Chemical properties, involving hydrophilicity and hydrophobicity, affect the contact angle of droplets on the surface. On the other hand, geometric morphology of micro- or nanostructures on the surface has a

great relationship with the droplet nucleation and growth. Understanding this relationship is crucial for us to design structure geometry on the surface, which will be of benefit to condensation control and industrial applications, such as ice formation on surfaces caused by the condensation of subcooled water. However, the nucleation and growth of droplets on surfaces are not well understood, and the effect of microstructures on the condensation behavior is not well studied. This study mainly discusses the effect of the microstructures of surfaces on the condensation by the numerical simulation. We could see the observable differences of condensation processes on surfaces with different calculated patterns. Some interesting phenomena that may not be easily obtained by experiments are observed in our simulation. In addition, by the simulation, we can economically obtain some important information for designing the structure geometry of microstructures without repeated experiments. In the paper, the lattice Boltzmann method (LBM) is used to simulate the condensation on surfaces. As a kind of computational fluid dynamic method, the LBM has been developed for simulating complex fluid flows in the last 2 decades.14 On the basis of the premise that macroscopic dynamics of fluid are the collective behavior of many microscopic particles, the basic idea of the LBM is to construct a kinetic model that involves microscopic processes. The macroscopic-averaged properties obtained by the LBM approximately obey the Navier−Stokes macroscopic equations. Because of its kinetic nature, the LBM is simple for programming and parallelism and can handle complex

Figure 6. Experimental result of condensation on a superhydrophobic structured surface. The side length of micropillars is 20 μm; the height is 20 μm; and the spacing is 30 μm. The focal plane is the top surface of the micropillars, and unclear black specks are condensation droplets at the bottom of the surface.

Figure 7. Condensation process on superhydrophobic surfaces with microridges. The spacing, height, and width of ridges are 6, 10, and 2, respectively. C


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Figure 8. Condensation process on superhydrophobic surfaces with microridges. The spacing and width of ridges are 16 and 2. The height of ridges on the upper surface is 10, and the height of ridges on the lower surface is 30. distribution function is given as

interfaces and geometry areas. The LBM has been successfully applied to such multiphase flows in which gas-liquid phase transitions occur. Among those developed multi-phase models, the pseudo-potential model15 is simple and widely used. In this study, we use a two-dimensional LBM (D2Q9) combined with a pseudo-potential model to simulate condensation on surfaces.

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f eq (u) =

NUMERICAL METHOD

f eq (u) =

(1)

f − f eq τ

(2πRT )

⎡ (ξ − u)2 ⎤ exp⎢ − ⎥ 2RT ⎦ ⎣

(3)

(2πRT )

⎛ ξ 2 ⎞⎡ (ξ·u) (ξ·u)2 u2 ⎤ + − exp⎜ − ⎥ ⎟⎢1 + 2 RT 2RT ⎦ 2(RT ) ⎝ 2RT ⎠⎣

+ Ο(u 3)

(4)

ρ D /2

To calculate the microscopic velocity in eq 1, we need appropriate discretization in space. In the study, a two-dimensional, nine-speed (D2Q9) model is used. In the model, the microscopic velocity has nine velocity directions, as shown in Figure 1. Then, the equilibrium distribution function with truncated small velocity expansion in eq 4 can be written as

where f(x, ξ, t) is the single-particle velocity distribution function, ξ is the microscopic velocity vector, a is acceleration caused by external forces, and Ω is the collision function. By the Bhatnagar−Gross−Krook (BGK) approximation, the collision function in eq 1 can be written as

Ω(f ) = −

D /2

where D is the dimension of the space, R is the idea gas constant, and ρ, u, and T are the density, velocity, and temperature, respectively. The equilibrium distribution function can be obtained by low Mach number approximation as follows

LBM. The Boltzmann equation that describes the statistical behavior of a thermodynamic system not in thermodynamic equilibrium is ∂f ∂f ∂f +ξ +a = Ω(f ) ∂t ∂x ∂ξ

ρ

⎡ (e ·u) (e ·u)2 u2 ⎤ ⎥ f ieq = ωiρ⎢1 + i 2 + i 4 − 2cs cs 2cs 2 ⎦ ⎣

(2)

where τ is the relaxation time as a result of collision and f eq is the Maxwell equilibrium distribution function. Assuming that the particle mass is m = 1, the particle density distribution function is mf and still written as f. The Maxwell equilibrium

(5)

where RT = cs2 = c2/3 and c = Δx/Δt is the velocity on the lattice of the D2Q9 model. In our simulation, dimensionless variables are used. The lattice spacing is Δx = Δy = 1 and the time step is Δt = 1. ω is the D


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Figure 9. Condensation process on superhydrophobic surfaces. The spacing, height, and width of microridges on the lower surface are 16, 10, and 2, respectively. The kinematic viscosity of the fluid is given by

weight coefficient

⎧ 4/9, i = 0 ⎪ ωi = ⎨1/9, i = 1, 2, 3, 4 ⎪ ⎩1/36, i = 5, 6, 7, 8

ν = cs 2(τ − 0.5)Δt = (6)

8

F(x , t ) = − Gψ (x , t ) ∑ ωiψ (x + eiΔt , t )ei i=1

(7)

ψ (ρ) = ψ0 exp(− ρ0 /ρ)

(8)

i=0

and

u=

1 ρ

8

∑ fi ei i=0

(9)

By the ideal gas equation of state, the macroscopic pressure is p = cs 2ρ =

ρ 3

(13)

where ψ0 and ρ0 are arbitrary constants. For attraction between fluid particles, G < 0, and if the density of particles is higher, the force is stronger. The interaction potential function has many forms, but according to Shan and Chen,15 this function must be increasing monotonically and bounded. The form of the interaction potential function that we use in eq 13 is a kind of widely used form that satisfies the above requirements. The interaction forces affect the velocity in the equilibrium distribution function, and the velocity used in the calculation of the equilibrium distribution function should be written as

8

∑ fi

(12)

where G is the interaction strength and ψ is the interaction potential function and written as follows

The density and velocity can be calculated by

ρ=

(11)

Pseudo-potential Model. To simulate multiphase fluids, longrange interactions between fluid particles are needed. Only considering forces between nearest-neighbor fluid particles, the interaction force can be written as16

Vectors ei are the discrete velocities given by ⎧(0, 0), i = 0 ⎪ ⎪ ⎪(cos θ , sin θ )c , θ = (i − 1)π , i = 1, 2, 3, 4 i i i ei = ⎨ 2 ⎪ (2i − 1)π ⎪ , i = 5, 6, 7, 8 ⎪ 2 (cos θi , sin θi)c , θi = ⎩ 4

1 (τ − 0.5) 3

u eq = u +

τF ρ

(14)

Therefore, the interaction between fluid particles is added to the LBM.

(10) E


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Figure 10. Condensation process on superhydrophobic surfaces. The spacing, height, and width of microridges on the lower surface are 6, 10, and 2, respectively.

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With the fluid−fluid interaction force, the equation of state of the real fluid is given as

p=

ρ ρ G G + ψ (ρ)2 = + [ψ0 exp(− ρ0 /ρ)]2 3 6 3 6

VALIDATION16 To verify the code, we simulate the static contact angle of droplets on solid surfaces. In our simulation, 200 × 200 lattices in x and y directions are used. The total time is t = 4000, and the relaxation time is τ = 1. The initial fluid velocity is set to 0, and the initial droplet radius is 20. In the simulation, parameters G = −120, ψ0 = 4, and ρ0 = 200 are used. The initial droplet density is 500, and the initial vapor density is 100. Under these conditions, the equation of state is non-monotonic and two phases can coexist. On the basis of the force balance rationale, when Gads = −46.534, −187.16, and −327.79, the contact angles are 180°, 90°, and 0°, respectively, as shown in Figure 3. Because of the liquid−vapor interaction, the size and density of the droplet have a small change before the droplet reaches a relatively stable state.

(15)

Similarly, the adhesive interaction between fluid particles and solid surfaces can be obtained in the same way. The fluid−surface force can be written as 8

Fabs(x , t ) = − Gadsψ (x , t ) ∑ ωis(x + eiΔt )ei i=1

(16)

where Gads is the intensity of the fluid−surface interaction and s is a “switch”; if the node at x + eiΔt is a solid, s = 1, and if not, s = 0. The hydrophilic or hydrophobic surface is determined by the intensity parameter Gads. The negative sign of Gads indicates “attractive”, and the absolute value of Gads indicates “strength”. When the absolute value of Gads is higher, adhesion forces are larger, which will make the surface more hydrophilic. In contrast, a low absolute value of Gads will lead to hydrophobic surfaces. Computational Model. Typical nature superhydrophobic surfaces are lotus leaf surfaces. There are many micro- and nanostructures distributed on lotus leaf surfaces.17 In our computational model, we use a two-dimensional LBM model, and it is hard to mimic actual threedimensional patterns accurately. The corresponding three-dimensional patterns of our calculated model are much like ridged/grooved surfaces.18 The computational model is shown in Figure 2. The upper and lower boundaries are solid walls with structures on surfaces. The left and right boundaries are periodic. There is a slight random perturbation of fluid density at the initial time. In our simulation, the total vapor is fixed.

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RESULTS AND DISCUSSION In this section, we define the superhydrophobic surface with Gads = −100 and the hydrophilic surface with Gads = −220, as shown in Figure 4. The initial fluid density is ρ = 150 + Δρ, where Δρ is the random perturbation with its value between [0, 1]. Effect of Spacing of Two Ridges on Superhydrophobic Surfaces. The upper and lower boundaries are superhydrophobic surfaces with microridges. The condensation process is shown in Figure 5. The spacing, height, and width of ridges are 16, 10, and 2, respectively. We can see that two small droplets nucleate within the unit cell, merge together, and grow. The size of droplets within cells is uniform. Droplets whose size is below the critical value F


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Figure 11. Condensation process on hydrophilic surfaces. The spacing, height, and width of microridges on the lower surface are 16, 10, and 2, respectively.

nucleate along the axial direction of ridges within the unit cell (t = 100), which is different from the upper surface, on which only one droplet nucleates. Small droplets merge together and grow as time passes. The size of droplets is larger within the lower cells than the upper cells. Droplets on the upper surface evaporate quickly, but droplets on the lower surface are maintained. Comparison of Condensation on Flat Surfaces and Structured Surfaces. To understand the difference of condensation on flat surfaces and structured surfaces, a model that has two kinds of surfaces is investigated. The upper surface is flat, and the lower surface is the structured surface (as shown in Figures 9 and 10). For large spacing of ridges, the condensation process is shown in Figure 9. We can see that the size of droplets on the flat surface is larger than that on the structured surface. Few droplets are maintained within the cell on the structured surface. On the contrary, some large droplets adhere to the flat surface. When the spacing of ridges is smaller, the droplets nucleate within the unit cell, fill, and grow beyond the unit cell on the structured surface, as shown in Figure 10. Note that, microridges make the structured surface more superhydrophobic, which lead to a little larger contact angle. In this case, droplets adhere to the ridge tips and the flat surface. Condensation on Hydrophilic Structured Surfaces. In this subsection, the condensation on hydrophilic surfaces with microridges is compared to that on flat hydrophilic surfaces,

cannot exist because the evaporation intensity is stronger than the condensation intensity if droplets are small enough. After some time, droplets within cells vanish. In this case, few droplets are on superhydrophobic surfaces. We have found that, even on a superhydrophobic surface, droplets can be trapped between microridges in the simulation, as shown in Figure 5 (t = 100). We also observed the same phenomenon in the experiments, as shown in Figure 6. In our experiment, the microstructure on surfaces is a regularly arranged array of square micropillars. The black specks indicate droplets trapped on the lower surface. Although patterns in calculations and experiments are not identical, the simulation results and experimental observations are qualitatively consistent. When the spacing of ridges becomes small, the condensation process is shown in Figure 7. At the beginning, droplets nucleate within the unit cell, subsequently fill the cell, and grow beyond the cell. Suspended droplets nucleate, merge together, and grow on the ridge tips. Droplets on the ridge tips are big enough that they will not disappear soon. On this occasion, some big droplets maintain on the solid surfaces. Effect of the Height of Ridges on Superhydrophobic Surfaces. In this subsection, we only change the height of microridges on the lower surface and keep other parameters unchanged, as shown in Figure 8. We can see that the condensation on the upper and lower surfaces is similar at t = 10. The lower surface has taller ridges and two small droplets G


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(8) Enright, R.; Miljkovic, N.; Al-Obeidi, A.; Thompson, C. V.; Wang, E. N. Condensation on superhydrophobic surfaces: The role of local energy barriers and structure length scale. Langmuir 2012, 28 (40), 14424−14432. (9) Cheng, Y.-T.; Rodak, D. E.; Angelopoulos, A.; Gacek, T. Microscopic observations of condensation of water on lotus leaves. Appl. Phys. Lett. 2005, 87 (19), 194112. (10) Chen, C.-H.; Cai, Q.; Tsai, C.; Chen, C.-L.; Xiong, G.; Yu, Y.; Ren, Z. Dropwise condensation on superhydrophobic surfaces with two-tier roughness. Appl. Phys. Lett. 2007, 90 (17), 173108. (11) Narhe, R.; Beysens, D. Growth dynamics of water drops on a square-pattern rough hydrophobic surface. Langmuir 2007, 23 (12), 6486−6489. (12) Chen, X.; Wu, J.; Ma, R.; Hua, M.; Koratkar, N.; Yao, S.; Wang, Z. Nanograssed micropyramidal architectures for continuous dropwise condensation. Adv. Funct. Mater. 2011, 21 (24), 4617−4623. (13) Rykaczewski, K.; Paxson, A. T.; Anand, S.; Chen, X.; Wang, Z.; Varanasi, K. K. Multimode multidrop serial coalescence effects during condensation on hierarchical superhydrophobic surfaces. Langmuir 2013, 29 (3), 881−891. (14) Chen, S.; Doolen, G. D. Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 1998, 30 (1), 329−364. (15) Shan, X.; Chen, H. Lattice Boltzmann model for simulating flows with multiple phases and components. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1993, 47 (3), 1815−1819. (16) Sukop, M. C.; Thorne, D. T. Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers; Springer: New York, 2007. (17) Patankar, N. A. Mimicking the lotus effect: Influence of double roughness structures and slender pillars. Langmuir 2004, 20 (19), 8209− 8213. (18) Narhe, R.; Beysens, D. Nucleation and growth on a superhydrophobic grooved surface. Phys. Rev. Lett. 2004, 93 (7), 076103.

as shown in Figure 11. Because of the surface adsorption, droplets nucleate around the ridges on the lower surface. As these droplets grow, they merge together within the unit cell and form liquid films inside the cells. When these liquid films connect together, they submerge the structures. On the upper surface, liquid films form at the beginning of the condensation. Later, liquid films cover both upper and lower surfaces.

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CONCLUSION In the paper, numerical simulations of droplet condensation on surfaces with different microstructures have been carried out. It can be seen that the geometry of these structures plays an important role in droplet nucleation and growth. We have investigated the morphology of droplets during the condensation on surfaces with different structures. It is shown that nucleation behavior depends upon the spacing and height of ridges on superhydrophobic surfaces. Microridges with large spacing and height can trap droplets between ridges and reduce large size droplets staying on the top of ridges. For microridges with small spacing, the large size droplets will form on the ridge tips. This interesting phenomenon may be useful for controlling the condensation process and industrial applications. However, for hydrophilic surfaces, microstructures have little effect on the condensation. Our study suggests that the phase change process can be controlled by optimized design of structured surfaces. Furthermore, the study has significant relevance to the ice formation on surfaces caused by the condensation of subcooled water.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project was funded by the National Science Foundation of China (Grants 11272176 and 11172156) and the National Basic Research Program (2012CB720101).

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REFERENCES

(1) Cao, L.; Hu, H.-H.; Gao, D. Design and fabrication of microtextures for inducing a superhydrophobic behavior on hydrophilic materials. Langmuir 2007, 23 (8), 4310−4314. (2) Bellanger, H.; Darmanin, T.; Taffin de Givenchy, E.; Guittard, F. Chemical and physical pathways for the preparation of superoleophobic surfaces and related wetting theories. Chem. Rev. 2014, 114 (5), 2694− 2716. (3) Dorrer, C.; Rühe, J. Condensation and wetting transitions on microstructured ultrahydrophobic surfaces. Langmuir 2007, 23 (7), 3820−3824. (4) Jung, Y.; Bhushan, B. Wetting behaviour during evaporation and condensation of water microdroplets on superhydrophobic patterned surfaces. J. Microsc. 2008, 229 (1), 127−140. (5) Forsberg, P. S.; Priest, C.; Brinkmann, M.; Sedev, R.; Ralston, J. Contact line pinning on microstructured surfaces for liquids in the Wenzel state. Langmuir 2009, 26 (2), 860−865. (6) Mishchenko, L.; Hatton, B.; Bahadur, V.; Taylor, J. A.; Krupenkin, T.; Aizenberg, J. Design of ice-free nanostructured surfaces based on repulsion of impacting water droplets. ACS Nano 2010, 4 (12), 7699− 7707. (7) Rykaczewski, K. Microdroplet growth mechanism during water condensation on superhydrophobic surfaces. Langmuir 2012, 28 (20), 7720−7729. H


Numerical Simulation of Condensation on Structured Surfaces

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