Detailed modelling of contact line motion in oscillatory wetting

Gustav Amberg

doi:10.1038/s41526-021-00186-0

Detailed modelling of contact line motion in oscillatory wetting

Amberg, Gustav 2022-01-19 00:00:00 www.nature.com/npjmgrav ARTICLE OPEN Detailed modelling of contact line motion in oscillatory wetting 1,2✉ Gustav Amberg The experimental results of Xia and Steen for the contact line dynamics of a drop placed on a vertically oscillating surface are analyzed by numerical phase ﬁeld simulations. The concept of contact line mobility or friction is discussed, and an angle-dependent model is formulated. The results of numerical simulations based on this model are compared to the detailed experimental results of Xia and Steen with good general agreement. The total energy input in terms of work done by the oscillating support, and the dissipation at the contact line, are calculated from the simulated results. It is found that the contact line dissipation is almost entirely responsible for the dissipation that sets the amplitude of the response. It is argued that angle-dependent line friction may be a fruitful interpretation of the relations between contact line speed and dynamic contact angle that are often used in practical computational ﬂuid dynamics. npj Microgravity (2022) 8:1 ; https://doi.org/10.1038/s41526-021-00186-0 INTRODUCTION On earth, the size of droplets that can be used is limited to a radius in the order of millimeters, but in microgravity, a larger A liquid spreading over a dry surface is a phenomenon that is parameter space can be investigated. In microgravity a droplet crucial to many natural processes and important in technology. size in the order of centimeters can be used instead, which will be However, the detailed description and understanding of dynamic 1,2 advantageous in several respects; one is that spatial dimensions wetting is still a complex and challenging problem . The fact that are larger, and the resonance frequency much lower, allowing for the continuum equations of ﬂuid mechanics exhibit a non- higher both spatial and temporal resolution. The larger droplet integrable singularity of the viscous stress at the contact line (CL) size also implies that the droplet and CL dynamics are even more shows that the detailed microscopic and nanoscopic features of dominated by inertia than in a millimeter-sized droplet on earth. It the liquid and the surface will be important for the macroscopic 11 was the intention of Paul Steen to perform such experiments , ﬂow. This introduces a host of different processes and phenomena and these have now been carried out. that need to be understood to predict and control wetting From a strictly thermodynamic point of view, a moving CL processes. should be associated with energetic losses of some kind . The In technology, in addition to such examples as spray painting, idea of a localized dissipation at the CL has been invoked in coating, etc., one particularly important ﬁeld is microﬂuidics .A different ways over the years. Following Hocking , Xia and Steen common challenge is to handle small volumes of liquid, often in introduce the concept of a CL mobility M as a phenomenological the form of small droplets, and one means for achieving this is to parameter that relates the deviation of the dynamic contact angle use wetting phenomena. Another area where surface tension and from equilibrium θ − θ to the CL speed U , e CL wetting become dominant is microgravity . In the absence of U ¼ MðÞ θ θ ; (1) CL e gravity, surface tension becomes dominant, and wetting will be important in any ﬂuid handling, from liquid fuel to many daily 8,13 see also refs. . In computational ﬂuid dynamics, more elaborate activities and needs of the astronauts. phenomenological relations between contact angle and CL speed Dynamic wetting driven by vibration is both of practical have been devised , which take the form θ ¼ fðU ; θ ;::Þ. CL e 15,16 importance and a convenient way to study the phenomenon. In Molecular Kinetic Theory (MKT) , dynamic wetting is The dynamic wetting on a glass plate dipping into a tank and described as an activated process on the molecular scale, and the oscillated vertically was studied in ref. , and damping of surface line friction ζ is given a phenomenological interpretation on the waves in a rectangular tank was investigated in ref. , revealing molecular scale. In its simplest linearized form, this can be written complex dependencies of damping rates on oscillation amplitude. as The dynamics of a sessile droplet on a vertically oscillating surface will be sensitive to the detailed conditions at the CL, such as U ¼ðÞ γ=ζðÞ cosθ cosθ sinθ ðÞ θ θ ; (2) CL e e e 8–10 9 ζ the presence of hysteresis or CL dissipation . Xia and Steen made careful experiments using droplets on a polydimethylsilox- where γ is the surface tension and ζ is the coefﬁcient of wetting- ane (PDMS)-covered substrate, which was oscillated vertically at line friction, which in MKT is estimated in terms of molecular frequencies near drop resonance. The resulting dynamics was quantities and thermal ﬂuctuations. examined through phase plots of the dynamic contact angle, the In the phase ﬁeld method, the ﬂuid is viewed as a mixture of CL position, and the CL speed. In particular, Xia and Steen used two immiscible species. The governing equations are derived from this information to measure the CL mobility. the thermodynamic potentials of such a system to yield typically 1 2 Flow Centre, Department of Engineering Mechanics, The Royal Institute of Technology, 100 44 Stockholm, Sweden. Södertörn University, Alfred Nobels allé 7, 141 89 Huddinge, Sweden. email: gustav.amberg@sh.se Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; G. Amberg 17,18 the Cahn–Hilliard equations . The interface now becomes a friction to vary with the dynamic contact angle. The nonlinear MKT diffuse region separating the two species, which has a deﬁnite theory gives one possibility and based on MD simulations width ε. The line friction appears as an energy dissipation Johansson and Hess formulated an expression for the angle associated with the CL displacement. Yue and Feng derive the dependence of line friction for water molecules on a surface with resulting equivalent condition relating CL speed and dynamic hydrogen bonds to the water. Other surfaces and liquids may contact angle as certainly show different behavior. 0 1 0 1 On this note, we evaluate the detailed experimental results of γ cosθ cosθ γ Xia and Steen , using phase ﬁeld simulations, with the aim of @ A @ A U ¼ ðÞ θ θ CL pﬃﬃ pﬃﬃ e (3) determining precisely what is required in the mathematical model, 2 2 sinθ 2 2 3Γε 3Γε in order to faithfully reproduce the experiments. We will study how the angle dependence of the CL friction should be chosen. here Γ is introduced as a rate parameter in the relaxation of the The model and the simulation are then used to draw conclusions dynamic contact angle boundary condition. on the source of the dissipation that is evident in the experiment. It is noted that the parameters in Eqs. (1)–(3) in the approximation of θ θ 1 can be identiﬁed by setting METHODS γ γ γ M ¼ sinθ ¼ ¼ pﬃﬃ The simulations are made using the Navier–Stokes–Cahn–Hilliard (4) ζ 2 2 μ 18,27,28 3Γε equations . These describe the two-phase system as two immiscible species and are motivated by the thermodynamics of In the last equality, we have introduced the line friction μ , f such a mixture. A phase ﬁeld variable is introduced that has which is essentially the same as the coefﬁcient of wetting-line different values in the two species, and the ﬂuid interface is friction used in ref. , except that it also absorbs the factor sinθ . e identiﬁed as the steep but continuous transition between those We note that ζ and μ have the dimensions of viscosity and that γ/ f values. μ is a velocity. Du 2 2 In many classical treatments, notably the Cox–Voinov law ,itis ρ ¼∇S þ μ∇ u C∇ϕ þ ρa €e (5) Dt presumed that the static contact angle applies right at the solid boundary and that the angle variations with the speed that are ∇ u ¼0(6) often observed are an “apparent” contact angle, which is attained a short distance away from the wall. It is also often assumed that DC ¼ κ∇ ϕ (7) there is a ﬂuid slip on the wall at the CL, which helps regularize the Dt singularity in stresses. In MKT, and inherent in the introduction of a 0 2 CL mobility, the contact angle is assumed to be different from the ϕ ¼ f ðÞ C σε∇ C (8) static value right at the wall, when viewed on molecular length 2 2 scales. In the phase ﬁeld model, mass diffusion will help regularize with the standard choices: fCðÞ¼ðÞ 1 C ðÞ 1 þ C =4, and the CL and there is no need for a ﬂuid slip. The introduction of gCðÞ¼ðÞ 2 þ 3C C =4. dissipation related to CL movement will cause the contact angle at Here u, C, and ϕ are the ﬂuid velocity, the phase ﬁeld variable, the wall to deviate from the static value. and the associated chemical potential, respectively. C is +1 in the It is far from clear what the actual conditions are for a given liquid and −1 outside the droplet. S is a reduced pressure such liquid spreading on a particular surface. For a system of decane that p ¼ S þ Cϕ is the actual pressure. μ and ρ are the local spreading on a surface covered with a thin layer of PDMS, it was viscosity and density, respectively, and κ is the phase ﬁeld demonstrated experimentally that the microscopic dynamic mobility. σ and ε are the phase ﬁeld parameters, where ε denotes contact angle is velocity dependent and deviates substantially the interface thickness, and σ gives the surface tension γ through p 2 2 from the equilibrium value also at very small but nonzero capillary γ ¼ σð2 2Þ=3. The function fCðÞ¼ðÞ 1 C ðÞ 1 þ C =4 represents numbers. In ref. , a theoretical model is developed that links the the standard choice in phase ﬁeld methods and gives a distribution of assumed nanoscopic geometrical surface defects to qualitatively reasonable thermodynamic behavior representative the line friction dissipation. Recent molecular dynamics (MD) of an immiscible mixture. simulations have also shown that for water molecules and a wall The simulations are performed in a cylindrical coordinate with hydrogen bonds with the water molecules, the ﬁrst layer of system that follows the oscillating substrate, giving rise to the water molecules are effectively bound to the surface, a no-slip acceleration term on the right-hand side of the momentum condition is appropriate, and the contact angle deviates from the equation, with atðÞ ¼ A sin 2πft denoting the vertical position of 21,22 equilibrium value . It is known that the local molecular the substrate. The boundary conditions on the solid wall express that the ﬂuid arrangements will be different in electrowetting and that this will 23,24 cannot penetrate through the wall and does not slip on the wall. alter the line friction . Also, a microscopic geometrical structure on the surface will change the dynamic wetting and can be ∇ϕ n ¼ 0 (9) described through an effective line friction . In an oil–water system, Rondepierre et al. demonstrated that CL friction was u ¼ 0 (10) responsible for a decrease in CL speed of three orders of magnitude, as a certain surfactant was added. One additional boundary condition is needed for the phase The actual nanoscopic cause of local dissipation at the CL can ﬁeld variable, which expresses the wetting conditions. thus be very different depending on the properties of the surface, ∂C surface roughness and structure, the liquid properties, the surface (11) εμ ¼σε∇C n þðÞ γ γ gðÞ C f 1 0 ∂t chemistry of the wet surface, etc. We conclude that we should not expect any universal answer to the question of what is causing CL Here σ and ε are the phase ﬁeld parameters as above, giving friction. Many different nanoscopic or microscopic processes can surface tension as γ ¼ σð2 2Þ=3. γ and γ are the surface 1 0 no doubt have this effect. However, whatever the origin, the effect energies of the dry and wet solid surface, respectively, so that the can be described as a single parameter, the CL friction. equilibrium contact angle θ is given byðÞ γ γ =γ ¼ cos θ . The e e 1 0 So far, the dependence of line friction on the contact angle has form of the function gCðÞ¼ðÞ 2 þ 3C C =4 is chosen in relation received limited attention, but there is clearly every reason for line to the function f in Eq. (4). npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; G. Amberg Yue developed a phase ﬁeld treatment of contact angle angle of θ = 101°. This will be the reference case for the hysteresis, where advancing and receding contact angels are simulations shown here. introduced in a piecewise continuous function on the right-hand The nondimensional numbers for this case are Oh = 0.00256, side of the equation corresponding to Eq. (11). The CL is then amplitude A = 0.047, and angular velocity of the driving ω = 4.41. allowed to move according to whether the value of the dynamic The low value of the Ohnesorge number indicates that the droplet contact angle exceeds (subceeds) the advancing (receding) angle. A dynamics are inertial and that viscosity plays a minor role. Near treatment inspired by this is also developed for level-set methods . the solid surface, we should expect a viscous Stokes layer of pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The left-hand side represents the dissipation associated with CL thickness μ=ðρπfÞ, which is approximately 0.07 mm for the motion, quantiﬁed by the parameter μ , which we will call CL experimental parameters, i.e., much less than the initial droplet friction. This has dimensions of viscosity. radius of 2.1 mm. By considering solutions to the Cahn–Hilliard equations near The order unity value of the angular velocity shows that the equilibrium at the CL, an explicit relation between CL speed U oscillations are reasonably matched with the inertial timescale, as is CL and dynamic contact angle θ can be derived from Eq. (11), see expected since the experiment aims at being near resonance. The ref. and Eq. (3): value of the line friction Ohnesorge number Oh is set to unity, which pﬃﬃﬃﬃﬃﬃﬃﬃ makes the reference value of the line friction μ ¼ ργR. We will f ;ref μ U cosθ cosθ f CL e return to the more detailed modeling of the line friction below. ¼ θ θ (12) γ sinθ Figure 1 shows the simulation results over one cycle for the M00 case. In Fig. 1a, the ﬁrst frame shows the position near the lower The dynamic contact angle is equal to the equilibrium angle if turning point when the droplet is compressed towards the μ = 0, in which case there is also no dissipation at the CL. The last substrate. In the second frame, it is extending upwards and is near approximate equality holds if θ is near θ . its most elongated state in the third frame. In the fourth frame, it is The above equations are made nondimensional using R as being again compressed as the substrate is moving upwards. length reference, the radius of the half-sphere (approximately the A Weber number based on the oscillation amplitude A and initial wet footprint radius), and an inertial capillary velocity 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ frequency f can be expressed as A ω , giving the value U ¼ γ=ðÞ ρR . 2 2 0:047 4:41 ¼ 0:043. We thus expect the ﬂow to be dominated by the surface tension. The nondimensional parameters that appear are Figure 1b shows the corresponding pressure ﬁelds, always pﬃﬃﬃﬃﬃﬃ Oh ¼ Ohnesorge number reﬂecting the curvature of the interface. The pressure is elevated ργR in the droplet due to the capillary pressure and ﬂuctuates as the pﬃﬃﬃﬃﬃﬃ Oh ¼ Line friction Ohnesorge number ργR interface curvature varies. In the most compressed state in the ﬁrst ρgR 2 frame of Fig. 1b, there is a concave shape at the top of the droplet, Bo ¼ Bond number, g ¼ 9:81 m=s <?tpb5pt?> γ qﬃﬃﬃﬃﬃﬃ 3 which causes a local low pressure there. The pressure in the air is ρR ω ¼ 2πf Nondimensional oscillation angular frequency. f is the almost constant, due to the low air density. Figure 1c shows the oscillation frequency in Hertz velocity ﬁelds for the points where the droplet is extending A= A /R Nondimensional oscillation amplitude upwards and being compressed (corresponding to frames two and four of Fig. 1a, b). In Fig. 2 are shown the time signals for the vertical substrate In addition to these, the Cahn–Hilliard equations use the position, together with the CL position and the droplet height. It is following parameters: noticed that the amplitude of the droplet height is about three qﬃﬃﬃﬃ UR γ R pﬃﬃ Pe ¼ ¼ Peclet number related to phase ﬁeld mobility. Set to times that of the substrate amplitude, as expected for a near 2 2κγ D ρR 3 ε 100 in the simulations. Reported here resonance situation. All signals quickly go into a periodic motion, Cn ¼ Cahn number, nondimensional interface width. Kept at with no sign of period-doubling or other more complex dynamics. 0.01 in the simulations. Reported here The responses in both the droplet height and the CL position are air nonlinear, however, with shapes departing from sinusoidal. Ratio between air and liquid density. Set to 0.01 here, liq for computational convenience The CL position signal deviates from a sinusoidal shape, and tends air towards a square wave, with ﬂat peaks. This is a signature of a stick- Ratio between air and liquid viscosity. Set to 0.03 here, liq slip motion of the CL; it is rather stationary at its extreme values but for computational convenience shifts quickly between them twice per cycle. We will see that this is built into the simulation through angle-dependent line friction. The simulations are carried out using an adaptive ﬁnite element solver, as in ref. . The adaptive grid is automatically reﬁned as Model for angle-dependent line friction needed down to an element size of 0.001. The air viscosity and In addition to the input data already discussed, angle-dependent line density are taken larger than those of air but much smaller than friction has been implemented in this simulation. The line friction in the values for the liquid. They are deemed small enough for the air Eqs. (11)and (12) is computed from a regularized well function: to have a negligible inﬂuence on the ﬂow inside the droplet. μ ¼ μ μ ðÞ θ (13) f f ;ref Reporting summary where the nondimensional μ ðÞ θ is Further information on research design is available in the Nature μ ðÞ θ ¼ μðÞ 1 HðÞ θþ μ HðÞ θ (14) Research Reporting Summary linked to this article. f 0 f 1 and RESULTS θ ðÞ θ þ dθ θ ðÞ θ dθ e e HðÞ θ ¼ 1 þ 1=2 tanh tanh Flow ﬁelds δ δ Xia and Steen report the most detailed results for their experiment (15) labeled M00, using a 20 µl water droplet oscillated at f = 61 Hz, with an amplitude of A = 0.1 mm. The substrate is a slightly Equation (14) thus describes nondimensional line friction that hydrophobic PDMS-covered silicon surface, with a static contact has a high value μf in the sticking region in the angle range θ ± 0 e Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 1 G. Amberg Fig. 1 One oscillation cycle for the M00 case. a The shape of the droplet at four instants. b The pressure ﬁeld. c Velocity ﬁeld at the instants corresponding to frame two and four of a and b. Fig. 2 Time signals for the contact line position r (blue), the CL droplet height z (yellow), and the substrate position Asinωt (gray). The substrate position is drawn around an equilibrium position of 1, to facilitate comparison with the CL position and the droplet height. dθ, and a low value μf outside of this range. δ smooths the corners of the well function and μ is a dimensional reference f,ref value that is used in the line friction Ohnesorge number Oh . In the pﬃﬃﬃﬃﬃﬃﬃﬃ present case, Oh = 1, and thus μ ¼ ργR. In the simulation in f ;ref Figs. 1–6, these values are: θ = 101°, dθ = 7°, δ = 1°, μf = 10, e 0 μf = 0.5 and Oh = 1. We could interpret θ + dθ = θ as 1 f e a approximating the advancing contact angle and θ − dθ = θ as e r the receding angle. Inserting the line friction according to Eq. (14) into Eq. (12), we readily obtain the dynamic contact angle as a function of CL speed (see Fig. 3). As expected, the high line friction near the equilibrium angle creates a region of near stick, with low velocities for angles in a region ≈±7° around the equilibrium. In this region, the well function H(θ) ≈ 0, and μ ðÞ θ will be constant and equal to the high Fig. 3 Model of angle-dependent line friction, according to Eqs. value μ . For CL speeds up to about 0.3, the angle is f0 (12)–(15). a The deviation of the dynamic contact angle from the approximately constant, between 7 and 10 degrees, establishing equilibrium angle θ − θ as a function of contact line speed. a “slip” region. For angles departing >10 degrees from equilibrium, b Nondimensional line friction as a function of θ − θ . Parameter or speeds larger than approximately 0.4, the well function H(θ)is1 values are the same as for the simulations in Figs. 1, 2, and 4: θ = and the line friction μ ðÞ θ becomes equal to the low value μ .To 101°, dθ = 7°, δ = 1°, μ = 10, and μ = 0.5. f1 f0 f1 npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA G. Amberg Fig. 4 Comparison of phase plots with Xia and Steen experiments for the case M00. Colored dots are the experimental results of Xia and Steen, and the black solid curves are the present simulations. a Dynamic contact angle departure from the equilibrium angle θ − θ as a function of contact line speed. b Contact line speed vs contact line position, as a departure from the mean position. c Dynamic contact angle θ − θ vs contact line position. d Contact line speed vs contact line position, as a departure from the mean position. Dynamic contact angle departure from the equilibrium angle multiplied by contact line positionðÞ r r ðÞ θ θ vs contact line position multiplied by contact line CL ref e speedðÞ r r U . CL ref CL summarize: in the windowjj θ θ < dθ, the line friction is magnitude of the sticking region in Fig. 4a, and the sticking region constant, equal to the high value μ , and for angles distinctly line friction, μ , sets the slope of the curve at the origin in Fig. 4a. f0 f0 outside this window,jj θ θ > dθ, the line friction tends to the The transition region width δ allows for the smooth curvature of constant value μ . Connecting these two regions is a velocity the TD at the end of the sticking region. The line friction far from f1 range where the dynamic angle is approximately constant, i.e. a equilibrium μ is then chosen to approximate the slope of the TD f1 region of “free slip”. in the “free slip” region. It should be noted that the numerical As seen in Fig. 3a, there is still some variation of angle with CL simulation accurately reproduces the theoretical curve in Fig. 3a speed also in the region of “free slip”. The slope of the curve in and that the CL velocities encountered in the experiment are Fig. 3a in this region is related to δ, the width of the transition always below about 0.3, so that only the “sticking” and the “free region in line friction in Fig. 3b, and the line friction far from slip” regions are visited. equilibrium μ . Approximating the contact angles and the CL Figure 4b shows a phase plot in terms of the CL speed vs the CL f1 speeds at the ends of the transition region as θ ¼ θ þ dθ þ δ, position. In addition to the experimental data shown as colored 0 e θ ¼ θ þ dθ δ,and U ¼ðθ θ Þ=μ , U ¼ðθ θ Þ=μ ,and circles, the black circles denote the position of the substrate. The 1 e 0 0 e 1 1 e f 0 f 1 assuming that μ μ , an estimate of the slope in Fig. 3a in the black line is the trajectory from the simulation. It captures both f 1 f 0 transition or “free slip” region can be obtained as Δθ=ΔU the elongation and the slight inclination of the experimental loop. dθ 2μ = 1 þ (angles in radians). For a very steep transition region The width of the simulated trajectory is slightly less than the f 1 (δ dθ) the slope will thus approach zero, while it becomes experimental one, but overall, the agreement is good. Figure 4c shows the dynamic contact angle vs CL position. Here comparable to μ for δ ~dθ. f1 The model for line friction in Eqs. (13)–(15) was constructed to the stick–slip character of the motions is evident; the shape of the be as conceptually simple as possible, characterized by a high trajectory is a quadrilateral, where the horizontal upper and lower friction region around equilibrium, and low friction further from parts show the rapid phase when the CL moves from one almost equilibrium. In order to test this against the experiments of Xia steady position to another, at a fairly constant contact angle. The and Steen the simulated results are overlaid on their experimental vertical sides represent the “stick” phase where the CL is nearly phase plots (their Figs. 4 and 5), see Fig. 4. stationary and the angle changes. Figure 4a shows the graph of dynamic contact angle vs CL Figure 4d was introduced by Xia and Steen to highlight the speed, the “traditional diagram” (TD) as it is referred to by Xia and “stick” and the “slip” parts of the motions and quantify those Steen. The colored circles are the experiments of Xia and Steen, separately. The graph shows dynamic contact angle departure and the black curve is obtained from the numerical simulation. If from the equilibrium angle multiplied by CL position the CL friction would be a constant this curve would always be a ðÞ r r ðÞ θ θ vs CL position multiplied by CL speed CL ref e straight line, thus the angle variation in the CL friction becomes ðÞ r r U . The nonlinearity that is introduced creates two CL ref CL important. The parameters dθ, δ, μ , and μ for the angle- loops, one for the receding and one for the advancing motion of f0 f1 dependent line friction are chosen to give a fair approximation of the CL. The simulated trajectory is again in fair agreement with the the TD, in the following manner. dθ is chosen to capture the experiment, even though the experiment extends somewhat Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 1 G. Amberg Fig. 5 Comparison of phase plots with Xia and Steen experiments for the case M00, using a closer ﬁt for the TD. Colored dots are the experimental results of Xia and Steen, and the black solid curves are simulations. a Dynamic contact angle departure from the equilibrium angle θ − θ as a function of contact line speed. b Contact line speed vs contact line position, as a departure from the mean position. c Dynamic contact angle θ − θ vs contact line position. d Contact line speed vs contact line position, as a departure from the mean position. Dynamic contact angle departure from the equilibrium angle multiplied by contact line positionðÞ r r ðÞ θ θ vs contact line position CL ref e multiplied by contact line speedðÞ r r U . CL ref CL further from the origin. In view of the scatter in the experimental experimental points show some differences between the advan- data the agreement in Fig. 4 was deemed sufﬁcient for the present cing and receding curves. A more complex mathematical discussion. expression for the angle dependence of the line friction was In terms of the model in Eqs. (13)–(15), we see that the sticking designed so that the TD is approximated more closely, see Fig. 5a. phase is characterized by a high constant line friction μ = 10. This f0 Overall the agreement in all four panels is somewhat improved shows up in the central part of Fig. 4d, where the simulated but not dramatically so. The conclusion is that the complexity of trajectory is steep and near the red experimental circles. The slope Eqs. (13)–(15), with the four parameters dθ, δ, μ , and μ is f0 f1 of the curve here is proportional to μ . The main source of CL f0 sufﬁcient to capture the essential features of the ﬂow. dissipation is at the slip phase, as the uncompensated Young’s stress at the receding or advancing contact angle multiplies the CL Energy dissipation speed. Here the CL slip speed becomes rather independent of the dynamic angle. In a macroscopic experiment where the overall CL It may be asked what the nature is of the dissipation that limits the speed is measured along with the contact angle, an overall CL response amplitude. In the simulation for the standard M00 case, friction for an advancing CL would be identiﬁed from Eq. (12)as the input energy per cycle was determined from the pressure and μ ¼ðÞ γ=UðÞ cosθ cosθ =sinθ . This would not be possible to speed of the substrate integrated over a cycle. Figure 6b shows a f CL e a a directly relate to material parameters, since U here would be CL graph of the total vertical force F from the substrate, vs the determined from the overall droplet dynamics, and the angle substrate position over a cycle. In the same manner, the would stay nearly constant, close to the advancing angle. To dissipation at the CL was determined from a graph of predict the wetting behavior, the CL friction would need to be 2πrðÞ cos θ cos θ vs r graph, see Fig. 6a. CL e CL modeled and the overall dynamics simulated. For angles further The total amount of work supplied to the drop by the oscillation away from the sticking region, μ ðÞ θ becomes μ and Eq. (12) f1 over a cycle is now the area inside the loop in Fig. 6b, which is reduces toðÞ μ U =γ ¼ðÞ cosθ cosθ =sinθ with a constant line f 1 CL e calculated as 0.132914. The area inside the loop in Fig. 6a giving friction. We do not however have any experimental data to verify the CL dissipation is 0.131779. As expected, the work input is if this is indeed the case. slightly larger than the CL dissipation, with a relative difference of The CL mobility parameter M (Eq. (1)), as identiﬁed by Xia and 0.8%. We should not overinterpret this small difference, in view of Steen is the inverse of the slope of the line formed by the blue other possible sources of inaccuracy, but this still shows that the circles in Fig. 4d. This is also approximately the same as the inverse CL dissipation is the completely dominating cause of damping in of the slope of the straight line obtained by connecting the two this case. There is an almost perfect match between the input regions of blue circles in the “wings” of Fig. 4a. energy and the energy dissipated at the CL over a cycle. The other The agreement with the experiment in Fig. 4 is fair, but there source of dissipation by bulk viscous dissipation is indeed are some differences, so more precise modeling of the experi- mental points in the TD in Fig. 4a was made. The model in Eqs. expected to be quite small, given the small value of the (13)–(15) is symmetric around the equilibrium angle, while the Ohnesorge number (Oh = 0.00256). npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA G. Amberg Fig. 6 Calculation of CL dissipation. a Uncompensated Young’s stress 2πrðÞ cos θ cos θ vs r over one cycle. The area inside the loop is CL e CL 0.131779. b Vertical force from the substrate on the drop vs substrate displacement atðÞ ¼ A sin ωt. The area inside the loop is 0.132914. DISCUSSION CODE AVAILABILITY The code used in the current study is available from the corresponding author on The good qualitative and quantitative agreement between reasonable request. simulation and experiment, in all the different aspects that can be accessed through the phase plots presented by Xia and Steen, Received: 1 August 2021; Accepted: 6 December 2021; indicates that the essential physical processes are well represented in the mathematical model. The experiment is quite sensitive since the response is ampliﬁed through the resonance of the droplet motion, and the damping that is present in the system is controlling the resulting periodic CL motion completely. As shown REFERENCES in Fig. 6, the damping is almost entirely due to the CL dissipation. 1. Bonn, D. et al. Wetting and spreading. Rev. Mod. Phys. 81, 739–805 (2009). Simulations were performed where the CL dissipation is removed, 2. Snoeijer, J. H. & Andreotti, B. 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Fluids 23, 012106 (2011). speciﬁcs of a particular surface and a particular ﬂuid to ﬁnd the 19. Lhermerout, R. et al. A moving contact line as a rheometer for nanometric line friction and its angle dependence and then hope that this interfacial layers. Nat. Commun. 7, 12545 (2016). line friction could be used with accuracy for all ﬂow situations, for 20. Perrin, H. et al. Defects at the nanoscale impact contact line motion at all scales. Phys. Rev. Lett. 116, 184502 (2016). that particular combination of ﬂuid and surface. 21. Lācis, U. et al. Steady moving contact line of water over a no-slip substrate. Eur. Phys. J. Spec. Top. 229, 1897–1921 (2020). 22. Johansson, P. & Hess, B. Molecular origin of contact line friction in dynamic DATA AVAILABILITY wetting. Phys. Rev. Fluids 3, 074201 (2018). 23. Johansson, P. & Hess, B. Electrowetting diminishes contact line friction in mole- The datasets generated during and/or analyzed during the current study are available cular wetting. Phys. Rev. Fluids. 5, 064203 (2020). from the corresponding author on reasonable request. Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 1 G. Amberg 24. Nita, S. et al. Electrostatic cloaking of surface structure for dynamic wetting. Sci. COMPETING INTERESTS Adv. 3, e1602202 (2017). The author declares no competing interests. 25. Wang, J. Y. et al. Surface structure determines dynamic wetting. Sci. Rep. 5,7 (2015). 26. Rondepierre, G. et al. Dramatic slowing down of oil/water/silica contact line ADDITIONAL INFORMATION dynamics driven by cationic surfactant adsorption on the solid. Langmuir 37, Correspondence and requests for materials should be addressed to Gustav Amberg. 1662–1673 (2021). 27. Yue, P., Zhou, C. & Feng, J. J. Sharp-interface limit of the Cahn–Hilliard model for Reprints and permission information is available at http://www.nature.com/ moving contact lines. J. Fluid Mech. 645, 279 (2010). reprints 28. Carlson, A., Do-Quang, M. & Amberg, G. Modeling of dynamic wetting far from equilibrium. Phys. Fluids 21, 121701 (2009). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims 29. Yue, P. Thermodynamically consistent phase-ﬁeld modelling of contact angle in published maps and institutional afﬁliations. hysteresis. J. Fluid Mech. 899, A15 (2020). 30. Zhang, J. & Yue, P. A level-set method for moving contact lines with contact angle hysteresis. J. Comput. Phys. 418, 109636 (2020). 31. Carlson, A., Do-Quang, M. & Amberg, G. Dissipation in rapid dynamic wetting. J. Open Access This article is licensed under a Creative Commons Fluid Mech. 682, 213–240 (2011). Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative ACKNOWLEDGEMENTS Commons license, and indicate if changes were made. The images or other third party This work was begun in discussion with Professor P.H. Steen and was interrupted only material in this article are included in the article’s Creative Commons license, unless by his untimely passing away. I gratefully acknowledge his role in formulating the indicated otherwise in a credit line to the material. If material is not included in the questions discussed here and all the contributions he made to science throughout article’s Creative Commons license and your intended use is not permitted by statutory his career. I wish to thank Yi Xia for clarifying details about the experiments and regulation or exceeds the permitted use, you will need to obtain permission directly Vanessa Kern for discussions. from the copyright holder. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/. FUNDING © The Author(s) 2022 Open access funding provided by Södertörn University. npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png npj Microgravity Springer Journals http://www.deepdyve.com/lp/springer-journals/detailed-modelling-of-contact-line-motion-in-oscillatory-wetting-pk6pok9zgp

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Publisher: Springer Journals
Copyright: Copyright © The Author(s) 2022
eISSN: 2373-8065
DOI: 10.1038/s41526-021-00186-0
Publisher site: See Article on Publisher Site

Abstract

www.nature.com/npjmgrav ARTICLE OPEN Detailed modelling of contact line motion in oscillatory wetting 1,2✉ Gustav Amberg The experimental results of Xia and Steen for the contact line dynamics of a drop placed on a vertically oscillating surface are analyzed by numerical phase ﬁeld simulations. The concept of contact line mobility or friction is discussed, and an angle-dependent model is formulated. The results of numerical simulations based on this model are compared to the detailed experimental results of Xia and Steen with good general agreement. The total energy input in terms of work done by the oscillating support, and the dissipation at the contact line, are calculated from the simulated results. It is found that the contact line dissipation is almost entirely responsible for the dissipation that sets the amplitude of the response. It is argued that angle-dependent line friction may be a fruitful interpretation of the relations between contact line speed and dynamic contact angle that are often used in practical computational ﬂuid dynamics. npj Microgravity (2022) 8:1 ; https://doi.org/10.1038/s41526-021-00186-0 INTRODUCTION On earth, the size of droplets that can be used is limited to a radius in the order of millimeters, but in microgravity, a larger A liquid spreading over a dry surface is a phenomenon that is parameter space can be investigated. In microgravity a droplet crucial to many natural processes and important in technology. size in the order of centimeters can be used instead, which will be However, the detailed description and understanding of dynamic 1,2 advantageous in several respects; one is that spatial dimensions wetting is still a complex and challenging problem . The fact that are larger, and the resonance frequency much lower, allowing for the continuum equations of ﬂuid mechanics exhibit a non- higher both spatial and temporal resolution. The larger droplet integrable singularity of the viscous stress at the contact line (CL) size also implies that the droplet and CL dynamics are even more shows that the detailed microscopic and nanoscopic features of dominated by inertia than in a millimeter-sized droplet on earth. It the liquid and the surface will be important for the macroscopic 11 was the intention of Paul Steen to perform such experiments , ﬂow. This introduces a host of different processes and phenomena and these have now been carried out. that need to be understood to predict and control wetting From a strictly thermodynamic point of view, a moving CL processes. should be associated with energetic losses of some kind . The In technology, in addition to such examples as spray painting, idea of a localized dissipation at the CL has been invoked in coating, etc., one particularly important ﬁeld is microﬂuidics .A different ways over the years. Following Hocking , Xia and Steen common challenge is to handle small volumes of liquid, often in introduce the concept of a CL mobility M as a phenomenological the form of small droplets, and one means for achieving this is to parameter that relates the deviation of the dynamic contact angle use wetting phenomena. Another area where surface tension and from equilibrium θ − θ to the CL speed U , e CL wetting become dominant is microgravity . In the absence of U ¼ MðÞ θ θ ; (1) CL e gravity, surface tension becomes dominant, and wetting will be important in any ﬂuid handling, from liquid fuel to many daily 8,13 see also refs. . In computational ﬂuid dynamics, more elaborate activities and needs of the astronauts. phenomenological relations between contact angle and CL speed Dynamic wetting driven by vibration is both of practical have been devised , which take the form θ ¼ fðU ; θ ;::Þ. CL e 15,16 importance and a convenient way to study the phenomenon. In Molecular Kinetic Theory (MKT) , dynamic wetting is The dynamic wetting on a glass plate dipping into a tank and described as an activated process on the molecular scale, and the oscillated vertically was studied in ref. , and damping of surface line friction ζ is given a phenomenological interpretation on the waves in a rectangular tank was investigated in ref. , revealing molecular scale. In its simplest linearized form, this can be written complex dependencies of damping rates on oscillation amplitude. as The dynamics of a sessile droplet on a vertically oscillating surface will be sensitive to the detailed conditions at the CL, such as U ¼ðÞ γ=ζðÞ cosθ cosθ sinθ ðÞ θ θ ; (2) CL e e e 8–10 9 ζ the presence of hysteresis or CL dissipation . Xia and Steen made careful experiments using droplets on a polydimethylsilox- where γ is the surface tension and ζ is the coefﬁcient of wetting- ane (PDMS)-covered substrate, which was oscillated vertically at line friction, which in MKT is estimated in terms of molecular frequencies near drop resonance. The resulting dynamics was quantities and thermal ﬂuctuations. examined through phase plots of the dynamic contact angle, the In the phase ﬁeld method, the ﬂuid is viewed as a mixture of CL position, and the CL speed. In particular, Xia and Steen used two immiscible species. The governing equations are derived from this information to measure the CL mobility. the thermodynamic potentials of such a system to yield typically 1 2 Flow Centre, Department of Engineering Mechanics, The Royal Institute of Technology, 100 44 Stockholm, Sweden. Södertörn University, Alfred Nobels allé 7, 141 89 Huddinge, Sweden. email: gustav.amberg@sh.se Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; G. Amberg 17,18 the Cahn–Hilliard equations . The interface now becomes a friction to vary with the dynamic contact angle. The nonlinear MKT diffuse region separating the two species, which has a deﬁnite theory gives one possibility and based on MD simulations width ε. The line friction appears as an energy dissipation Johansson and Hess formulated an expression for the angle associated with the CL displacement. Yue and Feng derive the dependence of line friction for water molecules on a surface with resulting equivalent condition relating CL speed and dynamic hydrogen bonds to the water. Other surfaces and liquids may contact angle as certainly show different behavior. 0 1 0 1 On this note, we evaluate the detailed experimental results of γ cosθ cosθ γ Xia and Steen , using phase ﬁeld simulations, with the aim of @ A @ A U ¼ ðÞ θ θ CL pﬃﬃ pﬃﬃ e (3) determining precisely what is required in the mathematical model, 2 2 sinθ 2 2 3Γε 3Γε in order to faithfully reproduce the experiments. We will study how the angle dependence of the CL friction should be chosen. here Γ is introduced as a rate parameter in the relaxation of the The model and the simulation are then used to draw conclusions dynamic contact angle boundary condition. on the source of the dissipation that is evident in the experiment. It is noted that the parameters in Eqs. (1)–(3) in the approximation of θ θ 1 can be identiﬁed by setting METHODS γ γ γ M ¼ sinθ ¼ ¼ pﬃﬃ The simulations are made using the Navier–Stokes–Cahn–Hilliard (4) ζ 2 2 μ 18,27,28 3Γε equations . These describe the two-phase system as two immiscible species and are motivated by the thermodynamics of In the last equality, we have introduced the line friction μ , f such a mixture. A phase ﬁeld variable is introduced that has which is essentially the same as the coefﬁcient of wetting-line different values in the two species, and the ﬂuid interface is friction used in ref. , except that it also absorbs the factor sinθ . e identiﬁed as the steep but continuous transition between those We note that ζ and μ have the dimensions of viscosity and that γ/ f values. μ is a velocity. Du 2 2 In many classical treatments, notably the Cox–Voinov law ,itis ρ ¼∇S þ μ∇ u C∇ϕ þ ρa €e (5) Dt presumed that the static contact angle applies right at the solid boundary and that the angle variations with the speed that are ∇ u ¼0(6) often observed are an “apparent” contact angle, which is attained a short distance away from the wall. It is also often assumed that DC ¼ κ∇ ϕ (7) there is a ﬂuid slip on the wall at the CL, which helps regularize the Dt singularity in stresses. In MKT, and inherent in the introduction of a 0 2 CL mobility, the contact angle is assumed to be different from the ϕ ¼ f ðÞ C σε∇ C (8) static value right at the wall, when viewed on molecular length 2 2 scales. In the phase ﬁeld model, mass diffusion will help regularize with the standard choices: fCðÞ¼ðÞ 1 C ðÞ 1 þ C =4, and the CL and there is no need for a ﬂuid slip. The introduction of gCðÞ¼ðÞ 2 þ 3C C =4. dissipation related to CL movement will cause the contact angle at Here u, C, and ϕ are the ﬂuid velocity, the phase ﬁeld variable, the wall to deviate from the static value. and the associated chemical potential, respectively. C is +1 in the It is far from clear what the actual conditions are for a given liquid and −1 outside the droplet. S is a reduced pressure such liquid spreading on a particular surface. For a system of decane that p ¼ S þ Cϕ is the actual pressure. μ and ρ are the local spreading on a surface covered with a thin layer of PDMS, it was viscosity and density, respectively, and κ is the phase ﬁeld demonstrated experimentally that the microscopic dynamic mobility. σ and ε are the phase ﬁeld parameters, where ε denotes contact angle is velocity dependent and deviates substantially the interface thickness, and σ gives the surface tension γ through p 2 2 from the equilibrium value also at very small but nonzero capillary γ ¼ σð2 2Þ=3. The function fCðÞ¼ðÞ 1 C ðÞ 1 þ C =4 represents numbers. In ref. , a theoretical model is developed that links the the standard choice in phase ﬁeld methods and gives a distribution of assumed nanoscopic geometrical surface defects to qualitatively reasonable thermodynamic behavior representative the line friction dissipation. Recent molecular dynamics (MD) of an immiscible mixture. simulations have also shown that for water molecules and a wall The simulations are performed in a cylindrical coordinate with hydrogen bonds with the water molecules, the ﬁrst layer of system that follows the oscillating substrate, giving rise to the water molecules are effectively bound to the surface, a no-slip acceleration term on the right-hand side of the momentum condition is appropriate, and the contact angle deviates from the equation, with atðÞ ¼ A sin 2πft denoting the vertical position of 21,22 equilibrium value . It is known that the local molecular the substrate. The boundary conditions on the solid wall express that the ﬂuid arrangements will be different in electrowetting and that this will 23,24 cannot penetrate through the wall and does not slip on the wall. alter the line friction . Also, a microscopic geometrical structure on the surface will change the dynamic wetting and can be ∇ϕ n ¼ 0 (9) described through an effective line friction . In an oil–water system, Rondepierre et al. demonstrated that CL friction was u ¼ 0 (10) responsible for a decrease in CL speed of three orders of magnitude, as a certain surfactant was added. One additional boundary condition is needed for the phase The actual nanoscopic cause of local dissipation at the CL can ﬁeld variable, which expresses the wetting conditions. thus be very different depending on the properties of the surface, ∂C surface roughness and structure, the liquid properties, the surface (11) εμ ¼σε∇C n þðÞ γ γ gðÞ C f 1 0 ∂t chemistry of the wet surface, etc. We conclude that we should not expect any universal answer to the question of what is causing CL Here σ and ε are the phase ﬁeld parameters as above, giving friction. Many different nanoscopic or microscopic processes can surface tension as γ ¼ σð2 2Þ=3. γ and γ are the surface 1 0 no doubt have this effect. However, whatever the origin, the effect energies of the dry and wet solid surface, respectively, so that the can be described as a single parameter, the CL friction. equilibrium contact angle θ is given byðÞ γ γ =γ ¼ cos θ . The e e 1 0 So far, the dependence of line friction on the contact angle has form of the function gCðÞ¼ðÞ 2 þ 3C C =4 is chosen in relation received limited attention, but there is clearly every reason for line to the function f in Eq. (4). npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA 1234567890():,; G. Amberg Yue developed a phase ﬁeld treatment of contact angle angle of θ = 101°. This will be the reference case for the hysteresis, where advancing and receding contact angels are simulations shown here. introduced in a piecewise continuous function on the right-hand The nondimensional numbers for this case are Oh = 0.00256, side of the equation corresponding to Eq. (11). The CL is then amplitude A = 0.047, and angular velocity of the driving ω = 4.41. allowed to move according to whether the value of the dynamic The low value of the Ohnesorge number indicates that the droplet contact angle exceeds (subceeds) the advancing (receding) angle. A dynamics are inertial and that viscosity plays a minor role. Near treatment inspired by this is also developed for level-set methods . the solid surface, we should expect a viscous Stokes layer of pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ The left-hand side represents the dissipation associated with CL thickness μ=ðρπfÞ, which is approximately 0.07 mm for the motion, quantiﬁed by the parameter μ , which we will call CL experimental parameters, i.e., much less than the initial droplet friction. This has dimensions of viscosity. radius of 2.1 mm. By considering solutions to the Cahn–Hilliard equations near The order unity value of the angular velocity shows that the equilibrium at the CL, an explicit relation between CL speed U oscillations are reasonably matched with the inertial timescale, as is CL and dynamic contact angle θ can be derived from Eq. (11), see expected since the experiment aims at being near resonance. The ref. and Eq. (3): value of the line friction Ohnesorge number Oh is set to unity, which pﬃﬃﬃﬃﬃﬃﬃﬃ makes the reference value of the line friction μ ¼ ργR. We will f ;ref μ U cosθ cosθ f CL e return to the more detailed modeling of the line friction below. ¼ θ θ (12) γ sinθ Figure 1 shows the simulation results over one cycle for the M00 case. In Fig. 1a, the ﬁrst frame shows the position near the lower The dynamic contact angle is equal to the equilibrium angle if turning point when the droplet is compressed towards the μ = 0, in which case there is also no dissipation at the CL. The last substrate. In the second frame, it is extending upwards and is near approximate equality holds if θ is near θ . its most elongated state in the third frame. In the fourth frame, it is The above equations are made nondimensional using R as being again compressed as the substrate is moving upwards. length reference, the radius of the half-sphere (approximately the A Weber number based on the oscillation amplitude A and initial wet footprint radius), and an inertial capillary velocity 2 2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ frequency f can be expressed as A ω , giving the value U ¼ γ=ðÞ ρR . 2 2 0:047 4:41 ¼ 0:043. We thus expect the ﬂow to be dominated by the surface tension. The nondimensional parameters that appear are Figure 1b shows the corresponding pressure ﬁelds, always pﬃﬃﬃﬃﬃﬃ Oh ¼ Ohnesorge number reﬂecting the curvature of the interface. The pressure is elevated ργR in the droplet due to the capillary pressure and ﬂuctuates as the pﬃﬃﬃﬃﬃﬃ Oh ¼ Line friction Ohnesorge number ργR interface curvature varies. In the most compressed state in the ﬁrst ρgR 2 frame of Fig. 1b, there is a concave shape at the top of the droplet, Bo ¼ Bond number, g ¼ 9:81 m=s <?tpb5pt?> γ qﬃﬃﬃﬃﬃﬃ 3 which causes a local low pressure there. The pressure in the air is ρR ω ¼ 2πf Nondimensional oscillation angular frequency. f is the almost constant, due to the low air density. Figure 1c shows the oscillation frequency in Hertz velocity ﬁelds for the points where the droplet is extending A= A /R Nondimensional oscillation amplitude upwards and being compressed (corresponding to frames two and four of Fig. 1a, b). In Fig. 2 are shown the time signals for the vertical substrate In addition to these, the Cahn–Hilliard equations use the position, together with the CL position and the droplet height. It is following parameters: noticed that the amplitude of the droplet height is about three qﬃﬃﬃﬃ UR γ R pﬃﬃ Pe ¼ ¼ Peclet number related to phase ﬁeld mobility. Set to times that of the substrate amplitude, as expected for a near 2 2κγ D ρR 3 ε 100 in the simulations. Reported here resonance situation. All signals quickly go into a periodic motion, Cn ¼ Cahn number, nondimensional interface width. Kept at with no sign of period-doubling or other more complex dynamics. 0.01 in the simulations. Reported here The responses in both the droplet height and the CL position are air nonlinear, however, with shapes departing from sinusoidal. Ratio between air and liquid density. Set to 0.01 here, liq for computational convenience The CL position signal deviates from a sinusoidal shape, and tends air towards a square wave, with ﬂat peaks. This is a signature of a stick- Ratio between air and liquid viscosity. Set to 0.03 here, liq slip motion of the CL; it is rather stationary at its extreme values but for computational convenience shifts quickly between them twice per cycle. We will see that this is built into the simulation through angle-dependent line friction. The simulations are carried out using an adaptive ﬁnite element solver, as in ref. . The adaptive grid is automatically reﬁned as Model for angle-dependent line friction needed down to an element size of 0.001. The air viscosity and In addition to the input data already discussed, angle-dependent line density are taken larger than those of air but much smaller than friction has been implemented in this simulation. The line friction in the values for the liquid. They are deemed small enough for the air Eqs. (11)and (12) is computed from a regularized well function: to have a negligible inﬂuence on the ﬂow inside the droplet. μ ¼ μ μ ðÞ θ (13) f f ;ref Reporting summary where the nondimensional μ ðÞ θ is Further information on research design is available in the Nature μ ðÞ θ ¼ μðÞ 1 HðÞ θþ μ HðÞ θ (14) Research Reporting Summary linked to this article. f 0 f 1 and RESULTS θ ðÞ θ þ dθ θ ðÞ θ dθ e e HðÞ θ ¼ 1 þ 1=2 tanh tanh Flow ﬁelds δ δ Xia and Steen report the most detailed results for their experiment (15) labeled M00, using a 20 µl water droplet oscillated at f = 61 Hz, with an amplitude of A = 0.1 mm. The substrate is a slightly Equation (14) thus describes nondimensional line friction that hydrophobic PDMS-covered silicon surface, with a static contact has a high value μf in the sticking region in the angle range θ ± 0 e Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 1 G. Amberg Fig. 1 One oscillation cycle for the M00 case. a The shape of the droplet at four instants. b The pressure ﬁeld. c Velocity ﬁeld at the instants corresponding to frame two and four of a and b. Fig. 2 Time signals for the contact line position r (blue), the CL droplet height z (yellow), and the substrate position Asinωt (gray). The substrate position is drawn around an equilibrium position of 1, to facilitate comparison with the CL position and the droplet height. dθ, and a low value μf outside of this range. δ smooths the corners of the well function and μ is a dimensional reference f,ref value that is used in the line friction Ohnesorge number Oh . In the pﬃﬃﬃﬃﬃﬃﬃﬃ present case, Oh = 1, and thus μ ¼ ργR. In the simulation in f ;ref Figs. 1–6, these values are: θ = 101°, dθ = 7°, δ = 1°, μf = 10, e 0 μf = 0.5 and Oh = 1. We could interpret θ + dθ = θ as 1 f e a approximating the advancing contact angle and θ − dθ = θ as e r the receding angle. Inserting the line friction according to Eq. (14) into Eq. (12), we readily obtain the dynamic contact angle as a function of CL speed (see Fig. 3). As expected, the high line friction near the equilibrium angle creates a region of near stick, with low velocities for angles in a region ≈±7° around the equilibrium. In this region, the well function H(θ) ≈ 0, and μ ðÞ θ will be constant and equal to the high Fig. 3 Model of angle-dependent line friction, according to Eqs. value μ . For CL speeds up to about 0.3, the angle is f0 (12)–(15). a The deviation of the dynamic contact angle from the approximately constant, between 7 and 10 degrees, establishing equilibrium angle θ − θ as a function of contact line speed. a “slip” region. For angles departing >10 degrees from equilibrium, b Nondimensional line friction as a function of θ − θ . Parameter or speeds larger than approximately 0.4, the well function H(θ)is1 values are the same as for the simulations in Figs. 1, 2, and 4: θ = and the line friction μ ðÞ θ becomes equal to the low value μ .To 101°, dθ = 7°, δ = 1°, μ = 10, and μ = 0.5. f1 f0 f1 npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA G. Amberg Fig. 4 Comparison of phase plots with Xia and Steen experiments for the case M00. Colored dots are the experimental results of Xia and Steen, and the black solid curves are the present simulations. a Dynamic contact angle departure from the equilibrium angle θ − θ as a function of contact line speed. b Contact line speed vs contact line position, as a departure from the mean position. c Dynamic contact angle θ − θ vs contact line position. d Contact line speed vs contact line position, as a departure from the mean position. Dynamic contact angle departure from the equilibrium angle multiplied by contact line positionðÞ r r ðÞ θ θ vs contact line position multiplied by contact line CL ref e speedðÞ r r U . CL ref CL summarize: in the windowjj θ θ < dθ, the line friction is magnitude of the sticking region in Fig. 4a, and the sticking region constant, equal to the high value μ , and for angles distinctly line friction, μ , sets the slope of the curve at the origin in Fig. 4a. f0 f0 outside this window,jj θ θ > dθ, the line friction tends to the The transition region width δ allows for the smooth curvature of constant value μ . Connecting these two regions is a velocity the TD at the end of the sticking region. The line friction far from f1 range where the dynamic angle is approximately constant, i.e. a equilibrium μ is then chosen to approximate the slope of the TD f1 region of “free slip”. in the “free slip” region. It should be noted that the numerical As seen in Fig. 3a, there is still some variation of angle with CL simulation accurately reproduces the theoretical curve in Fig. 3a speed also in the region of “free slip”. The slope of the curve in and that the CL velocities encountered in the experiment are Fig. 3a in this region is related to δ, the width of the transition always below about 0.3, so that only the “sticking” and the “free region in line friction in Fig. 3b, and the line friction far from slip” regions are visited. equilibrium μ . Approximating the contact angles and the CL Figure 4b shows a phase plot in terms of the CL speed vs the CL f1 speeds at the ends of the transition region as θ ¼ θ þ dθ þ δ, position. In addition to the experimental data shown as colored 0 e θ ¼ θ þ dθ δ,and U ¼ðθ θ Þ=μ , U ¼ðθ θ Þ=μ ,and circles, the black circles denote the position of the substrate. The 1 e 0 0 e 1 1 e f 0 f 1 assuming that μ μ , an estimate of the slope in Fig. 3a in the black line is the trajectory from the simulation. It captures both f 1 f 0 transition or “free slip” region can be obtained as Δθ=ΔU the elongation and the slight inclination of the experimental loop. dθ 2μ = 1 þ (angles in radians). For a very steep transition region The width of the simulated trajectory is slightly less than the f 1 (δ dθ) the slope will thus approach zero, while it becomes experimental one, but overall, the agreement is good. Figure 4c shows the dynamic contact angle vs CL position. Here comparable to μ for δ ~dθ. f1 The model for line friction in Eqs. (13)–(15) was constructed to the stick–slip character of the motions is evident; the shape of the be as conceptually simple as possible, characterized by a high trajectory is a quadrilateral, where the horizontal upper and lower friction region around equilibrium, and low friction further from parts show the rapid phase when the CL moves from one almost equilibrium. In order to test this against the experiments of Xia steady position to another, at a fairly constant contact angle. The and Steen the simulated results are overlaid on their experimental vertical sides represent the “stick” phase where the CL is nearly phase plots (their Figs. 4 and 5), see Fig. 4. stationary and the angle changes. Figure 4a shows the graph of dynamic contact angle vs CL Figure 4d was introduced by Xia and Steen to highlight the speed, the “traditional diagram” (TD) as it is referred to by Xia and “stick” and the “slip” parts of the motions and quantify those Steen. The colored circles are the experiments of Xia and Steen, separately. The graph shows dynamic contact angle departure and the black curve is obtained from the numerical simulation. If from the equilibrium angle multiplied by CL position the CL friction would be a constant this curve would always be a ðÞ r r ðÞ θ θ vs CL position multiplied by CL speed CL ref e straight line, thus the angle variation in the CL friction becomes ðÞ r r U . The nonlinearity that is introduced creates two CL ref CL important. The parameters dθ, δ, μ , and μ for the angle- loops, one for the receding and one for the advancing motion of f0 f1 dependent line friction are chosen to give a fair approximation of the CL. The simulated trajectory is again in fair agreement with the the TD, in the following manner. dθ is chosen to capture the experiment, even though the experiment extends somewhat Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA npj Microgravity (2022) 1 G. Amberg Fig. 5 Comparison of phase plots with Xia and Steen experiments for the case M00, using a closer ﬁt for the TD. Colored dots are the experimental results of Xia and Steen, and the black solid curves are simulations. a Dynamic contact angle departure from the equilibrium angle θ − θ as a function of contact line speed. b Contact line speed vs contact line position, as a departure from the mean position. c Dynamic contact angle θ − θ vs contact line position. d Contact line speed vs contact line position, as a departure from the mean position. Dynamic contact angle departure from the equilibrium angle multiplied by contact line positionðÞ r r ðÞ θ θ vs contact line position CL ref e multiplied by contact line speedðÞ r r U . CL ref CL further from the origin. In view of the scatter in the experimental experimental points show some differences between the advan- data the agreement in Fig. 4 was deemed sufﬁcient for the present cing and receding curves. A more complex mathematical discussion. expression for the angle dependence of the line friction was In terms of the model in Eqs. (13)–(15), we see that the sticking designed so that the TD is approximated more closely, see Fig. 5a. phase is characterized by a high constant line friction μ = 10. This f0 Overall the agreement in all four panels is somewhat improved shows up in the central part of Fig. 4d, where the simulated but not dramatically so. The conclusion is that the complexity of trajectory is steep and near the red experimental circles. The slope Eqs. (13)–(15), with the four parameters dθ, δ, μ , and μ is f0 f1 of the curve here is proportional to μ . The main source of CL f0 sufﬁcient to capture the essential features of the ﬂow. dissipation is at the slip phase, as the uncompensated Young’s stress at the receding or advancing contact angle multiplies the CL Energy dissipation speed. Here the CL slip speed becomes rather independent of the dynamic angle. In a macroscopic experiment where the overall CL It may be asked what the nature is of the dissipation that limits the speed is measured along with the contact angle, an overall CL response amplitude. In the simulation for the standard M00 case, friction for an advancing CL would be identiﬁed from Eq. (12)as the input energy per cycle was determined from the pressure and μ ¼ðÞ γ=UðÞ cosθ cosθ =sinθ . This would not be possible to speed of the substrate integrated over a cycle. Figure 6b shows a f CL e a a directly relate to material parameters, since U here would be CL graph of the total vertical force F from the substrate, vs the determined from the overall droplet dynamics, and the angle substrate position over a cycle. In the same manner, the would stay nearly constant, close to the advancing angle. To dissipation at the CL was determined from a graph of predict the wetting behavior, the CL friction would need to be 2πrðÞ cos θ cos θ vs r graph, see Fig. 6a. CL e CL modeled and the overall dynamics simulated. For angles further The total amount of work supplied to the drop by the oscillation away from the sticking region, μ ðÞ θ becomes μ and Eq. (12) f1 over a cycle is now the area inside the loop in Fig. 6b, which is reduces toðÞ μ U =γ ¼ðÞ cosθ cosθ =sinθ with a constant line f 1 CL e calculated as 0.132914. The area inside the loop in Fig. 6a giving friction. We do not however have any experimental data to verify the CL dissipation is 0.131779. As expected, the work input is if this is indeed the case. slightly larger than the CL dissipation, with a relative difference of The CL mobility parameter M (Eq. (1)), as identiﬁed by Xia and 0.8%. We should not overinterpret this small difference, in view of Steen is the inverse of the slope of the line formed by the blue other possible sources of inaccuracy, but this still shows that the circles in Fig. 4d. This is also approximately the same as the inverse CL dissipation is the completely dominating cause of damping in of the slope of the straight line obtained by connecting the two this case. There is an almost perfect match between the input regions of blue circles in the “wings” of Fig. 4a. energy and the energy dissipated at the CL over a cycle. The other The agreement with the experiment in Fig. 4 is fair, but there source of dissipation by bulk viscous dissipation is indeed are some differences, so more precise modeling of the experi- mental points in the TD in Fig. 4a was made. The model in Eqs. expected to be quite small, given the small value of the (13)–(15) is symmetric around the equilibrium angle, while the Ohnesorge number (Oh = 0.00256). npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA G. Amberg Fig. 6 Calculation of CL dissipation. a Uncompensated Young’s stress 2πrðÞ cos θ cos θ vs r over one cycle. The area inside the loop is CL e CL 0.131779. b Vertical force from the substrate on the drop vs substrate displacement atðÞ ¼ A sin ωt. The area inside the loop is 0.132914. DISCUSSION CODE AVAILABILITY The code used in the current study is available from the corresponding author on The good qualitative and quantitative agreement between reasonable request. simulation and experiment, in all the different aspects that can be accessed through the phase plots presented by Xia and Steen, Received: 1 August 2021; Accepted: 6 December 2021; indicates that the essential physical processes are well represented in the mathematical model. The experiment is quite sensitive since the response is ampliﬁed through the resonance of the droplet motion, and the damping that is present in the system is controlling the resulting periodic CL motion completely. As shown REFERENCES in Fig. 6, the damping is almost entirely due to the CL dissipation. 1. Bonn, D. et al. Wetting and spreading. Rev. Mod. Phys. 81, 739–805 (2009). Simulations were performed where the CL dissipation is removed, 2. Snoeijer, J. H. & Andreotti, B. 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Fluid Mech. 645, 279 (2010). reprints 28. Carlson, A., Do-Quang, M. & Amberg, G. Modeling of dynamic wetting far from equilibrium. Phys. Fluids 21, 121701 (2009). Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims 29. Yue, P. Thermodynamically consistent phase-ﬁeld modelling of contact angle in published maps and institutional afﬁliations. hysteresis. J. Fluid Mech. 899, A15 (2020). 30. Zhang, J. & Yue, P. A level-set method for moving contact lines with contact angle hysteresis. J. Comput. Phys. 418, 109636 (2020). 31. Carlson, A., Do-Quang, M. & Amberg, G. Dissipation in rapid dynamic wetting. J. Open Access This article is licensed under a Creative Commons Fluid Mech. 682, 213–240 (2011). Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative ACKNOWLEDGEMENTS Commons license, and indicate if changes were made. The images or other third party This work was begun in discussion with Professor P.H. Steen and was interrupted only material in this article are included in the article’s Creative Commons license, unless by his untimely passing away. I gratefully acknowledge his role in formulating the indicated otherwise in a credit line to the material. If material is not included in the questions discussed here and all the contributions he made to science throughout article’s Creative Commons license and your intended use is not permitted by statutory his career. I wish to thank Yi Xia for clarifying details about the experiments and regulation or exceeds the permitted use, you will need to obtain permission directly Vanessa Kern for discussions. from the copyright holder. To view a copy of this license, visit http://creativecommons. org/licenses/by/4.0/. FUNDING © The Author(s) 2022 Open access funding provided by Södertörn University. npj Microgravity (2022) 1 Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA

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Detailed modelling of contact line motion in oscillatory wetting

Detailed modelling of contact line motion in oscillatory wetting

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Detailed modelling of contact line motion in oscillatory wetting

Detailed modelling of contact line motion in oscillatory wetting

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