Applied Thermal Engineering

Applied Thermal Engineering 29 (2009) 372 379 Contents lists available at ScienceDirect Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng Numerical simulations of the equilibrium shape of liquid droplets on gradient surfaces Qiang Liao *, Yong Shi, Yong Fan, Xun Zhu, Hong Wang Institute of Engineering Thermophysics, Chongqing University, Chongqing 400030, PR China article info abstract Article history: Received 23 January 2008 Accepted 6 March 2008 Available online 13 March 2008 Keywords: Equilibrium droplet shape Gradient surface Horizontal and inclined Principle of energy minimum Finite element method The equilibrium shape of liquid droplets on horizontal and inclined plates that have a surface energy gradient is simulated numerically by applying a finite element method based on the principle of energy minimum in the present study. The numerical results show that the liquid droplet shape changes with locations under the influence of the unbalanced surface tension created by the gradient surface. It is shown that the contact angle reaches the maximum value at the one end of the droplet (2D), but it becomes minimum at the other end; the triple-phase contact line deforms toward the region with a smaller contact angle. It is further shown that the length of the liquid droplet increases with an increase in the surface energy gradient on the surface. More interestingly, an inflexion point appears when the droplet length varies with the center contact angle of the droplet, where the liquid droplet just locates at the transition region from the hydrophilic side to the hydrophobic side. It shifts to the hydrophilic side with the increase in the surface energy gradient. On the inclined gradient surface, the gravity induces a significant deformation of the equilibrium droplet shape towards the bottom of the surface. And the surface energy gradient further enhances the deformation when the unbalanced surface tension is directed to the bottom of the surface. However, the droplet shrinks back when the unbalanced surface tension is opposite to the component of gravity. Ó 2008 Elsevier Ltd. All rights reserved. 1. Introduction Materials with gradient surface, being as a new type of materials, have been widely applied to mechanical engineering, chemical engineering and bioengineering et al. in recent years [1]. In contrast to ordinary homogeneous surface, surface energy gradient is created on the surface of the materials. Such a unique feature makes the materials present distinct performances. For example, when vapor condenses on a gradient surface, the difference in surface energy along the surface will give rise to a drive to the condensed liquid droplets, which may be large enough to overcome the friction resistance. As a result, we can observe that liquid droplets move spontaneously on the gradient surface even without any other external forces, e.g. gravity. This provides a new means to enhance condensation heat transfer, especially for condensation on the horizontal surfaces or under condition of zero gravity. Another example is micro/nanofluidic devises [2]. Using materials with the gradient surface for micro/nanochannel, deformation and movement of the liquid droplets can be accurately controlled, satisfying various desires of the micro/nanofluidic devises applications. Generally, gradient surface can be fabricated by chemical vapor deposition method [3,4], contact printing [5], photoirradiation [6]. * Corresponding author. Tel./fax: +86 023 65102474. E-mail address: lqzx@cqu.edu.cn (Q. Liao). The motion mechanism of liquid droplet on the gradient surface was first identified by Greenspan [7] in 1978, and analyzed by Greenspan [7] and Brochard [8] later. The phenomenon of droplets self-motion on the solid surface with gradient wettability was demonstrated experimentally by Chaudhury and Whitesides [3] where the velocity of the droplet was hundreds of times faster than the classical Marangoni flows. Furthermore, Daniel et al. [4] demonstrated that the peak speed of the droplets could reach up to 1.5 m/s under condensation condition and the condensation heat transfer coefficient could remarkably improve on the horizontal gradient surface as compared with that on horizontal general homogenous surface. Liao et al. [9] did visualization experiments on the movement of liquid droplet on the gradient surfaces. They recorded the details of the droplets movement, classified such a movement into a acceleration period and a deceleration period, and discussed the creep behavior of the droplet. These studies made us know more clearly about the droplets movement on the gradient surfaces. However, they have not reflected the overall mechanisms of droplets on the gradient surfaces in different applications. To this end, it is necessary to develop a reliable numerical algorithm for simulating liquid droplets behavior on the gradient surfaces. In the present study, based on the principle of energy minimum, the finite element method developed by Iliev [10] is applied to study on the equilibrium shape of liquid droplets on the gradient surfaces, and the changes of the 1359-4311/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2008.03.003

Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 373 Nomenclature a surface energy gradient ( /mm) b ratio of gravity to surface tension, (q q g )g/c (mm 2 ) g gravity acceleration (m/s 2 ) L length of droplet (mm) n outwards unit vector of gradient surface at triple-phase contact line N i unit vector of virtual displacement p i triangle node r boundary liquid element r 0 boundary gas element U energy of total system (J) V volume of liquid droplet (mm 3 ) Greek symbols a inclination angle of inclined gradient surface ( ) dr virtual displacement of boundary liquid element r at triple-phase contact line dr 0 virtual displacement of boundary gas element r 0 at triple-phase contact line e i module of virtual displacement c surface tension (kg/m) c sl solid liquid interfacial tension (kg/m) c gs solid gas interfacial tension (kg/m) h contact angle ( ) h c contact angle at droplet center ( ) h a advancing contact angle ( ) h R receding contact angle ( ) q density of liquid (kg/m 3 ) q g density of gas (kg/m 3 ) C gas liquid interface C discrete liquid gas interface C sl liquid solid interface C sl discrete liquid solid interface R gradient energy surface X total volume of system X l volume of liquid droplet X l total volume of system involving a discrete liquid surface equilibrium droplet shapes on both horizontal and inclined surfaces under different conditions are discussed. The rest of the article is organized as follows: in Section 2, we briefly introduce the principle of energy minimum as well as corresponding finite element method for determination of the droplet shape on the gradient surface; in Section 3, we apply the finite element method to simulation of the equilibrium droplet shape on both horizontal and inclined gradient surfaces, and discuss the deformation of the liquid droplet under various conditions; finally we draw out remarkable conclusion in Section 4. 2. Mathematic model and finite element method 2.1. Mathematic model Fig. 1 shows a liquid droplet with volume X l on a horizontal gradient surface R, which is the typical scenario of the present simulation. The simulation domain consists of the liquid droplet, the gradient surface and the ambient gas. It is assumed that the liquid droplet is incompressible and the gradient surface is large enough that the droplet always moves on it. Thus the total energy of this system can be expressed as U ¼ U p þ U int ; ð1þ where Z Z Z Z Z Z U p ¼ ðq q g ÞgzdX l þ q g gzdx; X l X Z Z Z Z Z Z U int ¼ cdc þ ðc sl c gs ÞdC sl þ c gs dr: C C sl R ð2þ ð3þ Fig. 1. Simulation domain for a liquid droplet on a horizontal gradient surface. In Eqs. (2) and (3), X, C, and C sl represent total volume of the system, gas liquid interface and liquid solid interface, respectively; g is gravity acceleration; q is density of the liquid; q g is density of gas; c is surface tension; c sl is solid liquid interfacial tension, and c gs is solid gas interfacial tension. For the gradient surface, these interfacial tensions change continuously along the surface. In the present study, q, q g and c are assumed to be constants. To determine the equilibrium shape of the liquid droplet on the gradient surface, it is needed to evaluate the minimum of the total system energy, U, by adopting the variation method. Note that the second integral term on the right side of Eq. (2) and the third integral term on the right side of Eq. (3) are constants, implying that we can obtain an identical equilibrium shape of the droplet by canceling these terms from the original function U. Following this idea, the variational method is directly employed to the following function Z Z Z Z Z Z Z U 0 ¼ ðq q g ÞgzdX l þ cdc þ ðc sl c gs ÞdC sl : ð4þ X l C C sl Actually, since the surface tension c is a constant, Eq. (4) can be further reduced to Z Z Z Z Z Z Z U ¼ bzdx l þ dc cos hdc sl ; ð5þ X l C C sl where U ¼ U 0 =c, b =(q q g )g/c is the ratio of gravity to surface tension and h is contact angle defined as cos½hðx; yþš ¼ c gsðx; yþ c sl ðx; yþ : ð6þ c On the gradient surface, the contact angle h is observed to linearly change with the wettability gradient, i.e., x direction in Fig. 1 [9]. As a result, the contact angle can be given as hðxþ ¼ax þ h 0 ; ð7þ where a is the gradient of the contact angle and h 0 is the contact angle at a reference point x 0.

374 Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 However, to determine the equilibrium shape of the droplet on the gradient surface, the reduced energy function, Eq. (5), should be further subjected to two conditions [10], i.e., incompressible condition Z Z Z X l dx l ¼ V ¼ constant; ð8þ and boundary conditions at the triple-phase contact line dr n P 0; dr 0 n P 0; ð9þ ð10þ where V is volume of the liquid droplet, dr is virtual displacement of the liquid element r, dr 0 is virtual displacement of the gas element r 0 and n is the outward unit normal vector of the gradient surface. Thus, the energy function, Eq. (5), and the corresponding conditions, Eqs. (8) (10) can be solved for the liquid droplets on the gradient surfaces. Then, we shall apply the finite element method [10] to simulation of the equilibrium shape of the liquid droplets on different gradient surfaces in the following sections. 2.2. Finite element method Due to the difficulty in obtaining an analytical solution to Eq. (5) with the conditions of Eqs. (8) (10) for a three-dimensional liquid droplet, the numerical method developed by Iliev [10] is applied to studying the shape of the liquid droplets on the gradient surfaces in the present study. We first discretize the free liquid gas interface C. The resulting surface C is a mesh of triangles T k (k = 1,2,...,R), as shown in Fig. 2. The corresponding triangle node is denoted as p i, where i = 1,2,...,M. Here R is the total number of the triangular surfaces and M is the total number of nodes on the discrete liquid gas interface. Now we move every node on the surface C with a finite displacement. The surface C then evolves into C D, which is C D ¼ C D ðp 1 ; p 2 ;...; p M ; Dp 1 ; Dp 2 ;...; Dp M Þ; ð11þ where Dp i is the virtual displacement of the node p i. Note that Dp i can be further expressed as Dp i ¼ e i N i ; with jn i j¼1; i ¼ 1; 2;...; M; ð12þ where e i and N i are the module and unit vector of the virtual displacement, respectively. With substitution of Eq. (12), Eq. (11) can be further represented as C D ¼ C D ðp 1 ; p 2 ;...; p M ; e 1 ; e 2 ;...; e M ; N 1 ; N 2 ;...; N M Þ: ð13þ It should be noted that although the liquid droplet is displaced, its volume is still a constant due to the incompressible condition, i.e., DV ¼ VðC D Þ VðCÞ ¼0: ð14þ Moreover, the boundary nodes at the triple-phase contact line are required to be kept at the triple-phase contact line even after the displacements, which leads their virtual displacements to being the vectors to the surface R. For the discrete liquid surface, the original internal energy U is approximated as Z Z Z Z Z Z Z UðCÞ UðCÞ ¼ bzdx l þ dc cos hdc sl ; ð15þ X l C C sl where X l is the total volume of the system involving the discrete liquid surface, and C, C sl are the discrete gas liquid interface and liquid solid interface, respectively. Then we turn to determine unit vector, N i, and the module, e i,of the displacement of the node, say p i. As shown in Fig. 3, the evaluation of N i for the nodes at the boundary can be expressed as N i ¼ ðpi p j Þ=jp i p j jþðp i p k Þ=jp i p k j jðp i p j Þ=jp i p j jþðp i p k Þ=jp i p k jj ; ð16þ where p i is the node at the triple-phase contact line, and p j, p k are its neighborhood at the triple-phase contact line. Fig. 4 shows the schematic of the interior nodes. It is assumed that an interior node, i.e., p i, is an intersection of surfaces with number l, which are marked as surfaces T i 1 ; T i 2 ;...; T i l 2 C. The corresponding neighborhood nodes are p 1, p 2,..., p l. As suggested in Ref. [10], a vector c j is introduced as c j ¼ p i p j ðj ¼ 1; 2;...; lþ: ð17þ With this c j, the outward unit normal vector of each surface T i j (j = 1,2,...,l), n j (j = 1,2,...,l), can be calculated as n 1 ¼ cl c 1 jc l c 1 j ; n2 ¼ c1 c 2 jc 1 c 2 j ; n3 ¼ c2 c 3 jc 2 c 3 j ; ; n l ¼ cl 1 c l jc l 1 c l j : ð18þ As a result, the N i of the interior node p i is then obtained in terms of n j as N i Xl n j = Xl n j : ð19þ j¼1 j¼1 Now we evaluate the module of each node. The condition of constant volume of the liquid droplet results in at least two nodes Fig. 3. Schematic of the nodes on the boundary. Fig. 2. The discrete surface C with a mesh of triangles. Fig. 4. Schematic of the interior nodes.

Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 375 being moved spontaneously during the process of the virtual displacement. Suppose the node p i makes a virtual displacement Dp i = e i N i. Therefore, the volume of the droplet is changed as DVðC; e i ; N i Þ¼VðC; e i ; N i Þ VðCÞ: ð20þ To keep the volume being a constant, another node p j (j i) must be moved spontaneously to remove the volume change calculated by Eq. (20), i.e., DVðC; e i ; N i ; e j ; N j Þ¼VðC; e i ; N i ; e j ; N j Þ VðC; e i ; N i Þ ¼ DVðC; e i ; N i Þ: ð21þ Combining Eqs. (20) with (21), we thus have e j ¼ DVðC; e i; N i Þ DVðC; e i ; N i ; N j Þ : ð22þ With the help of Eqs. (16), (19) and (22) the equilibrium state of the liquid droplet on the gradient surface can be simulated by solving Eq. (15). The corresponding iterative process is listed as follows: 1. Discretize the interface of the original liquid droplet C 0 to C 0 ðp 1 0 ; p2 0 ; ; pm 0 Þ, and choose the initial step of the simulation e 0, the minimum step of the simulation e c 2. Calculation in the kth iterative process 2.1 Choose the node p i,16i6m, and make a corresponding virtual displacement Dp i = e k N i. 2.2 Choose the node p j, j i,16 j 6 M, and calculate e j by using Eq. (22). Make a virtual displacement of p j as Dp j = e j N j. 2.3 Calculate DU ¼ DUðC k 1 ; e k ; N i ; e j ; N j Þ DUðC k 1 Þ. 2.3.1 If DU < 0, save the results and replace C k 1 by the newly obtained surface C0 k 1 ðc; e k ; N i ; e j ; N j Þ. 2.3.2 If DU P 0, go back to Step 2.2 and rechoose the node p j, redo the corresponding calculation based on the new chosen node p j. 2.3.3 If DU P 0 for all p j, j i,, go back to Step 2.1, and rechoose the node p i, redo the corresponding calculation based on the new chosen node p i. 2.3.4 If for all pair (p i,p j ), DU P 0, save the results, and go to Step 3. 3. Determine the magnitude of e k 3.1 If e k < e c, iteration is over and save the results. 3.2 If e k P e c, set e k+1 = e k /2, then go back to Step 2 and start the k + 1th iteration. 3. Numerical results and analysis In Section 2, the mathematic model and the corresponding numerical algorithm are described for simulation of the liquid droplets on the gradient surfaces, and they will be applied to studying the equilibrium state of the liquid droplets on horizontal and inclined gradient surfaces under various conditions. Fig. 5. The liquid droplet on the gradient surface with a = 6 /mm, h 0 =90, b = 1mm 2, V =1lL. L ¼jx 1 x 2 j; ð23þ with x 1 and x 2 being the x coordinates of two end points of the triple-phase contact line (2D). The other parameter is the contact angle at the droplet center, which is given as x 1 þ x 2 h c ; 0 2 ¼ aðx 1 þ x 2 Þ þ h 0 ; 2 ð24þ and is used to be an indication of the wettability of where the droplet locating. Figs. 6 and 7 show the droplets shape and the corresponding triple-phase contact lines for different centre contact angle, h c, on the gradient surface, where a = 5 /mm, b = 0.5 mm 2 and V =2lL. It is seen that the contact angle of each droplet reaches the maximum value at the left end point while the minimum value at the right end point at the triple-phase contact line, cf. Fig. 6. The height of the liquid droplet gradually decreases when the droplet deforms from the hydrophobic side to the hydrophilic side. These numerical results clearly demonstrate that the change of surface energy affects the equilibrium state of the liquid droplets. It causes the liquid droplet to present different equilibrium shape when it locates at different positions on the gradient surface. Figs. 8 and 9 show the variation in the droplet length L as a function of the centre contact angle for cases of a =2 /mm, 5 / mm, 8 /mm, where the conditions of b = 0.13 mm 2, V =1lL and b = 0.9 mm 2, V =2lL are set, respectively. It is shown the droplet length decreases rapidly when the centre contact angle increases for all cases. This is consistent with our understanding that the droplet shrikes when it locates from the hydrophilic side to the hydrophobic side. Moreover, it is also found that a large wettability gradient induces a long liquid droplet, especially in the zone with large h c. However, it should be noted that the length of the droplet 3.1. Liquid droplets on horizontal gradient surfaces We first simulate a liquid droplet with volume of 1 ll on a horizontal gradient surface. For this case, we choose a = 6 /mm, h 0 =90, b =1mm 2, and set x coordinate to follow the gradient of surface energy and y coordinate normal to the gradient surface. Fig. 5 shows the resulting equilibrium droplet shape on the gradient surface. It is seen that the droplet deforms toward the hydrophilic side along x coordinate due to the influence of the unbalance surface tension created by the gradient surface, while it still keeps symmetrical with respect to y coordinate. To more clearly reveal the deformation of the droplet caused by the wettability gradient, we introduce two parameters. One is the length of the droplet, L, which is defined as Fig. 6. The droplet shape on the gradient surface with different h c when a = 5 / mm, b = 0.5 mm 2, and V =2lL.

376 Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 Fig. 10. Variation of the droplet length with the centre contact angle for different b when a =5 /mm, V =2lL. Fig. 7. The triple-phase contact line of the droplet on the gradient surface with different h c when a = 5 /mm, b = 0.5 mm 2, V =2lL. when the liquid droplet locates in the transition region from the hydrophilic side to the hydrophobic side. Specifically speaking, it is found that the advancing contact angle h a <90 while the receding contact angle h R >90 when the droplet locates in this transition region. The inflexion point becomes more noticeable for a large a and shifts to the hydrophilic side with increasing a. We also investigate the influence of b on the equilibrium shape of the droplets. Figs. 10 and 11 show the numerical results of the droplet length varying with the centre contact angle for the cases of (a =5 /mm, V =2lL) and (a =8 /mm, V =1lL) with Fig. 8. Variation of the droplet length with the centre contact angle for different a when b = 0.13 mm 2, V =1lL. Fig. 11. Variation of the droplet length with the centre contact angle for different b when a =8 /mm, V =1lL. Fig. 9. Variation of the droplet length with the centre contact angle for different a when b = 0.9 mm 2, V =2lL. with different a are almost identical when h c <35, indicating that the wettability gradient has rather weak influence on the equilibrium droplet shape when the droplet locates on highly hydrophilic surface. More interestingly, Figs. 8 and 9 also show that an inflexion point appears in the variation of the droplet length when h c increases. The results show that this inflexion point always appears Fig. 12. Variation of the droplet length with the centre contact angle for different V when a =5 /mm, b = 0.13 mm 2.

Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 377 b = 0.13 mm 2, 0.50 mm 2 and 1.00 mm 2, respectively. It can be found that an increase in b enhances the effect of gravity on the equilibrium state of the droplets. Under the same condition, a large b leads to a long droplet. Even for the region with a small h c, the effect of gravity is quite noticeable, leading to different equilibrium droplet shape for different b. However, the corresponding inflexion points in the variation of the droplet lengths appear at the locations with almost same h c for different b for all cases, and appear at the locations with different h c for different a and V. In addition, the effect of the droplet volume on the shape is studied. Figs. 12 and 13 give the numerical results of the droplet length varying with the centre contact angle for different droplet volume under the conditions of (b = 0.13 mm 2, a =5 /mm) and (b =1mm 2, a =2 /mm), respectively. It can be found that the variation of the droplet length with h c for different droplet volume is quite similar to our previous discussion. The location of the inflexion point in the variation of L shifts to the region with a small h c, cf. Fig. 12. However, the inflexion points almost disappear for the case under the condition of b =1mm 2, a =2 /mm, cf. Fig. 13. 3.2. Liquid droplets on inclined gradient surfaces In this section, we shall apply the virtual displacement method to simulation of liquid droplets on inclined gradient surfaces. It should be pointed out that a liquid droplet can not stand on an inclined gradient surface due to the gravity if we just use the method described in previous section. It, as rigid body, will move constantly along the inclined surface. Therefore, we will develop a new constraint for the liquid droplets on the inclined gradient surfaces. We suppose a liquid droplet at equilibrium state standing on the inclined gradient surface with inclination angle a, as shown in Fig. 14. Now the liquid droplet, as a rigid body, moves along the inclined surface with a distance of dx. The change of the total potential energy is thus given as du ¼ qgvdx sin a: ð25þ Meanwhile, it is seen that the corresponding triple-phase contact line also moves along the inclined gradient surface with a distance of dx, cf. Fig. 15. This movement causes that the original gas solid interface on the front side of the triple-phase contact line has been replaced by the liquid solid interface. At the same time, the liquid solid interface on the back side of the triple-phase contact line has been replaced by the gas solid interface. It is known this interface transformation will change the energy of the system. Therefore, we have a constraint condition for the droplet movement on the inclined gradient surface as qgvdx sin a > dw f þ dw b ; δ Γ sl δ Γ gs Fig. 15. Comparison of the triple-phase contact line of the droplet before and after its movement on the inclined gradient surface. ð26þ where dw f ¼ c a d jc sl j; dw b ¼ c a d jc gs j; ð27þ ð28þ and c a = c + c gs c sl, dw f and dw b are the work done on the front side and the back side, respectively. For more general cases, we can further obtain du > gd jc sl jþgd jc gs j; ð29þ where g = c a /c, which can be evaluated by Young s equation: g ¼ 1 þ cos h: ð30þ In summary, for the liquid droplets on the inclined gradient surfaces, we have two conditions, i.e., Fig. 13. Variation of the droplet length with the centre contact angle for different V when a =2 /mm, b =1mm 2. du < 0; du > gðd jc sl jþd jc gs jþ: ð31aþ ð31bþ Fig. 14. Movement of the liquid droplet on the inclined surface. We shall apply these conditions to simulation of the equilibrium state of the liquid droplets on the inclined gradient surfaces. Fig. 16 shows the numerical results of the equilibrium shape of the liquid droplet with volume of 1 ll on the inclined gradient surface with inclination angle of 30, a = 4 /mm, and h 0 =75 when b =1mm 2. It can be found that the droplet on the gradient surface has an asymmetrical equilibrium shape due to the effects of both gravity and unbalance surface tension created by the gradient surface. To clearly reveal the gravity effect, Figs. 17 and 18 show the comparison of the droplets shape and the corresponding triplephase contact lines on the gradient surfaces with inclination angle of 0, 30, 60, respectively, when V =1lL, b =1mm 2, a =2 /mm and h 0 =90. It is seen that the droplet has started to deform since

378 Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 Fig. 19. Comparison of the shape of the droplets on the inclined surface with different b. Fig. 16. The equilibrium shape of the liquid droplet on an inclined gradient surface. Fig. 20. Comparison of the triple-phase contact line of the liquid droplets on the inclined surface with different b. Fig. 17. Comparison of the triple-phase contact line of the liquid droplets on the inclined surfaces with different inclination angles. Figs. 19 and 20 discuss the effect of the ratio of gravity to surface tension, b, on the equilibrium droplet shape on the inclined gradient surface, where the droplets with V = 1 ll on the inclined gradient surface with a =45, a = 4 /mm, and h 0 =90 are simulated. It can be found that when b is very small, the effect of gravity is neglected while the surface tension takes control of the equilibrium droplet shape. The shape and the triple-phase contact line are almost symmetrical. However, when b increases, the front part of the droplet begins to deform towards the bottom of the surface. Similar results with those in Figs. 17 and 18 are also observed. This indicates that the increases in b and a enhance the deformation of the droplet due to the effect of gravity. Fig. 18. Comparison of the shape of the droplets on the inclined surfaces with different inclination angles. the inclination angle increases from 0. Specifically, although the back parts of the droplets in three cases are still identical, the front parts of the droplets differ from each other, cf. Fig. 17. They move towards the bottom of the inclined gradient surface, and such a movement varies with the inclination angles. It is found that a large inclination angle induces a small curvature for the front part of the droplet and a long droplet length. In Fig. 18, we can also find that the droplet height has been changed with the increase in the inclination angle. The symmetry of the droplet shape has been broken when the inclination angle increases. The height for the back part of the droplet gradually decreases while that for the front part increases with increasing inclination angle. Fig. 21. Comparison of the triple-phase contact line of the liquid droplets on the inclined surfaces with different a.

Q. Liao et al. / Applied Thermal Engineering 29 (2009) 372 379 379 Fig. 22. Comparison of the shape of the droplets on the inclined surfaces with different a. Furthermore, the effect of the surface energy gradient on the droplet shape is investigated. Figs. 21 and 22 give the comparison of the triple-phase contact lines and the droplets shape for three cases with a = 4, 0, 4 /mm respectively when the droplet with V =1lL locates on the inclined gradient surfaces with h 0 =90, b =1mm 2, and a =45. It can be seen that the surface energy gradient shows remarkable effects on the droplet shape. When a is positive, e.g., a = 4 /mm, the unbalance surface tension is directed to the bottom of the surface, which is consistent with the effect of the gravity. As a result, the droplet deforms more significantly towards the bottom of the surface with lower droplet height than that on the homogenous inclined surface (a = 0 /mm). On the contrary, when a is negative, e.g. a = 4 /mm, the unbalance surface tension is opposite to the component of the gravity. For this condition, the surface energy gradient will counteract partly the effect of the gravity on the liquid shape, resulting in the droplet shrinking back towards the upside of the surface with high droplet height. 4. Conclusion In the present study, the finite element method based on the principle of energy minimum is applied to the simulation of equilibrium shape of the liquid droplets on the horizontal and the inclined gradient surfaces. The salient conclusions are summarized as follows: 1. On the horizontal gradient surfaces, the liquid droplet shape changes with locations. The contact angle reaches the maximum value at the one end of the droplet (2D), but it becomes minimum at the other end. The corresponding triple-phase contact line deforms toward the region with a small contact angle. 2. On the horizontal gradient surfaces, the length of the liquid droplet increases with an increase in the surface energy gradient. However, the length of the liquid droplet becomes insensitive to the surface energy gradient when h c <35. Moreover, it is interesting to note that an inflexion point appears in the variation of the droplet length as a function of the center contact angle when the liquid droplet locates in the transition region from the hydrophilic side to the hydrophobic side. It becomes more noticeable for a large a and shifts to the hydrophilic side with increasing a. 3. On the inclined gradient surfaces, the gravity, represented by the parameters a and b, induces a noticeable deformation of the equilibrium droplet shape. It leads the droplet to deform towards the bottom of the surface. 4. On the inclined gradient surfaces, the surface energy gradient enhances the deformation of droplet towards the bottom of the surface with lower droplet height when the unbalance surface tension is directed to the bottom of the surface. However, the droplet shrinks back towards the upside of the surface with high droplet height when unbalance surface tension is opposite to the x component of the gravity. Acknowledgements Authors are grateful for the support of National Natural and Science Foundation of China (No. 50276072), NCET-07-0912 and CSTC-2007BB0188. References [1] H. Elwing, S. Welin, A. Askendal, I. Lundström, Adsorption of fibrinogen as a measure of the distribution of methyl groups on silicon wafers, Journal of Colloid and Interface Science 123 (1988) 306. [2] M. Grunze, Surface science: driven liquids, Science 283 (5398) (1999) 41. [3] M.K. Chaudhury, G.M. Whitesides, How to make water run uphill, Science 256 (6) (1992) 1539. [4] S. Daniel, M.K. Chaudhury, J.C. Chen, Fast drop movement resulting from the phase change on a gradient surface, Science 291 (1) (2001) 633. [5] S.-H. Choi, B.-M.Z. Newby, Micrometer-scaled gradient surfaces generated using contact printing of octadecyltrichlorosilane, Langmuir 19 (18) (2003) 7427. [6] K. Ichimura, S.-k. Oh, et al., Light-driven motion of liquids on a photoresponsive surface, Science 288 (5471) (2000) 1624. [7] H.P. Greenspan, On the motion of a small viscous droplet that wets a surface, Journal of Fluid Mechanics 84 (1) (1978) 125. [8] F. Brochard, Motions of droplets on solid surfaces induced by chemical or thermal gradient, Langmuir 5 (1989) 432. [9] Q. Liao, H. Wang, X. Zhu, et al., Experimental study on movement characteristics of droplet on the surface with gradient surface energy, Journal of Engineering Thermophysics 28 (1) (2007) 134. [10] S.D. Iliev, Iterative method for the shape of static drops, Computer Methods in Applied Mechanics and Engineering. 126 (3 4) (1995) 251.