Toward a Better Understanding of Hemiwicking: A Simple Model to

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Towards a Better Understanding of Hemi-wicking: Simple Model to Comprehensive Prediction Huadong Chen, Hang Zang, Xinlei Li, and Yanping Zhao Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b03611 • Publication Date (Web): 23 Jan 2019 Downloaded from http://pubs.acs.org on January 26, 2019

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Towards a Better Understanding of Hemi-wicking: Simple Model to Comprehensive Prediction Huadong Chen, Hang Zang, Xinlei Li*, and Yanping Zhao* MOE Key Laboratory of Laser Life Science & Institute of Laser Life Science, College of Biophotonics, South China Normal University, Guangzhou 510631, China ABSTRACT The hemi-wicking state has attracted much interest due to numerous important potential applications in inking, printing, boiling heat transfer and condensation. However, the mechanism of the emergence of hemi-wicking has not been well understood, especially the effects of geometry of patterned surfaces on the hemi-wicking state has not been systematically investigated. Here we presented a new method to study the critical conditions for hemi-wicking on patterned surfaces. By minimizing the variation of the free energy, we obtain the corresponding stable height of the hemi-wicking film, and find that it is easier for a droplet to be in the hemi-wicking state if the pillar surface has small spacing, large radius and height, and a small intrinsic contact angle. Our established model is applied to flat-topped cylindrical pillar-patterned surface and the modeling results are in well agreement with experiments and other existing theories. Besides, our model is also applied to other kinds of patterned surfaces including hemispherical-topped cylindrical and conical pillars about which the other existing theories are deficient. Our theoretical results not only are in well agreement with the experimental observations but also provide some important predictions, which implies that the established model could 1

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be applicable to understanding the basic physical mechanism of hemi-wicking state and be useful in guiding the design and fabrication of hemi-wicking surfaces.

2


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INTRODUCTION With the development of nano- or microtechnology, the wettability of liquid droplets can be modified by decorating the surfaces with nano- or micropatterns, which has broad range of applications such as self-cleaning, microfluidic devices, water proofing and so on. In order to improve surface wettability, the theoretical estimation of wetting properties of patterned surfaces has been of interest to researchers in recent years.1-5 Traditionally, when a liquid droplet is deposited on a patterned surface, it will sit on top of the surface topography without penetration, which is called Cassie-Baxter state,6 or wet the cavities of the surface topography, which is called Wenzel state.7 However, a new state was proposed recently, i.e. hemi-wicking state.8 The main difference between Wenzel and hemi-wicking states is that the hemi-wicking droplet has a liquid film around. So far, the hemi-wicking state has attracted much interest due to its applications in inking, printing, boiling heat transfer9-10 and condensation11, etc. Since the hemi-wicking state was firstly proposed by J. Bico et al,8 it becomes a relatively new area of interest in wetting research. C. W. Extrand et al.12 discussed the design of an optimized hemi-wicking surface based on creating a network of capillary channels which can enhance spreading. Dinesh Chandra, Shu Yang13 and Chang Quan Lai et al.14 investigated the dynamics of droplets spreading on wicking surfaces and found that the spreading follows some power laws. Ciro Semprebon et al.15 used the method of energy landscapes to assess the robustness of the pinning mechanism during wicking progress, and their results showed that energy barriers may affect the 3


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criteria for wicking. Additionally, Beom Seok Kim et al.9-10 came to the conclusion that hemi-wicking leads to a greater than 100% improvement of CHF compared to no wicking after theoretical and experimental analysis. Despite the existing studies of hemi-wicking, the mechanism of the emergence of hemi-wicking has not been well understood and there are still some issues to be solved. For example, existing studies main considered the hemi-wicking on the surface with flat-topped pillars, and other kinds of partnered surfaces were rarely studied. Meanwhile, the height of the hemi-wicking film was assumed to be the same as the pillar height in the existing theoretical studies.8, 16-17 The assumption is merely based on experimental observations and lacks theoretical support. In reality, the non-flat-topped pillar surfaces are more common, such as the hemispherical-topped cylindrical or conical pillar surfaces, and the hemi-wicking phenomenon will also occur on such surfaces. However, the assumption that the hemi-wicking film has the same height as that of the pillar is invalidated in these cases. So far, we have had no theoretical ideas to determine in which case we could obtain hemi-wicking phenomenon on these patterned surfaces. Meanwhile, there have been not any quantitative theories to address the origin of hemi-wicking phenomenon. For the sake of solving these problems, in this paper, we establish a thermodynamic theory to address the hemi-wicking state on nano- or microtextured surfaces.

Taking

three

typical

surfaces,

i.e.

flat-topped

cylindrical,

hemispherical-topped cylindrical and conical pillar-patterned surfaces, as examples, we study the physical mechanism of hemi-wicking by calculating the variation of the 4


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free energy of the system and obtain the critical conditions for hemi-wicking. We find that it will be easier for a droplet to be in the hemi-wicking state if the pillar surface has small spacing, large radius and height, and a small intrinsic contact angle. Based on our model, we obtain the corresponding stable height of the hemi-wicking film by minimizing the variation of the free energy when the droplet is in the hemi-wicking state. Our theoretical results not only are in well agreement with the experimental observations but also predict some new results which have not been observed or done. We expect that our model can offer some guidance to the design and fabrication of the hemi-wicking surfaces, and allow one to study the hemi-wicking for other diverse surfaces.

THEORETICAL MODEL When a smooth surface is patterned with regular nano- or micropillars, the surface may be easier for hemi-wicking. As a result, if a liquid droplet is deposited on such a surface, the droplet will spread around, and it will be observed that a liquid film is ahead of the three-phase contact line during spreading, as shown in Figure 1 (a). Figure 1 (b) shows the microcosmic view of a small area of the edge of the film. Based on thermodynamics theory, the variation of the free energy of the system is given by E  PdV   i dSi   dL ,18 where the first term is a bulk contribution when i

the volume V of the droplet is varied, the second term is a surface contribution when the area of any interface Si is varied, and the final term is a line contribution when the length of the contact line L is varied. Considering the droplet is like a spherical 5


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cap with a round brim, we divide the droplet into two parts: the hemi-wicking film with a radius of R f and a height of h f , and the bulk which is defined as the island part of the droplet with the radius of R b (see Figure 1 (a) and (b)).19 In this situation, the interface area of the droplet S can also be divided into two parts, which are the interface area of the hemi-wicking film S f and the interface area of the bulk Sb , i.e. S  S f  Sb . Consequently, the variation of the system free energy can be rewritten as

dE  PdV   j d ( S f ) j   k d ( Sb ) k   dL j

(1)

k

where the liquid-solid, liquid-vapor, and solid-vapor interfaces should be taken into consideration in the summations. Therefore, Eq. (1) can be used to predict the wetting behaviors of the droplet by considering the stable state with the lowest free energy. In our model, the evaporation of the droplet can be neglected. Therefore, the volume of the droplet is unvaried and the contribution of the first term in Eq. (1) is zero ( PdV  0 ). Also, the line tension τ is controversial and generally accepted on a small order of 1012 to 1010 J / m ,20 which is very small relative to interface energy, so we also neglect the contribution of the final term in Eq. (1). On the basis of the above considerations, we should only consider the variation of the system free energy caused by the change of the surface area during wetting process. During wetting process of a droplet, the spreading of the droplet results in the shrink of bulk and the expanse of film. In other words, the volume of the bulk will decrease and the volume of the film will increase during spreading process. The change of film volume is the same as that of the bulk base on volume conservation. It is worth mentioning that the variations of the free energies of the film and bulk, i.e. 6


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the second and third terms in Eq. (1), are the functions of the volume of transferred liquid between the film and bulk. The above considerations allow us to rewrite Eq. (1) into the following succinct formula: dE   f dVt  b dVt

(2)

where Vt represents the volume of the transferred liquid between the film and bulk,

 f and b are defined as the variations of the free energies of the film and bulk caused by the transformation of per unit volume. The negative sign of the second term means the shrink of bulk. Consequently, the first term in Eq. (2) is corresponding to the second term of Eq. (1), and the second term is corresponding to the third term of Eq. (1). Thus, when dE > 0, the free energy of the system will increase, it indicates that the liquid won’t be transferred from the bulk to the film, so the droplet will not be in the hemi-wicking state. Conversely, when dE < 0, it’s beneficial for the liquid to be transferred from the bulk to the film and the droplet may be in the hemi-wicking state as a result. When the transformation from bulk to film occurs, the increase of film volume will result in two possible changes. The first one is that the radius R f increases in x direction, i.e. the advancing direction of the three-phase contact line. The other one is that the film height h f increases in z direction, i.e. the direction along the pillars, as shown in Figure 1 (a) and (b). Thus the variation of the free energy of the film can be expressed as  f  x if R f varies, or  f  z if h f varies, accordingly, the variation of the system free energy, i.e. Eq. (2), can also be expressed as

7


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 (  f  x  b )dVt dE     f  z  b  dVt

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(3)

By using Eq. (3), we can identify in which case a droplet will be in the hemi-wicking state. More specifically, if  f  x  b and  f  z  b , i.e. dE is always larger than zero, the hemi-wicking state is unfavorable for the droplet. On the other hand, we can analyze the variation of the film when dE < 0, in which the increase of the film height will be more favorable if  f  z   f  x , and the film radius R f will increase if  f  z   f  x . Meanwhile, we can obtain the stable height hstable of the film in the case that  f  z   f  x . In the following sections, we will take three kinds of patterned surfaces with a regular array of flat-topped cylindrical pillars, hemispherical-topped cylindrical pillars, and conical pillars as examples to investigate the critical condition for the emergence of hemi-wicking and the evolution of film at hemi-wicking stage. Figure 1 (c), (d) and (e) show the side views of the unit cell of the three kinds of pillars, in which P is the centre-to-centre spacing between the regular pillars, R is the radius of each pillar and H is the pillar height. Analytical Model for the Flat-topped Cylindrical Pillar-Patterned Surface As the three-dimensional (3D) model shown in Figure 1, when a droplet is wetting on a flat-topped cylindrical pillar patterned surface with the hemi-wicking state, the shape of the film around the droplet is actually like a doughnut with a thickness of h f . For simplicity, we assume that the sidewall of edge of film is normal to the substrate and the area of the sidewall is neglected because the area of the sidewall is much smaller than the total area of film. Based on thermodynamics, the 8


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free energy of a droplet on a solid substrate is E    sl   sv  Asl   lv Alv , where Alv and Asl are the areas of liquid-vapor interface and solid-liquid interface.21 Therefore, by analyzing the 3D interface geometry of the droplet system, the free energy of the hemi-wicking film E𝑓 is calculated as Ef 

  R f 2  Rb 2 

where

P2

 ls ,  sv









  ls   sv  P 2   R 2  2 Rh f   lv P 2   R 2   

and

 lv

(4)

are respectively the liquid-solid, solid-vapor and

liquid-vapor interface energies,21-22 Rb , R f and h f (h f  H ) represent the radius of bulk, the radius of film and the height of film separately. Using Eq. (4) and Young’s equation23 that cosY    sv   ls  /  lv , where Y is the equilibrium contact angle of the droplet on a smooth surface, we can obtain the variation of the free energy of the film in x direction ( R f increases to R f  dR f ) and in z direction ( h f increases to h f  dh f ) by derivation as that

 2 R f 2 2 2 2  2  lv  P   R  cosY P   R  2 Rh f  dR f  P dE f    R f 2  Rb 2   lv  2 RcosY  dh f  P2













(5)

Also, the volume of film can be calculated by analyzing the 3D interface geometry of the droplet system. If the distribution of cylindrical pillar is square, the film volume is calculated to be V f    R f 2  R b 2  P 2   R 2  h f P 2 , and dV f can be expressed as a function of dR f or dh f by deriving V f , as

2 R f  2 2 dR f  P   R hf P2  dV f   2 2  R f  R b P 2   R 2 dh f 2  P

 

 



(6)



If the distribution is not square, we can also calculate the film volume, for example, 9


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Vf

is equal to

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





V f  2 R f 2  R b 2 3 3P 2 2  2 R 2 h f 3 3P 2

when the

distribution is hexagonal. Combining Eq. (5) and Eq. (6), the variation of the free energy of the film per increased unit volume of the film dE f / dV f , i.e.  f , can be calculated by

 f x 

 f z 

 lv  P 2   R 2   cosY  P 2   R 2  2 Rh f  



hf P2   R2

(7)



 lv  2 RcosY  P2   R2

(8)

when h f  H . Using the same derivation process, when h f =H,  f  x and  f  z can be expressed as follows:

 f x 

 lv  P 2   R 2   cosY  P 2   R 2  2 RH  

(9)

H  P 2   R 2 

 f  z  lim

 lv R 2 1  cosY 

hf H

h

f

(10)

 H  P2

When h f  H ,

 f x 

 lv  P 2  cosY  P 2  2 RH  

(11)

H (P2   R2 )   hf  H  P2

 f z  0

(12)

For the bulk of the droplet, when the apparent contact angle is  * , the variation of the free energy of the bulk per increased unit volume is

b 

2 lv 3

  S sl 2 3sin3 *   cos   Y 2 * Sp 1  cos    1  cos * 2  cos * 





2 3



 1  V 3  b 

(13)

where Ssl is the area of the solid-liquid interface of the bulk in an unit cell, and S p is the projected area of the unit cell. Here S sl   R 2  2 R  H  h f 10




and S p  P 2 .

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Detailed derivation process can be found in the Supporting Information. Substituting Eqs. (7)-(13) into Eq. (3), we can get the relation of dE / dVt with the geometry of patterned surface and intrinsic contact angles. Furthermore, we can predict that droplet will be in the hemi-wicking state when dE / dVt  0 . Analytical Model for Hemispherical-topped Cylindrical Pillar-Patterned Surface Similarly, when a droplet with a hemi-wicking film is wetting on a hemispherical-topped cylindrical pillar patterned surface, as shown in Figure 1(d),

 f  x and  f  z can be replaced by

 f x 

 f z 

 lv  P 2   R 2   cosY  P 2   R 2  2 Rh f  



hf P2   R2



 lv  2 RcosY  P2   R2

(14)

(15)

when h f   H  R  ,

 f x

 P2   R2   h  H  R 2     f       lv   cosY  P 2   R 2  2 R  H  R   2 R  h f  H  R        2   H  R  P 2   R 2   h f  H  R   P 2   R 2   h f  H  R   3  



 f z 



2 lv  h f  H  R  RcosY  P2   R2    hf  H  R 

(16)

(17)

2

when H  R  h f  H , and

 f x





 lv P 2  cosY  P 2   R 2  2 R  H  R    2  H  R   P 2   R 2   RP 2  R3   h f  H  P 2

(18)

3

 f z  0

(19)

When h f  H . The expression of b for the hemispherical-topped cylindrical pillar patterned 11


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surface is the same as Eq. (13). Analytical Model for Conical Pillar-Patterned Surface Analogously, for conical pillar patterned surface,  f  x and  f  z are

2     H  h f 2    H  hf   2 2 2 2 2 R    cosY  P   R   R H  R 1   P         H     H      lv 2   H  hf   2 2 P hf  R H   R   H  h f  3   H  



 f x



(20)  2 R  H  h f 

 f z 



 2 2   H R cos  Y H2  2  H  hf  2 P   R  H 

 lv 



R 

(21)

when h f  H , and

 f x 





 lv  P 2  cosY P 2   R 2   R H 2  R 2  

P2 H 



3

R2 H   hf  H  P2

 f z  0

(22)

(23)

when h f  H . Also, the formula of b is the same as Eq. (13), and the variation of the system free energy can also be obtained by combining Eqs. (20)-(23) and Eq. (13). RESULTS AND DISCUSSION In order to analyze the influence of the geometries of the pillars and the coatings of the surfaces on the wetting states clearly, we take distilled ionized water droplets on different micropillar-patterned silicon surfaces coated with varied coatings as 12


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examples. In our calculations, the liquid-vapor interface energy is 73mJ / m 2 .24 The droplet volume V = 6.3 μL, and the intrinsic contact angles Y of droplets on the three kinds of smooth surfaces are selected as 34º, 58º and 109º in order to compare with experimental results.25 Hemi-wicking State on Flat-topped Cylindrical Pillar-Patterned Surface For the flat-topped cylindrical micropillar-patterned surface (Fig. 1 (c)), in our calculations, the geometrical parameters P, R and H are selected as 40 μm, 8 μm and 20 μm, respectively based on experimental conditions. The volume of droplet used in experiments is of the order of several microliters. In this case, the bulk volume ( Vb ) is approximately equal to V, and the variation of the free energy of bulk caused by the change of per unit volume ( b ) is extremely small relative to that of film (  f ) because of its large volume. Therefore, we can neglect the contribution of bulk energy (Eq. (13)) to total energy in Eq. (3) (Supporting Information). So we only consider the contribution of film energy, i.e.  f . Figures 2 (a)-(c) show the variation of the free energy of the film per increased unit volume when it extends to the direction of x, i.e.  f  x , and z, i.e.  f  z , by using Eqs. (7)-(12) in cases of that Y is equal to 34º, 58º and 109º. As shown in Figure 2 (a), when Y  34 ,  f  x and  f  z are changed with the film height h f . Clearly, when h f  H ,  f  z is always less than 0 and  f  x , which indicates the thickness of

film can firstly increases in order to desire lower free energy. However, when h f

exceeds the height of micropillars, in other words, the film overflows the micropillars,

 f  z has an abrupt change and becomes larger than 13


 f  x (  f  x  0 ), which

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suggests that the spread in x direction becomes favorable and the height of film remains a stable height, hstable  H . The calculated results above show the droplet will have a hemi-wicking film on the micropillar-patterned surface, and the stable height of film is just equal to the height of micropillars. The modeling results are in agreement with experimental observations by Yu et al.25 They found that distilled ionized water droplets on the micropillar-patterned surfaces with the same morphology as used in Figure 2 (a) have a hemi-wicking film. In Figure 2 (b), when the Y increases to 58º, although the variation  f  z is always less than both zero and  f  x in the case that h f  H , the variation  f  x is always greater than 0, which suggests that the film will not extend to the x direction. In other words, the droplet will not have a hemi-wicking film. Interestingly, Yu et al had also observed that the droplet is in the Wenzel state in this case.25 Figure 2 (c) shows that both the variations  f  z and

 f  x are greater than zero when Y  109 , so that the

droplet will also not be in the hemi-wicking state, which had been observed to be in the Cassie-Baxter state in this case by Yu et al.25 As a result, we can know from Figures 2 (a)-(c) that the droplet is in the hemi-wicking state if the intrinsic contact angle Y is small enough. And the reason for the result is that the variation free energy is affected by the liquid-vapor and liquid-solid interface energies.26-28 More specifically, the free energy is expressed by

E   lv Alv    sl   sv  Asl , where Alv and Asl are the areas of the liquid-vapor interface and the liquid-solid interface, respectively. When the film spreads, the solid-vapor interface will be replaced by liquid-solid and liquid-vapor interface. If the 14


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intrinsic contact angle is smaller than a critical value, the increase of the liquid-vapor and liquid-solid interface energies will be smaller than the decrease of the solid-vapor interface energy, which lead to the decrease of the total free energy and facilitates the emergence of hemi-wicking film. However, if the intrinsic contact angle is larger than the critical value, the increase of the liquid-vapor and liquid-solid interface energies will become large, and the hemi-wicking film will not occur because the total free energy increases. Besides the effects of the intrinsic contact angle, the geometry of surface also has a strong influence on the wetting behaviors. In other word, the parameters of pillars also determine the wetting state. For comparison, the value of centre-to-centre spacing P used in Figure 2 (a) was replaced by 100 μm in Figure 2 (d)-(f). Differently, R = 8 μm and H = 20 μm in Figure 2 (d), R = 38 μm and H = 20 μm in (e), and R = 38 μm and H = 4 μm in (f). From the figures, we can find that the droplets corresponding to Figures 2 (d) and (f) are no wicking, while the droplet corresponding to Figure 2 (e) is in the hemi-wicking state. In addition, Figures 2 (a) and (d) show that the droplet can’t be in the hemi-wicking state if the center-to-center spacing between the regular pillars P is large enough. Also, by comparing Figures 2 (d) and (e) or Figures 2 (e) and (f), we can deduce that the droplet will be in the hemi-wicking state if the pillar has large radius or height. Reasonably, a large spacing, a small radius or a small height makes the increased liquid-vapor interface be larger, so that the free energy of system will increase, which denotes that the hemi-wicking film is not available. Importantly, the modeling results agree well with experimental observations by Yu et 15


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al,25 in which the droplets with small intrinsic contact angle on the surfaces with small spacing, large height and radius are in hemi-wicking states. Conversely, the droplets with large intrinsic contact angle on the surfaces with large spacing, small height and radius are not in hemi-wicking states. . We can note that the graphs of functions  f  z in Figure 2 are horizontal straight lines, that is to say,  f  z has nothing to do with the pillar height, which can be judged from Eq. (8) clearly. What the reason is that the sidewall of the flat-topped cylindrical pillar is normal to the liquid-vapor interface, which means that the liquid-vapor interface area remains a constant and the rate of change of solid-liquid interface area remain unvaried with increasing h f . For x direction,  f  x decreases with the increase of h f . It is because the increase of liquid-vapor interface area remains unvaried, while the increase of solid-liquid interface area becomes larger and larger when h f increases. Therefore,  f  x becomes smaller and smaller. In this situation, if the spacing is small enough, or the radius and height are large enough,  f  x will decrease to a value smaller than zero before h f increases to H, so that the droplet will be in the hemi-wicking state. As a summary, if a droplet wetting on a flat-topped cylindrical pillar-patterned surface is in the hemi-wicking state, the intrinsic contact angle Y

and the

centre-to-centre spacing between the regular pillars P should be small enough, and the pillar radius and height should be large enough. More importantly, we have gotten the result that the stable height of the hemi-wicking film hstable is H. Although many studies have mentioned or observed that the stable height of the 16


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hemi-wicking film is H for flat-topped pillar-patterned surface,8,

16-17

they can’t

explain well the physical mechanism behind it. Our results clearly prove that the stable height is equal to H. Based on the discussions above, we can obtain that the criteria of the emergence of hemi-wicking should be as follows:  f  z is equal to  f  x when h f reaches a certain value ( hstable ), and  f  x is less than zero when h f  hstable . So we can obtain the conditions for hemi-wicking by using Eqs. (8) and (9), i.e. Y   c , with

1 s P2   R2  cos c  2 2 P   R  2 RH r   s

(24)

where  c is a critical value,  s is the fraction of the solid-vapor interface above the hemi-wicking film, and 𝑟 is the roughness factor defined as the ratio of the total surface area to the projected surface area.7 Interestingly, Bico et al. also deduced the similar expression of critical contact angle using a different model in which the variation of free energy caused by the advance of the three-phase contact line was analyzed.8 However, the results in the paper of Bico et al were calculated based on the premise that hstable is equal to H without detailed analysis, which can’t provide a better understanding of hemi-wicking. Hemi-wicking State on Hemispherical-topped Cylindrical Pillar-Patterned Surface Analogously, we obtained the graphs of  f  x and  f  z corresponding to droplets wetting on hemispherical-topped cylindrical micropillar-patterned surfaces. Figures 3 (a)-(c) show the changing curves of  f  x and  f  z as functions of h f when P = 40 μm, R = 8 μm and H = 20 μm and the intrinsic contact angles Y are 17


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different. Certainly, when h f  H  R , the curves in Figures 3 (a)-(c) have the same trends of change as that in Figures 2 (a)-(c), respectively, because they have the same geometrical parameters when h f  H  R . However, when h f  H  R , there are intersections of the two curves of

 f  x and  f  z in the Figures 3 (a) and (b) but

not in (c). As mentioned in the previous sections, the intersection point in Figure 2 (a) corresponds to the stable height of film ( hstable ) which is located in the part of hemispherical cap. However, the stable height in Figure 3 (b) does not exist in reality because  f  x is larger than zero when h f  hstable , which indicates the free energy of the system will increase in this case. In other words, because  f  x is always larger than zero, the droplet will not spread in x direction and be not in the hemi-wicking

 f x  0

state. By contrast, in Figure 3 (a),

when

h f  hstable , thus the

corresponding droplet is in the hemi-wicking state. Similarly, the droplet corresponding to Figure 3 (c) is also no wicking in the case of large intrinsic contact angle ( Y = 109º). Based on the analysis above, we can obtain the conclusion that small intrinsic contact angle is favorable to the emergence of hemi-wicking. For the purpose of studying the effects of centre-to-centre spacing on the wetting state, we set P = 100μm instead of P = 40μm in Figure 3 (a) when Y  34 as shown in Figure 3 (d). From Figure 3 (d), we can judge that the corresponding droplet becomes no wicking. So the increase of centre-to-centre spacing is detrimental to hemi-wicking. Furthermore, the effects of radius and height are also studied. Compared with Figure 3 (a), the pillar radius in Figure 3 (e) is 2 μm, and the pillar height in Figure 3 (f) is 10 μm. The results of Figures 3 (e) and (f) show that the both 18


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corresponding droplets are also not in the hemi-wicking state. As a result, a large pillar radius or a large height is necessary for the emergence of hemi-wicking. We can know from Figure 3 that the stable height hstable is between H-R and H when the droplet is in the hemi-wicking state on a hemispherical-topped cylindrical pillar-patterned surface. In this case,

 f  z is always smaller than  f  x when

h f  H  R , which can be judged from Eqs. (14) and (15). But when h f  H  R , the

value of  f  z will gradually increase and then exceed  f  x with increasing h f , and  f  x is always smaller than zero. So there will be a stable height hstable that makes  f  x   f  z . The reason is that the increased liquid-vapor interface area is larger than the increased liquid-solid interface area with the increase of the film height when H  R  h f  H , and if the height ℎ𝑓 is high enough, the total liquid-vapor interface area will be larger than the liquid-vapor interface area, and  f  z will become larger than zero. Under these discussions, we can get the stable height hstable by making Eq. (16) equal to Eq. (17), i.e.  f  x   f  z , which is an implicit function of Y , P, R, and H, as expressed follows: 2  P2   R2    h   stable  H  R       cosY  P 2   R 2  2 R  H  R   2 R  hstable  H  R     2  H  R  P 2   R 2   hstable  H  R   P 2   R 2   hstable  H  R   3    H  R  RcosY  2  h  2 stable 2 P   R 2    hstable  H  R 





(25)

To clearly discuss, we drew the graph of hstable as a function of Y according to Eq. (25), as shown by the black line in Figure 4, in which P = 40 μm, R = 8 μm and H = 20 μm. Figure 4 shows that the stable height increases with the increase of intrinsic 19


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contact angle Y , it’s no doubt that the line always locates between h f  H  R and h f  H . However, when Y is larger than a critical value  c , the droplet will not be

in the hemi-wicking state, because  f  x always is larger than zero in this situation. Although the black dotted line in Figure 4 illustrates that the film has a stable height when Y   c , it does not exist in reality, because it will make the free energy of the system increase. From the results, we can find that the stable height hstable is not just equal to H or H-R for a hemispherical-topped cylindrical pillar-patterned surface, which is different from the case of the flat-topped cylindrical micropillar-patterned surface. To the best of our knowledge, there are no theories to study the stable height on the surface so far. Our model (Eq. (25)) can predict the stable film height for the hemispherical-topped pillars. Evidently, by using  f  x   f  z  0 we can also get the value of  c , so if the droplet is in the hemi-wicking state, Y should meet the condition that Y   c , where



2

cos c 



 P 2   R 2  2 R  H  R    4 R 2 P 2   R 2   P 2   R 2  2 R  H  R   2 R 2

(26) As reflected by region I in Figure 4, the droplet is in the hemi-wicking state, but region II implies that the droplet will be no wicking. Thus Eq. (26) is useful in predicting the wetting state for the hemispherical-topped pillars surfaces. Hemi-wicking State on Conical Pillar-Patterned Surface Besides the flat-topped cylindrical micropillar-patterned surface and the hemispherical-topped

cylindrical

micropillar-patterned 20


surfaces,

the

conical

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pillar-patterned surface is also a typical patterned surface. It’s also important to study the hemi-wicking state on the conical pillar-patterned surface. Unfortunately, there have been few the relevant reports on the droplets wetting on such surfaces with the hemi-wicking state so far. Using the established the model, we can theoretically investigate the hemi-wicking state in advance. As depicted in Figures 5 (a)-(c), the graphs of the variation of the system free energy have different characteristics when θY = 34º, 58º and 109º. Figure 5 (a) shows that the droplet has a large stable height which is marked by the point of intersection of  f  x and  f  z , as shown in the inset in Figure 5 (a). So the droplet is in the hemi-wicking state in this case. However, when the angle Y increases to 58º and 109º, the droplet can’t be in the hemi-wicking state, which is inferred from Figures 5 (b) and (c). Then we study the effects of geometrical parameters on wetting state. When the centre-to-centre spacing between the regular pillars P increases to 100 μm, the pillar radius R decreases to 4 μm or the pillar height H decreases to 10 μm in Figure 5 (a), the droplet will not be in the hemi-wicking state, because the film does not have a stable height, as show in Figures 5 (d)-(f). As mentioned above, the free energy of the system is affected by the liquid-solid and liquid-vapor interface areas. In Figure 5 (a), the increase of film volume will result in the increases of liquid-solid and liquid-vapor interface areas in the form of extension in x or z direction. According to the results and discussion for the flat-topped cylindrical pillar-patterned surface, we know that when the thickness of film is small, the increased liquid-vapor interface area is slightly smaller than the 21


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increased liquid-solid interface area if the film extends in the x direction, which leads to  f  x  0 . However, the increased liquid-solid interface area is much larger than the increased liquid-vapor interface area if the film extends in the z direction, which means that  f  z  0 . As the result, the film will extends to the z direction and the film height will increase in order to achieve the lower free energy when the thickness of film is small. However, when the thickness of film exceeds to a certain value, the increase of the liquid-vapor interface area caused by the volume change will become much smaller than the increased liquid-solid interface area if the film extends in the x direction, which can lead to  f  x  0 . In this case, both the increase of liquid-solid interface area and the increase of liquid-vapor interface area caused by the volume change become smaller than that in the case of small h f . But the change of liquid-solid interface area is more obvious than that of liquid-vapor interface area. Therefore,  f  z in the case of large h f is higher than that in the case of small h f , in other words,  f  z increases with increasing h f . When h f exceeds H, i.e. the film overflows the cylindrical micropillars,  f  z is equal to zero. Therefore, there is an intersection point between the lines of  f  x and

 f  z , as shown in Figure 5 (a).

According to the analysis on Figure 5 (a), we know that  f  x will decrease to a smaller value than both zero and the value of  f  z when the thickness exceed certain values, which means that the film will have a stable height and the droplet can be in the hemi-wicking state. On the contrary, when the intrinsic contact angle and spacing increase, or the radius and height decrease, the increase of the liquid-vapor interface area caused by the volume change can’t become much smaller than the increased 22


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liquid-solid interface area if the film extends in the x direction, which suggests that

 f  x is always larger than zero. Therefore there is no intersection point between the lines of  f  x and

 f  z , and the hemi-wicking film will not emerge, as shown in

Figure 5 (b)-(f). Analogous to the above sections for the flat-topped and hemispherical-topped cylindrical micropillar-patterned surfaces, by using

 f x   f z

( h f  H ) we

obtained the stable height hstable , that is 2  2   H  hstable   P R        H         2  2    H h    2 2 2 2 2 stable   cosY  P   R   R H  R   R H  R  H       2   H  hf   P 2 hstable   R 2 H   R   H  hstable   3   H    2 R  H  hstable   R  H 2  R 2 cosY   2 H   2  H  hstable  P2    R H  







(27)



And we plot the graph of hstable as a function of Y when P = 40 μm, R = 8 μm and H = 20 μm, as shown in Figure 6. Obviously, there is a stable height of film ( hstable ) on if the intrinsic contact angle Y is smaller than  c . Meanwhile, hstable increases with the increase of Y when Y   c . When Y   c , the droplet will not be in the hemi-wicking state. Therefore, region I and II in Figure 6 are divided by the critical contact angle  c . It should be known that the droplet is in the hemi-wicking state only if the intrinsic contact angle is within the limits of region I. Also, the critical angle 𝜃𝑐 can be determined by  f  x   f  z  0 . Surely,  f  x and  f  z must below 0 if the droplet is in the hemi-wicking state. Therefore, for the conical 23


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pillar-patterned surface, if the droplet is in the hemi-wicking state, the intrinsic contact angle must comply with Y   c while

cos c 

P2 P  R  R H  R 2

2

2

2



1 r

(28)

Dettre and Johnson also found the similar relations with Eq. (28), and they expected that the droplet will wick if the cosine of the intrinsic contact angle exceeds the inverse of the roughness factor. 29 Interestingly, our calculated results are in agreement with the experimental observations and other theoretical results by Srinivas Bengaluru Subramanyam et al.30 They found that an impregnating lubricant with a positive spreading coefficient which has a larger intrinsic contact angle underwater will submerge the entire texture. While a lubricant with a negative spreading coefficient which has a small intrinsic contact angle will result in the tops of the texture being exposed. Unfortunately, few researchers have fully studied the hemi-wicking state of droplet on conical pillar surfaces, so our results can only make a prediction of it, and further investigation need to be done to verify our results. Comparison among three kinds of pillar-patterned surfaces Finally, we compare the critical contact angles for hemi-wicking of the three kinds of pillar-patterned surfaces using Eqs. (24), (26) and (28), as illustrated in Figure 7. We can find that a small spacing, a large radius or height will lead to a large critical contact angle, which is easier for a liquid droplet to be in the hemi-wicking state. The result is in agreement with that mentioned in the sections above (Figures 2 (d)-(f), Figures 3 (d)-(f), and Figures 5 (d)-(f)). More importantly, from Figure 7 24


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(a)-(c) we know that the flat-topped cylindrical pillar-patterned surface has the largest critical contact angle compared with the other two surfaces when the pillars of the three patterned surfaces have the same height, radius, and spacing, which suggests the flat-topped cylindrical pillar-patterned surface is the most easily one to result in the hemi-wicking state. We can also find that the conical pillar-patterned surface has the smallest critical contact angle, which indicates it is the most difficult one to result in the hemi-wicking state. As a reason, the conical pillar has a pointed top, so the hemi-wicking film on a conical pillar-patterned surface will have larger liquid-vapor interface area and smaller solid-liquid interface area than the other two kinds of surfaces, which restricts to the decrease of free energy. According to the results shown in Figure 7, we know that it is easier for a flat-topped cylindrical pillar-patterned surface to result in hemi-wicking than a hemispherical one. Interestingly, the theoretical prediction agrees well with experimental observations by Fan et al.31 In their experiments, they found that the droplets were in Wenzel state if the nanorod surfaces have small roughness, as shown by the red circles in Figure 8. However, if the roughness is large, the contact angles can’t be interpreted by the Wenzel equation, so they guessed that the droplets were in the hemi-wicking state, as shown by the blue dots in Figure 8. In order to verify the rationality of their guess, we plotted the critical line of hemi-wicking for the flat-topped cylindrical pillar-patterned surface by applying Eq. (24), as shown by the yellow line in Figure 8. It is obviously that the yellow line can’t distinguish well the results of Wenzel state (the red circles in Figure 8). Through carefully observing the 25


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SEM images of nanorod in the paper of Fan et al., we can find that the top of nanorod was not flat. So we considered that the nanorod can be treated as a hemispherical-topped cylindrical pillar, and the corresponding critical line was plotted as the black line in Figure 8 by applying Eq. (26). Clearly, the black line can distinguish the wetting states very well. As a result, it makes more sense to think of the nanorod as a hemispherical-topped cylindrical pillar. In addition, the results match better with Fan et al.’s supposition and our model provides a new method to predict hemi-wicking for the conical pillar-patterned surface. Application in circular truncated cone pillar-patterned surface Based on the discussion above, our model behaves well in the predictions for hemi-wicking of the three typical pillar-patterned surfaces. Besides the three patterned surfaces, our model can be also applied to other kinds of pillar-patterned surfaces, such as a circular truncated cone pillar-patterned surface. In experiments, D C Pham et al.32 measured the contact angles of water droplets on circular truncated cone pillar-patterned surfaces with different pillar spacings, radii and heights. They found that the droplets were in Wenzel state if the spacing is large, as the red circles plotted in Figure 9. However, the droplets on the surface with small spacing, large radius and height were not found to be in Wenzel state. The droplets were thought to be in the hemi-wicking state according to the values of apparent contact angles, as shown by the blue dots in Figure 9. Using our established model, we plotted the critical line of hemi-wicking on the circular truncated cone pillar-patterned surface which is determined by spacing, radius and height, as shown by the black line in Figure 9. It is 26


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readily apparent that the critical line behaves well in the distinction between the two wetting states. So our results are in accord with D C Pham et al.’s conjectures.32 To sum up the above discussions, the present model provides a better understanding of hemi-wicking. In the previous study of hemi-wicking, some other researchers had also proposed some praiseworthy models. For example, Bico et al8 proposed a simpler model that predicts the critical angle  c from the dry surface solid fraction  s and roughness r. In the simpler model, the hemi-wicking on surface with flat-topped pillars was studied based on the assumption that the stable height of film hstable is equal to the pillar height H. however, the assumption lacks theoretical support. In the present model, the hemi-wicking on surface with flat-topped pillars is studied through the analysis of the height of film h f , and the results prove that hstable is equal to H. Although the critical angle  c calculated by the present model has the same expression with that in the simpler model, which means the present model works equally well, the present model provides a theoretical support and makes it easier to understand hemi-wicking. Besides, it seems that the simple model proposed by Bico et al8 can be used for random microstructured surfaces. However, the simple model has some limits, such as the surface should be patterned with flat-topped pillars. For example, if a surface is patterned with conical pillars, the height of hemi-wicking film is not equal to the height of pillar, which implies that the simpler model can’t work. However, the present model can be applied into other non-flat-topped patterned surface, such as hemispherical-topped cylindrical and conical pillars. Meanwhile, the present model 27


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only takes ordered microstructured surfaces as examples, but the random microstructured surfaces can also be studied using the present model.

CONCLUSIONS In conclusion, we have presented a new method to study the hemi-wicking states of liquid droplets on nano- or micropillar-patterned surfaces based on the thermodynamic theory. Three kinds of typical pillars, i.e. flat-topped cylindrical, hemispherical-topped cylindrical and conical pillars, were considered particularly. By analyzing the variation of the free energy, we obtained the expressions of the stable height of the hemi-wicking film. We find that the stable height of film is determined by the intrinsic contact angle and the geometry of patterned surfaces. When the intrinsic contact angle is larger than a critical value (  c ), the stable height of film does not exist, which indicates the droplet is not in hemi-wicking state in this case. On the other hand, the stable height of film also does not exist on the patterned surfaces with a large spacing, a small radius or height. It is only when the droplet with a smaller intrinsic contact angle than  c wets on the patterned surface with a small spacing, a large radius or height that it is in hemi-wicking state. Our established model is applied to flat-topped cylindrical and the modeling results are in well agreement with experiments and other existing theories. Besides, our model can be applied to other patterned surfaces including hemispherical-topped cylindrical and conical pillars about which the other existing theories are deficient. Our theoretical results not only are in well agreement with the experimental 28


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observations but also provide some important predictions. Though our established model seems simple, it can help us have a better understanding of hemi-wicking and provide comprehensive prediction.

ASSOCIATED CONTENT Supporting Information Further details of theoretical methods and calculations. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] (X.L.). *E-mail: [email protected] (Y.Z.). ORCID Xinlei Li: 0000-0002-9294-0459 Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS This research was supported by Natural Science Foundation of Guangdong Province (Grants No. 2017A030313389 and 2018A030313125), and the Science and Technology Project of Guangzhou (Grant No. 201805010002). 29


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REFERENCES (1) Yao, C.-W.; Alvarado, J. L.; Marsh, C. P.; Jones, B. G.; Collins, M. K., Wetting behavior on hybrid surfaces with hydrophobic and hydrophilic properties. Applied Surface Science 2014, 290, 59-65. (2) Jansen, H. P.; Bliznyuk, O.; Kooij, E. S.; Poelsema, B.; Zandvliet, H. J., Simulating anisotropic droplet shapes on chemically striped patterned surfaces. Langmuir 2011, 28, 499-505. (3) Raspal, V.; Awitor, K.; Massard, C.; Feschet-Chassot, E.; Bokalawela, R.; Johnson, M., Nanoporous surface wetting behavior: The line tension influence. Langmuir 2012, 28, 11064-11071. (4) Bormashenko, E., Progress in understanding wetting transitions on rough surfaces. Advances in colloid and interface science 2015, 222, 92-103. (5) Zhao, Y.; Lin, R.; Tran, T.; Yang, C., Confined wetting of water on CNT web patterned surfaces. Applied Physics Letters 2017, 111, 161604. (6) Cassie, A.; Baxter, S., Wettability of porous surfaces. Transactions of the Faraday society 1944, 40, 546-551. (7) Wenzel, R. N., Resistance of solid surfaces to wetting by water. Industrial & Engineering Chemistry 1936, 28, 988-994. (8) Bico, J.; Thiele, U.; Quéré, D., Wetting of textured surfaces. Colloids and Surfaces A: Physicochemical and Engineering Aspects 2002, 206, 41-46. (9) Kim, B. S.; Lee, H.; Shin, S.; Choi, G.; Cho, H. H., Interfacial wicking dynamics and its impact on critical heat flux of boiling heat transfer. Applied Physics Letters 2014, 105, 191601. (10)Kim, B. S.; Choi, G.; Shim, D. I.; Kim, K. M.; Cho, H. H., Surface roughening for hemi-wicking and its impact on convective boiling heat transfer. International Journal of Heat and Mass Transfer 2016, 102, 1100-1107. (11)Anand, S.; Paxson, A. T.; Dhiman, R.; Smith, J. D.; Varanasi, K. K., Enhanced condensation on lubricant-impregnated nanotextured surfaces. ACS nano 2012, 6, 10122-10129. 30


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(12)Extrand, C.; Moon, S. I.; Hall, P.; Schmidt, D., Superwetting of structured surfaces. Langmuir 2007, 23, 8882-8890. (13)Chandra, D.; Yang, S., Dynamics of a droplet imbibing on a rough surface. Langmuir 2011, 27, 13401-13405. (14)Lai, C. Q.; Mai, T. T.; Zheng, H.; Lee, P.; Leong, K.; Lee, C.; Choi, W., Droplet spreading on a two-dimensional wicking surface. Physical Review E 2013, 88, 062406. (15)Semprebon, C.; Forsberg, P.; Priest, C.; Brinkmann, M., Pinning and wicking in regular pillar arrays. Soft Matter 2014, 10, 5739-5748. (16)Ahn, H. S.; Park, G.; Kim, J.; Kim, M. H., Wicking and spreading of water droplets on nanotubes. Langmuir 2012, 28, 2614-2619. (17)Murakami, D.; Kobayashi, M.; Moriwaki, T.; Ikemoto, Y.; Jinnai, H.; Takahara, A., Spreading and structuring of water on superhydrophilic polyelectrolyte brush surfaces. Langmuir 2013, 29, 1148-1151. (18)Weijs, J. H.; Marchand, A.; Andreotti, B.; Lohse, D.; Snoeijer, J. H., Origin of line tension for a Lennard-Jones nanodroplet. Physics of fluids 2011, 23, 022001. (19)Gaillard, P.; Saito, Y.; Pierre-Louis, O., Imbibition of solids in nanopillar arrays. Physical review letters 2011, 106, 195501. (20)Shao, M.; Wang, J.; Zhou, X., Anisotropy of Local Stress Tensor Leads to Line Tension. Scientific reports 2015, 5, 9491. (21)Wang, C.; Yang, G., Thermodynamics of metastable phase nucleation at the nanoscale. Materials Science and Engineering: R: Reports 2005, 49, 157-202. (22)Xiao, K.; Zhao, Y.; Ouyang, G.; Li, X., An analytical model of nanopatterned superhydrophobic surfaces. Journal of Coatings Technology and Research 2017, 14, 1297-1306. (23)Young, T., III. An essay on the cohesion of fluids. Philosophical transactions of the royal society of London 1805, 95, 65-87. (24)Huang, Y.; Zhang, X.; Ma, Z.; Zhou, Y.; Zheng, W.; Zhou, J.; Sun, C. Q., Hydrogen-bond relaxation dynamics: resolving mysteries of water ice. Coordination Chemistry Reviews 2015, 285, 109-165. 31


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(25)Yu, D. I.; Doh, S. W.; Kwak, H. J.; Kang, H. C.; Ahn, H. S.; Park, H. S.; Kiyofumi, M.; Kim, M. H., Wetting state on hydrophilic and hydrophobic micro-textured surfaces: Thermodynamic analysis and X-ray visualization. Applied Physics Letters 2015, 106, 171602. (26)Ouyang, G.; Yang, G.; Sun, C.; Zhu, W., Nanoporous structures: smaller is stronger. small 2008, 4, 1359-1362. (27)Ouyang, G.; Wang, C.; Yang, G., Surface energy of nanostructural materials with negative curvature and related size effects. Chemical reviews 2009, 109, 4221-4247. (28)Xiao, K.; Zhao, Y.; Ouyang, G.; Li, X., Modeling the Effects of Nanopatterned Surfaces on Wetting States of Droplets. Nanoscale research letters 2017, 12, 309. (29)Johnson, R. In Contact Angle Hysteresis I. Study of an Idealized Rough Surfaces, Advances in Chemistry, Ser., 1964; p 112. (30)Subramanyam, S. B.; Azimi, G.; Varanasi, K. K., Designing Lubricant ‐ Impregnated Textured Surfaces to Resist Scale Formation. Advanced Materials Interfaces 2014, 1, 1300068. (31)Fan, J.; Tang, X.; Zhao, Y., Water contact angles of vertically aligned Si nanorod arrays. Nanotechnology 2004, 15, 501. (32)Pham, D.; Na, K.; Piao, S.; Cho, I.; Jhang, K.; Yoon, E., Wetting behavior and nanotribological properties of silicon nanopatterns combined with diamond-like carbon and perfluoropolyether films. Nanotechnology 2011, 22, 395303.

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Figure captions Figure 1. (a) Schematic diagram showing a droplet which is in the hemi-wicking state spreading on a nano- or microtextured surface. R f and Rb indicate the radii of the film and the bulk. (b) The microcosmic view of a small area of the edge of the film. h f is the film height. Side views of three typical kinds of nano- or microtextured

surfaces patterned with: (c) flat-topped cylindrical pillars, (d) hemispherical-topped cylindrical pillars, (e) conical pillars. The top views of the unit cell of the surfaces are all square, and P, R and H are respectively the centre-to-centre spacing between the regular pillars, the radius and height of each pillar.

Figure 2. Evolutions of the variation of the free energy of the system caused by the transformation of per unit volume of liquid from bulk to film as the function of the film height h𝑓 on the patterned surface with flat-topped cylindrical micropillars. The parameters are from Ref. 25, in which P=40μm, R=8μm, H=20μm when Y = 34º (a) (hemi-wicking), 58º (b) (no wicking) and 109º (c) (no wicking). And P=100μm,

Y

= 34º when R=8μm, H=20μm (d) (no wicking), R=38μm, H=20μm (e)

(hemi-wicking) and R=38μm, H=4μm (f) (no wicking).

Figure 3. Evolutions of the variation of the free energy of the system caused by the transformation of per unit volume of liquid from bulk to film as the function of the film height h f

on the patterned surface with hemispherical-topped cylindrical

micropillars, in which P=40μm, R=8μm and H=20μm when

33


Y

= 34º (a)

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(hemi-wicking), 58º (b) (no wicking) and 109º (c) (no wicking). Compared with (a), P is replaced by 100μm (d) (no wicking), R is replaced by 2μm (e) (no wicking) and H is replaced by 10μm (f) (no wicking).

Figure 4. The stable height of the hemi-wicking film

hstable

for the

hemispherical-topped cylindrical micropillar-parttened surface as a function of the intrinsic contact angle Y when P=40μm, R=8μm and H=20μm.

Figure 5. Evolutions of the variation of the free energy of the system caused by the transformation of per unit volume of liquid from bulk to film as the function of the film height h f on the patterned surface with conical micropillars, in which P=40μm, R=8μm and H=20μm when Y = 34º (a) (hemi-wicking), 58º (b) (no wicking), and 109º (c) (no wicking). Compared with (a), P is replaced by 100μm (d) (no wicking), R is replaced by 4μm (e) (no wicking) and H is replaced by 10μm (f) (no wicking).

Figure 6. The stable height of the hemi-wicking film

hstable for the conical

micropillar-parttened surface as a function of the intrinsic contact angle Y when P=40μm, R=8μm and H=20μm.

Figure 7. The critical contact angles  c as functions of (a) the centre-to-centre spacing P when R=8μm and H=20μm, (b) the pillar radius R when P=40μm and

34


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H=20μm and (c) the pillar height H when P=40μm and R=8μm, for surfaces patterned with flat-topped cylindrical, hemispherical-topped cylindrical and conical pillars.

Figure 8. Comparison of the theoretical results with experimental data for hemispherical and cylindrical pillars (Ref. 31).

Figure 9. Comparison of the theoretical results with experimental data for circular truncated cone pillars (Ref. 32).

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Figure 1

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Figure 2 Y

10

=34°

(b) f-x f-z

5 0 -5

H 0

10

dE/dVt ( 103J/m3)

dE/dVt ( 103J/m3)

(a)

20

Y

10

f-z

5 0 -5

H 0

10

Y

25

=109°

f-z

15 10 5

H

0 0

R=8 m, H=20 m

(d) 10 f-x

20

10

f-x

8

f-z

6 4 2

H

0

20

0

10

hf ( m) R=38 m, H=20 m

10

f-x f-z

5 0 -5

H 0

10

20

hf ( m) (f)

dE/dVt ( 103J/m3)

(e)

20

hf ( m) dE/dVt ( 103J/m3)

dE/dVt ( 103J/m3)

(c)

=58° f-x

hf ( m)

dE/dVt ( 103J/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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f-x

15

f-z

10 5 0 -5

20

R=38 m, H=4 m

20

0

hf ( m)

2

hf ( m)

37


H 4

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Figure 3 Y

10

=34°

(b)

f-x f-z

5 0 -5

H-R 0

10

H

dE/dVt ( 103J/m3)

dE/dVt ( 103J/m3)

(a)

20

Y

10

f-z

5 0 -5

H-R 0

Y

25

=109°

(d) f-x

20

f-z

15 10 5

H-R

0 0

10

H 20

f-x

4

f-z

2 0 -2

H-R 0

f-z

6 4 2 0 -2

(f)

0

H-R H 10 20

dE/dVt ( 103J/m3)

f-x

8

10

H 20

hf ( m)

R=2 m

10

20

P=100 m

6

hf ( m) (e)

10

H


dE/dVt ( 103J/m3)

(c)

=58°

f-x

hf ( m)

dE/dVt ( 103J/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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H=10 m

10 8

f-x

6

f-z

4 2 0

H-R

-2 0

hf ( m)

H 5

hf ( m)

38


10

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Figure 4

25

I

II

(wicking)

20

hf ( m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(no wicking)

H

15

H-R

10 hstable

5 0

c

0

50

100 Y

(degree)

39


150

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Figure 5 Y

10

=34°

(b) f-x

0.05

f-z

0

5

-0.05 -0.1 19.2

19.6

20

0 H 0

10

dE/dVt ( 103J/m3)

dE/dVt ( 103J/m3)

(a)

Y

f-x f-z

5 0

H 0

20

Y

25

=109°

(d)

20

f-x

15

f-z

10 5 H

0 0

10

f-x f-z

4 2 0

20

H 0

2 0 H 0

10

dE/dVt ( 103J/m3)

f-z

4

-2

20

H=10 m

(f) 10 f-x

6

10

hf ( m)

R=4 m

8

20

P=100 m

6

hf ( m) (e)

10


dE/dVt ( 103J/m3)

(c)

=58°

10

hf ( m)

dE/dVt ( 103J/m3)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8

f-x

6

f-z

4 2 0 -2

20

0

5

hf ( m)

hf ( m)

40


H 10

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Figure 6

25 20

hf ( m)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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I

II

(wicking)

(no wicking)

H

15 hstable

10 5 0

c

0

50

100 Y

(degree)

41


150

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Figure 7

(degree)

(a) 90

cylindrical pillars hemispherical pillars conical pillars

c

60 30 0

2R

0

30

60

90

120

P ( m)

(degree)

(b) 90


c

60 30 0

P/2

0

5

10

15

20

R ( m)

(degree)

(c) 90

c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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60 30 0


R

0

30

60

90

H ( m)

42


120

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Figure 8

hemispherical pillars cylindrical pillars hemi-wicking no wicking

12 10

H/R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8 6 4 2 0

2

3

4

5

6

P/R

43


7

8

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Figure 9

2.5 2

H/R

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.5 1 circular truncated cone hemi-wicking no wicking

0.5 0

2

6

10

14

P/R

44


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GRAPHIC ABSTRACT

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Toward a Better Understanding of Hemiwicking: A Simple Model to

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