pubs.acs.org/Langmuir © 2009 American Chemical Society
Simulation of Drop Movement over an Inclined Surface Using Smoothed Particle Hydrodynamics Arup K. Das and Prasanta K. Das* Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India Received April 3, 2009. Revised Manuscript Received August 17, 2009 Smoothed particle hydrodynamics (SPH) is used to numerically simulate the movement of drops down an inclined plane. Diffuse interfaces have been assumed for tracking the motion of the contact line. The asymmetric shape of the three-dimensional drop and the variation of contact angle along its periphery can be calculated using the simulation. During the motion of a liquid drop down an inclined plane, an internal circulation of liquid particles is observed due to gravitational pull which causes periodic change in the drop shape. The critical angle of inclination required for the inception of drop motion is also evaluated for different fluids as a function of drop volume. The numerical predictions exhibit a good agreement with the published experimental results.
Introduction Sliding of drops over inclined planes is not only a topic of fundamental interest but al so relevant in many technological and biological applications. While fast droplet movement over wind shields, solar panels, and greenhouse covers is desirable, effective utilization of pesticides calls for a greater stability of droplets over plant foliages.1,2 Stability of droplets is crucially important in the transition from dropwise to filmwise condensation3 over an inclined surface, and it is also important in printing and coating techniques. Besides, manipulation of droplet movement over terrained4 surfaces is of great interest in droplet microfluidics. In recent years, there has been interest in building superhydrophobic surfaces.5 The basic understanding of droplet sliding can provide useful information about the design and development of such surfaces. Sliding of droplets is also important for the design of condensing surfaces. In general, the shape, stability, and motion of the liquid drops over solid surfaces rely on a number of physicochemical phenomena, all of which are not well understood. An explanation of certain drop behaviors needs the consideration of interactions even at a molecular level. Nevertheless, tireless efforts have been made by the researchers over the years to describe the same in terms of macroscopic forces and properties. One such effort can be traced back to the pioneering Young-Laplace equation6 given as follows: σlg cos θ ¼ σsg - σ sl
ð1Þ
where σab is the surface tension between phase a and phase b. Solid, liquid, and gases are expressed as s, l, and g in eq 11, respectively. The contact angle at the triple line is represented by angle θ. However, the situation becomes different if the solid plate makes an *To whom correspondence should be addressed: Indian Institute of Technology, Kharagpur 721302, India. Telephone: þ91-03222-282916. Fax: þ91-03222-282278. E-mail:
[email protected]. (1) Pieters, J. G.; Deltour, J. M.; Debruyckere, M. J. Agric. For. Meteorol. 1997, 85, 51–62. (2) Wirth, W.; Storp, S.; Jacobsen, W. Pestic. Sci. 1991, 33, 411–422. (3) Oron, A.; Bankoff, S. G. Phys. Fluids 2001, 13, 1107–1117. (4) Abdelgawad, M.; Freire, S. L. S.; Yang, H.; Wheeler, A. R. Lab Chip 2008, 8, 672–677. (5) Osawa, S.; Yabe, M.; Miyamura, M.; Mizuno, K. Polymer 2006, 47, 3711– 3714. (6) Young, T. Philos. Trans. R. Soc. London 1805, 95, 65–87.
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angle with the direction of gravity. Surface force needs to balance the component of the gravitational pull along the plane. Equation 11 is to be modified to accommodate the effect of inclination and gravity. As a consequence, the drop no longer remains axisymmetric, and eventually, it begins to slide down with the increase in plate inclination. Numerous efforts7-17 have been made to determine the drop shape over an inclined plane, the inception of its sliding, and its sliding velocity. In this manner, contact angle hysteresis18 is a well-accepted fact. Contact angle hysteresis refers to the difference in contact angles at the advancing (θa) and receding (θb) front of the drop resting on an inclined plane. For a drop resting on the inclined plane the following condition is satisfied: θa < θ < θb. Beyond a critical inclination, the drop starts sliding when this relationship is violated.18 While the contact angle hysteresis is being treated, the surface is assumed to be homogeneous in this work. It may be noted that for a heterogeneous surface the local contact angle also depends on the surface condition in addition to the plate inclination. Most of the experimental and theoretical studies7-17 aim to analyze the static or at best the quasi-static drop shapes just at the yield point. On the basis of the asymptotic theory,18 drop movement after the yield condition has also been investigated. Different postulations have been made to explain drop sliding over inclined planes. Frenkel19 explained the drop sliding considering the “pouring of the liquid” from the rear edge of the drop to its front edge along the fluid-fluid interface. He further stated that the drop leaves a thin unstable liquid film behind in the case of perfect (7) Dussan, V. E. B.; Chow, R. T. P. J. Fluid Mech. 1983, 137, 1–29. (8) Podgorski, T.; Flesselles, J. M.; Limat, L. Phys. Rev. Lett. 2001, 87, 036102– 036105. (9) Le Grand, N.; Daerr, A.; Limat, L. J. Fluid Mech. 2005, 541, 293–315. (10) Larkin, B. K. J. Colloid Interface Sci. 1967, 23, 305–312. (11) Brown, R. A.; Orr, F. M.; Scriven, L. E. J. Colloid Interface Sci. 1980, 73, 76–87. (12) Lawal, A.; Brown, R. A. J. Colloid Interface Sci. 1982, 89, 332–345. (13) Rotenberg, Y.; Boruvka, L.; Neumann, A. W. J. Colloid Interface Sci. 1984, 102, 424–434. (14) Durbin, P. J. Fluid Mech. 1988, 197, 157–169. (15) Iliev, S. D. J. Colloid Interface Sci. 1997, 194, 287–300. (16) Kim, H. Y.; Lee, H. J.; Kang, B. H. J. Colloid Interface Sci. 2002, 247, 372– 380. (17) Thiele, U.; Neuffer, K.; Bestehorn, M.; Pomeau, Y.; Velarde, M. G. Colloids Surf., A 2002, 206, 87–104. (18) Krasovitsky, B.; Marmur, A. Langmuir 2005, 21, 3881–3885. (19) Frenkel, Y. I. J. Exp. Theor. Phys. 1948, 18, 659–668.
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wetting. Though heterogeneity of the solid surface is thought18,20-22 to be the cause of a drop being pinned on an inclined surface, there have also been alternate explanations.23 One of the initial attempts to predict the non-axisymmetric shape of liquid drops over a sloped solid surface was made by Larkin.10 He imposed a polar function of contact angle along the triple line to yield a partial differential equation of capillarity. Brown et al.11 assumed the contact line to be circular and employed a Galerkin finite element technique for predicting the shape of the drop on an inclined plane. Following the same procedure, effort12 has also been made to evaluate the drop profile considering an oval contact line. Rotenberg et al.13 used an experimental functional relationship between the spatial contact angle and contact line velocity as the lower boundary. They solved a system of finite element equations which was obtained by minimizing the surface energy potential of a liquid drop. Durbin14 considered velocity slip near the contact line and assumed interface yield stress as the shear stress near the triple line. An equilibrium variation approach is used by Iliev15 to track the virtual motion of the threephase contact line. He modeled the effect of “drop holdup” on the contact surface by establishing the counteraction between the separating media. Kim et al.16 reported the sliding velocity of a liquid drop of known wetting characteristics by using a scaling law and established the distortion of free surface during its movement. Thiele and co-workers17,24 combined diffused interface theory and long wave approximation to study one-dimensional periodic drop profiles sliding down an inclined plane. They also extrapolated their model for two-dimensional drops and noticed front instabilities of the advancing drop front. Though a large volume of work exists on drop shape, its spreading, and its stability on a tilted plane, insufficient effort25 has been spent in analyzing the movement of the drop down the plane. Gao and McCarthy25 postulated two mechanisms for drop motion. Droplets can move by sliding where the particles near the solid-liquid interface exchange their position with those at the gas-liquid interface while the bulk of the fluid remains unaffected. The particle movement along the gas-liquid and solid-liquid interfaces is similar to the motion of a tread of a caterpillar tank. On the other hand, there could be rolling motion where the entire fluid mass undergoes a circulatory movement. They did not rule out a combination of these two modes of motion in the case of an actual drop movement. However, this postulation has not been explicitly demonstrated. It is worth mentioning that many of the analytical models15-17 simulate two-dimensional drop or idealized the three-dimensional drop as a planar drop.18 The simulation of a three-dimensional drop constitutes a considerable level of complexity. In general, the problem can be solved only by a suitable numerical scheme. Apart from the three dimensionality of the problem, a common difficulty experienced by such techniques is the stress singularity at the triple line. A number of remedies have been prescribed for this. As the singularity arises from the small length scale associated with the triple line, incorporation of a precursor film,22 surface tension relaxation,26 and slip length based level set27 methods have been suggested. Modified VOF was used by Huang et al.28 to investigate (20) Dettre, R. H.; Jhonson, R. E. Adv. Chem. Ser. 1964, 43, 136–144. (21) Joanny, J. F.; de Gennes, P. G. J. Chem. Phys. 1984, 81, 552–562. (22) de Gennes, P. G. Rev. Mod. Phys. 1985, 57, 827–863. (23) Roura, P.; Fort, J. Phys. Rev. E 2001, 64, 0116011–0116015. (24) Thiele, U.; Velarde, M. G.; Neuffer, K.; Bestehorn, M.; Pomeau, Y. Phys. Rev. E 2001, 64, 0616011–0616012. (25) Gao, L.; McCarthy, T. J. Langmuir 2006, 22, 6234–6237. (26) Shikhmurzaev, Y. D. J. Fluid Mech. 1997, 334, 211–249. (27) Splet, P. D. M. J. Comput. Phys. 2005, 207, 389–404. (28) Huang, H.; Meakin, P.; Liu, M. Water Resour. Res. 2005, 41, W12413.1– W12413.12.
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the wetting of fracture walls by imposing contact angles near the contact lines. Liu et al.29 used dissipative particle dynamics to model flow of wetting and nonwetting liquids through porous media by controlling the fluid-fluid and fluid-solid interaction strengths. Alternately, the sharp interface is replaced by a diffuse interface in a number of attempts.30-32 In this work, we have tried to simulate the deformation, stability, and motion of a three-dimensional liquid drop over a solid surface. Hybrid diffuse interface-smoothed particle hydrodynamics (DI-SPH) is used for the discretization of flow field into particles in both phases. The technique of smoothed particle hydrodynamics (SPH) was conceived by Lucy33 and further developed by Gingold and Monaghan34 for treating astrophysical problems. Because of its lagrangian nature, SPH is subsequently used in different types of fluid dynamics problems like flow of viscous fluids,35 evolution of a free surface,36 and low-Reynolds number incompressible flow.37 Different modifications have also been made from time to time in the basic SPH methodology to tackle the real physics behind the continuum mechanics. Chen et al.38 proposed a density reinitialization approach for a class of problems where density gradient is much smaller than that of the smoothing kernel. This approach is known as the corrective smoothed particle method. It normalizes the kernel and particle approximation to reduce the possible error in approximating the continuum into discrete particles. The field function and its derivatives are calculated more accurately by a matrix equation. Symmetrical surface boundary condition has been modeled by Libersky and Petschek39 using ghost particles along and outside the boundary. The continuum surface force concept has been incorporated by Morris40 for simulation of surface tension acting at an interface between two fluids. Tartakovsky and Meakin41 combined pairwise fluid-fluid and fluid-solid particle-particle interactions along with standard SPH equations to simulate three-phase contact line dynamics. Melean et al.42 used SPH with a modified stress tensor to model a circular van der Walls liquid drop. Their model eliminated the unphysical clustering of particles near the interface due to tensile instability. Later, Liu and Liu43 proposed a new approach based on Taylor series expansion for kernel approximation of a function to restore particle inconsistency. Recently, Liu and Liu44 undertook a very extensive review of the subject and its subsequent developments for different types of flow situations. To date, the description of the SPH methodology assumes the interface as a surface of zero thickness separating two fluids (sharp interface). Another concept of treating the interface is to allow it to stretch over a narrow region characterized by a smooth but rapid variation of physical properties between the bulk values (29) Liu, M.; Meakin, P.; Huang, H. Water Resour. Res. 2006, 43, W04411.1– W04411.12. (30) Jacqmin, D. J. Comput. Phys. 1996, 155, 96–127. (31) Anderson, D. M.; McFadden, G. B.; Wheeler, A. A. Annu. Rev. Fluid Mech. 1998, 30, 139–165. (32) Ding, H.; Spelt, P. D. M. J. Fluid Mech. 2007, 576, 287–296. (33) Lucy, L. B. J. Astron. 1977, 88, 1013–1024. (34) Gingold, R. A.; Monaghan, J. J. Mon. Not. R. Astron. Soc. 1977, 181, 375– 389. (35) Takeda, H.; Miyama, S. M.; Sekiya, M. Prog. Theor. Phys. 1994, 92, 939– 960. (36) Monaghan, J. J. J. Comput. Phys. 1994, 110, 399–406. (37) Morris, J. P.; Fox, J. P.; Yi, Z. J. Comput. Phys. 1997, 136, 214–226. (38) Chen, J. K.; Beraun, J. E.; Jih, C. J. Comput. Mech. 1999, 23, 279–287. (39) Libersky, L. D.; Petschek, A. G. J. Comput. Phys. 1992, 109, 67–75. (40) Morris, J. P. Int. J. Numer. Methods Fluids 2000, 33, 333–353. (41) Tartakovsky, A.; Meakin, P. Phys. Rev. E 2005, 72, 206301-1–206301-9. (42) Melean, Y.; Sigalotti, L. D. G.; Hasmy, A. Comput. Phys. Commun. 2004, 157, 191–200. (43) Liu, M. B.; Liu, G. R. Appl. Numer. Math. 2006, 56, 19–36. (44) Liu, G. R.; Liu, M. B. Smoothed Particle Hydrodynamics: A mesh free particle method; World Scientific: Singapore, 2003.
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of the two fluids. It is commonly known as diffuse interface (DI) and has successfully been incorporated in different grid-based techniques,45,46 but diffuse interface is not well-established for particle-based techniques because of its inherent diffusive properties inside the domain of influence. The object of this study is to simulate asymmetric shapes and contact line dynamics of a liquid drop over an inclined plane using a hybrid DI-SPH methodology. The focus is on the estimation of the sliding limit of the inclination angle for drops of different sizes and fluid-solid combinations. The numerical results have been compared well with experimental data taken from various sources. Efforts have also been made to study the internal fluid structure of the drop over the solid surface during the rolling motion.
continuous. For the boundary condition of the color (C), no flux condition near the wall and diffusively controlled local equilibrium at the interface are taken into account. SPH is used for the numerical discretization of the equations. In SPH, the fluid continuum is assumed as an assemblage of interconnected particles having individual mass and thermophysical properties. The properties of any particle are influenced by its neighbor by using an appropriate smoothing kernel, W(x, h),44 and any change in these properties obeys the conservation laws. Here, x is the spacing between the particle of interest and its neighboring particle. The span of the area of influence for a particle is commonly known as the smoothing length (h). Using the smoothing function, any property f(x) can readily be expressed in the following form:44
Model Development
Z
Sliding of a liquid drop over an inclined plane can be described by a pair of lagrangian mass and momentum equations for each of the phases. The concept of DI47 has been invoked to achieve a smooth variation of properties across the interface. DI modeling has been done through the Cahn Hilliard30 equation suitably modified for the particulate system. Liquid and gas particles are distinguished by a color code (C) which assumes a value of þ1 for liquid and -1 for gas. The color (C) of the solid boundary is assigned as -2. As there is no mass transfer between the phases, the continuity equation can be written without any source term as follows: DF ¼ -Frv Dt
ð2Þ
Chemical potential (φ) is adopted in the momentum equation to incorporate the concept of diffuse interface. Chemical potential φ is calculated for each phase as given below: φ ¼ rðC þ 1Þ2 ðC - 1Þ2 - ðCn2 r2 CÞ
ð3Þ
The second term in the definition of chemical potential signifies the affinity of the fluid phase for a solid surface. Different values of C can be used to simulate different affinities of the liquid and gas phase for the solid. The Cahn number Cn is defined as the ratio of the mean interfacial thickness (Δi) to the equivalent spherical radius of the drop. For this problem, Δi is taken to be 10 μm. Incorporating surface energy at the fluid-fluid interface, the conservation equation for the momentum can be written as Dv C ¼ -rpþrτþFgz rφ ð4Þ F Dt CaCn h R i Rβ Dv Dvβ 2 where τRβ ¼ μi Dx is the shear stress tensor. β þ DxR - 3 ðrvÞδ Approximating interfacial diffusion fluxes proportional to the chemical potential gradients, the conservation equation for color (C) can be written as follows. DC ¼ kr2 φ Dt
ð5Þ
A precise boundary condition is required near the solid wall and at the gas-liquid interface. For the velocity boundary condition, no slip and no penetration at the wall are assumed. Near the interface, the velocity and stress are considered to be (45) Braun, R. J.; Mcfadden, G. B.; Coriell, S. R. Phys. Rev. E 1994, 49, 4336– 4352. (46) Koplik, J.; Banavar, J. R. Phys. Rev. Lett. 1998, 80, 5125–5128. (47) Antanovskii, L. K. Phys. Fluids 1995, 7, 747–753.
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f ðxÞ ¼
Ω
f ðx0 ÞWðx - x0 , hÞ dx0
ð6Þ
After performing kernel approximation44 and particle approximation,44 we can write a function associated with the ith particle as f ðxi Þ ¼
N X mj j ¼1
Fj
f ðxj ÞWðxi - xj , hÞ
ð7Þ
Following the same argument, the first derivative of a function f(x) can be written as Df ðxÞ ¼ Dx
Z
f ðx0 Þ DWðx - x0 , hÞ 0 Fðx Þ dx0 0 Dx Ω Fðx Þ
ð8Þ
Next, we describe the computation of different properties like density, chemical potential, color code, and viscosity based on a particle approach. In principle, the information regarding density can be obtained from the continuity equation through a “continuity density approach”.44 However, there is an alternate procedure for the calculation of density. On the basis of eqs 66-88, the density field (F) of the particulate system is evaluated using the summation density approach:44 N P
Fi ¼
j ¼1
mj Wij ð9Þ
N P mj F Wij
j ¼1
j
where mj is the mass and Fj is the corresponding density of the jth particle of a system of N particles. Though the summation density approach requires more computational power, it strictly ensures the conservation of mass. We have adopted this approach in this work. Chemical potential (φei) has been calculated for the particlebased system in the following manner: φi ¼
N X m 3 j ðCi þ 1Þ2 ðCi - 1Þ2 - ðCj þ 1Þ2 ðCj - 1Þ2 DWij FF jrij j Drij j ¼1 i j
-
N X m 3 j ðCni 2 þ Cnj 2 ÞðCi - Cj Þ D2 Wij jrij j FF Drij 2 j ¼1 i j
ð10Þ
where rij is the spacing between particles i and j. Equating the summation of the gradient and bulk energy of the fluids with the DOI: 10.1021/la901172u
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mobility of fluid particles, we can write the transport equation of C as follows: N X dCi m 3 j kðφi - φj Þ D2 Wij ¼ dt FF Drij 2 jrij j j ¼1 i j
Frij
where k is the mobility of the ith particle and assumed to be 10 for this problem. Using kernel approximation44 and particle approximation,44 the momentum equation for the continuum reduces to ! N N X Dvi R X σi Rβ þ σ j Rβ DWij mj þ ξij þ mj g ¼ Dt Fi Fj Dxi β j ¼1 j ¼1 Cj Ci N X m 3 j Cai Cni þ Caj Cnj ðφi - φj Þ DWij ð12Þ jrij j FF Drij j ¼1 i j or ! N N Dvi R X σ i Rβ σ j Rβ DWij X ¼ mj þ þ ξ þ mj g ij Dt Fi 2 Fj 2 Dxi β j ¼1 j ¼1 Cj Ci N X m 3 j Cai Cni þ Caj Cnj ðφi - φj Þ DWij ð13Þ jrij j FF Drij j ¼1 i j where σiRβ = piδRβ þ μi[∂viβ/∂xjR þ ∂vjR/∂xiβ - 2/3(rvi)δRβ], Ca is the Capillary number (Caij = viμij/σij), and ξij is the Neumann-Richtmyer artificial viscosity.48 The symbol pi is the pressure of particle i, and μi can be calculated on the basis of the directional spacing (xij) of the particles in the following manner: X hvij xij ð14Þ μi ¼ rij 2 þ η2 j where η2 = 0.001h2 and vij is the relative velocity of two particles i and j having velocities vi and vj, respectively. No flux condition along the interfaces and a local equilibrium condition at the triple line form30 the necessary boundary conditions for the chemical potential. The following conditions are generated using these
j ¼1
Fi Fj jrij j
Drij
¼0
ð15Þ
and N X m 3 j φi - φj DWij F F jrij j Drij j ¼1 i j
" # N X m 3 j 3kwi 3kwj 2 2 þ ð1 -Ci Þ ð1 - Cj Þ ¼ 0 F F 4Cni 4Cnj j ¼1 i j
ð16Þ
In eq16, kwi is the wetting coefficient of particle i which can be estimated from the cosine of the instantaneous contact angle at the triple line obeying the modified Young’s Laplace equation.49 Smoothening length is varied dynamically as the density of a particle changes under the influence of its neighbor. At the solid (48) Monaghan, J. J.; Gingold, R. A. J. Comput. Phys. 1983, 52, 374–389. (49) Harvey, D. Int. J. Mod. Phys. B 1999, 13, 3255–3259.
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2
ð11Þ -6
N X m 3 j φi - φj DWij
boundary, two additional layers of particles50 are placed to impose impermeability. These particles pose a resistive force to the interior particles which takes the following form: r0 ¼ D4 rij
!12
r0 rij
!6 3 5 xij for rij < r0 rij 2
ð17Þ
¼ 0 for rij gr0 where r0 is the minimum spacing of the particles. D is assumed to be the multiplication of 120 times the initial spacing of the particles and gravitational acceleration. The time step is determined so that the physical rate of propagation of information is slower than the numerical propagation rate.44 Cubic spline is considered as a kernel function for particle to particle interaction. The nearest neighbor particle searching (NNPS) technique44 is used to identify the immediate neighbors of a particle under consideration. The continuum surface force method (CSF)40 has been applied to calculate force per unit volume, Fsi, which follows the following equation: Fsi ¼ -σlv r2 Ci
n δs jnjs
ð18Þ
where σlv is the liquid gas interface surface tension and δs is the delta function which is unity only at the interface and zero elsewhere. n can be calculated as n = rCi.
Results and Discussion The hybrid DI-SPH model is used to investigate numerically the movement of a drop over an inclined plane. Most of the simulations have been conducted with a water droplet (σlv = 0.072 N/m, F = 1000 kg/m3, and μ = 10-3 Pa/s) sliding over a steel surface (contact angle of 83°). The gas-liquid interface is simulated as a free surface; i.e., pressure differences between two phases are generated only due to interfacial curvature. The particles along the triple line were identified using the NNPS technique,44 and their immediate neighbors along the liquid-gas interface were redistributed to maintain the dynamic contact angle (θd) as described by the modified Young’s equation:49 σ θd ¼ cos -1 cos θ Rσ lv
ð19Þ
where θ is the as placed contact angle and σ is the line tension and has a value of 1 μJ/m in this context. R is the radius of curvature of the contact point. The model has been validated against available theoretical and experimental results. Simulation has been started considering a static drop over a horizontal surface. The inclination of the surface has then been gradually increased until the drop starts sliding over the surface. The volume of the water drop is taken to be 1.75 mm3. At every inclination, the drop is allowed to take its stable shape before proceeding further. In Figure 1, the shapes of the stable drop for different inclinations have been depicted by the liquid particles at the interface. It is clear from the figure that drop loses its symmetric shape as the solid surface at its base makes an inclination with the horizontal. At any inclined position of the solid surface, the contact angle of the drop changes from point to point along its periphery. From Figure 1, one can see that the drop (50) Liu, M. B.; Liu, G. R.; Lam, K. Y. Shock Waves 2002, 12, 181–195.
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Figure 1. Asymmetric drop shape at various angles of inclination.
thins down at the top and assumes a blunt shape toward the bottom due to the effect of gravity. As a result, one obtains a substantial difference in the contact angles between the lowest and highest point of the droplet. The contact angle at the lower portion is called the advancing contact angle (θa) which exceeds the nominal contact angle of the fluid-solid pair. On the other hand, the contact angle associated with the upper corner or the receding contact angle (θr) of the drop decreases compared to the nominal contact angle. An increase in the advancing contact angle and a simultaneous decrease in the receding contact angle continue as the inclination of the solid surface increases. After a certain degree of inclination, the asymmetry in drop shape becomes conspicuous due to contact angle hysteresis. With the increase in the angle of inclination, this effect becomes strong enough to surpass the frictional resistance offered by the solid surface and the drop starts sliding down. The magnitude of the guiding force (Fc) is as follows: Fc ¼ πRσ lv ðcos θa - cos θr Þ Langmuir 2009, 25(19), 11459–11466
ð20Þ
In Figure 1, drop shape is shown for six different inclinations of the solid surface along with the analytical solution of Iliev.15 While the drop is stationary for the first five angles of inclination, it is sliding at 60°. A very good match can be seen between this numerical simulations and an analytical simulation.15 Study of the drop footprint at various inclinations is made in conducted to investigate the contact line dynamics. As the inclination of the solid surface increases, the drop starts losing its azimuthal symmetry. In Figure 2, the contact line is depicted for different angles of inclination. It is evident from the figure that the contact line remains more or less symmetric up to 30° inclination. Beyond that, asymmetry becomes prominent, and at an inclination of >45°, drop movement begins. Analytical results of Iliev,15 also presented in Figure 2, show the good predictability of the model. Once the angle of inclination exceeds 45°, the drop slides down the inclined plane. The unique CFD simulation adopted for this study provides an opportunity to probe the interesting hydrodynamics during sliding of the drop. It is clear from the simulation DOI: 10.1021/la901172u
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Figure 2. Drop footprint at different angles of inclination.
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results that the liquid drop undergoes a continuous change in its shape during its movement. The local interface curvature of the drop changes to produce a peristaltic motion similar to that observed in the movements of worms. Spatiotemporal ripples are generated at the interface and move from one end of the drop to the other. The bulk movement of the drop generates these alternate swelling-deswelling surface ripples. The surface waves are also associated with an internal circulation. In Figure 3, snapshots of the drop sliding down a 60° inclination are shown. The continuous change of the droplet shape is obvious from this figure. Additionally, a bunch of marked fluid particles are traced during sliding. It can be noted from Figure 3 that near the interface a clustering of liquid particles occurs. The tensile instability at the interface may be a probable cause. Melean et al.42 addressed this problem and suggested a method for eradicating it by imposing an artificial stress in the governing momentum equation. The application of the same methodology may also produce some improvement in our case. However, this needs further investigation.
Figure 3. Motion of the marked particles inside a drop moving downward. 11464 DOI: 10.1021/la901172u
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Figure 5. Evolution of advancing and receding contact angles as the inclination varies.
Figure 4. Velocity vector of all the particles at different angles of inclination.
Figure 6. Sliding curve for different fluids over a PTFE surface.
As shown in Figure 3, initially the marked particles are located just below the liquid-gas interface. During sliding, they move down under the effect of gravity along the interface and reach the advancing front. In the next course of their motion, they move up along the inclined solid surface. It may be noted that as the internal circulation of the drop does not contain a strong vortex the marked particle bunch retains its shape more or less intact during the downward motion. However, while climbing up the inclined plane, the bunch is stretched along the surface. This implies while the fluid lump rolls down the gas-liquid interface it undergoes a deformation near the solid-liquid contact surface due to the presence of a strong shear field. After reaching the receding end, the particles form a lump once again due to the sudden velocity drop. A bookkeeping of the particle numbers has also been meticulously conducted during each step of the movement. It has been noticed that near the advancing end one particle disappears from the twodimensional plane presented in the figure. The decrease in the number of marked particles, while passing near the advancing front, signifies the bulging of the leading tip of the drop. The missing particles shift to the other plane adjacent to the reported one, and this leads to the asymmetry of the drop. The observation described above can also be supported by instantaneous velocity fields inside the droplets. In Figure 4,
velocity vectors are depicted for 50° and 60° inclinations for the same volume of the liquid drop. An internal circulation is clear from the velocity vector plot for both the inclination angles. Circulation becomes stronger as the angle of inclination increases from 50° to 60°. The internal liquid motion obtained by the numerical simulation supports some of the earlier conjecture19,25 regarding the mechanism for movement of a drop down a tilted plane. Of two mechanisms, namely, sliding and rolling, postulated by Gao et al.,25 Figures 3 and 4 present a picture very close to the description of rolling. For the range of parameters considered in this study, we did not come across a sliding motion in which only the liquid adjacent to the solid surface is affected. This will be investigated further for a wide variety of fluid properties. The variation of contact angles for both the advancing (θa) and receding corner of the drop (θr) due to the change in the inclination angle is plotted in Figure 5. With a 0° inclination, the drop is symmetric which makes both contact angles the same (83°). With the increase in the inclination angle, a steep change in the advancing contact angle is noticed. The receding contact angle is also decreased with the increase in inclination angle, but its change is not as sharp. At 60° inclination there is a 90° variation in contact angle between the advancing and receding front. The difference between the contact angles generates a force as
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Article
Das and Das
results of Ryley and Ismail51 plotted in the same figure show a good agreement with our simulation. Similar studies have been conducted for other fluids. We can see that the sliding curve of kerosene (contact angle of 20°; less viscous than water) over a PTFE surface shifts up while the curve for glycerine (contact angle of 105°; more viscous than water) shifts down in comparison with that of water. As the contact between mercury and the PTFE surface (contact angle of 150°) is almost nonwetting, the sliding curve for this solid-liquid pair indicates that a mercury drop will instantaneously roll down a PTFE surface at a nominal angle of inclination. Another validation of the hybrid DI-SPH model for the prediction of a sliding curve is presented in Figure 7. A simulation using a water drop over an alkyl ketene dimer (AKD) surface (θ = 165°) is matched satisfactorily along with experimental results of Pierce et al.52
Conclusion Figure 7. Sliding curve of water over an AKD surface.
described in eq 14 which helps the rolling motion of the fluid particles inside the drop. The drop moves down the surface when the force generated due to the difference in contact angle exceeds the pinning force of the solid-liquid contact. Simulations have been done by increasing the inclination angle continuously to reach the limiting inclination angle beyond which a drop of a specified volume starts moving. A curve signifying this limit in inclination angle versus volume plane is termed the sliding curve. The sliding curve for a water drop over a polytetrafluoroethylene (PTFE) surface (contact angle of 109°) is reported in Figure 6. The limiting inclination angle decreases with drop volume, which can also be conceived intuitively. Experimental
11466 DOI: 10.1021/la901172u
The shape of three-dimensional drops over an inclined plane is studied in detail with the help of a combined DI-SPH methodology. The diffuse interface concept is applied along with modified Young’s equation to ensure an accurate modeling of contact line dynamics. The simulation can capture the internal circulation of the drop and depict a rolling motion of the fluid particles down the fluid-fluid interface and a shear-driven movement of the fluid-solid interface. The simulation can also predict the advancing and receding contact angles. The drop shape and the sliding curve predicted by this method agree very well with the experimental observations. (51) Ryley, D. J.; Ismail, M. S. B. J. Colloid Interface Sci. 1978, 65, 394–396. (52) Pierce, E.; Carmona, F. J.; Amirfazli, A. Colloids Surf., A 2008, 323, 73–82.
Langmuir 2009, 25(19), 11459–11466
