Pinning and depinning of Wenzel-state droplets around inclined steps

Colloid and Interface Science Communications 35 (2020) 100238

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Pinning and depinning of Wenzel-state droplets around inclined steps a,⁎

b

Umut Ceyhan , Aslı Tiktaş , Mert Özdoğan a b

b

T

Department of Mechanical Engineering, İzmir Kâtip Çelebi University, İzmir 35620, Turkey Graduate School of Natural and Applied Sciences, İzmir Kâtip Çelebi University, İzmir 35620, Turkey

ARTICLE INFO

ABSTRACT

Keywords: Hysteresis Wetting Sliding droplet Lubrication theory Interfaces

Droplets moving over inclined heterogenous substrates may pin around rough spots. A simplified model of one of these spots having a backward facing step geometry over a chemically homogeneous inclined substrate reveals that the step is responsible from the pinning-depinning of the leading edge of the droplet in Wenzel-state. We consider a two-dimensional partially wetting droplet and model its motion by the evolution equation with a precursor film model. The phase diagrams formed define pinning-depinning transition curves determined by a simple force balance of retention forces due to hysteresis, step and gravitational force. For fixed Bond number and inclination, we show that it is the slope of the step that determines transition. The existence of the multiple steps, however, alters this transition as the receding contact line may pin at the steps and leave residual droplets behind.

1. Introduction The understanding of the droplet motion on surfaces is crucial either to obtain clean surfaces from stains such as solar panels and car windshields [1–3], to increase the heat transfer efficiency [4,5], to keep droplets in place in some industrial processes such as printing of electronics [6], or to promote the uphill motion of droplets [7]. Besides, many researchers have studied droplet motion in nature which they have become later an inspiration for the design of new materials or structures [8–12]: for instance, Namib Desert beetles harvest droplets from the fog-laden by tilting their body towards the wind. Their wing surface consists of combined hydrophilic and hydrophobic bumps which allow them to coalesce small droplets into the bigger ones inside the fog to obtain drinking water [13,14]. Similarly, the structure of the mosquito eyes (C. pipiens) has superhydrophobic anti-fogging property for maintaining a clear vision in dark areas. The re-entrant structures observed on the surface of lotus leaves have been discovered to be responsible from the unexpectedly high repellency. Surfaces having reentrant curvatures [15–17] allowing oleophobicity encounters pinning and depinning of the contact lines around some topographical heterogeneities. These structures let even the very low-energy liquids suspend and generate superrepellent substrates which can be used for oil-water separation, electronics cooling by nucleate boiling, etc. In most of the aforementioned examples, the motion of droplets is susceptible to topographical and/or chemical heterogeneities which result in non-unique contact angles the droplet makes with the surface.

⁎

To understand the behaviour of contact lines around structured substrates, we study the motion of Wenzel-state droplets around inclined steps. The simple step structure allows us to mimic troughs and peaks. When the substrate is ideal (i.e. atomically smooth and chemically homogeneous), the equilibrium contact angle, θe, droplet makes with the substrate is unique and given by its Young [18] value as

cos

e

=

sv

sl lv

(1)

where γsv, γsl and γlv are the surface energies between solid-vapor, solidliquid and liquid-vapor interfaces, respectively. Later, Dupré [19] defines a similar equation concerning the work of adhesion for wetting. When the substrate is not ideal, however, the contact angle deviates from its Young value. The measured advancing contact angle differs from the receding one, this difference is termed as contact angle hysteresis [20–22]. This can be due to chemical heterogeneity, roughness and adsorption into the surface which may alter the surface properties [23,24]. Cassie and Baxter [25] developed an equation to evaluate equilibrium contact angles over chemically heterogeneous surfaces. Similarly, Wenzel [26] derived an equation for chemically homogeneous but rough surfaces by introducing surface roughness factor. It is observed, however, that for the same roughness factor, the measured contact angles may differ due to different topography. The advancing and receding of the contact lines over pillars do not behave in the same way as pored structures [27]. The classical Wenzel-state or CassieBaxter state relations are not sufficient to explain this discrepancy as

Corresponding author. E-mail address: [email protected] (U. Ceyhan).

https://doi.org/10.1016/j.colcom.2020.100238 Received 22 October 2019; Received in revised form 11 January 2020; Accepted 23 January 2020 2215-0382/ © 2020 Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).

Colloid and Interface Science Communications 35 (2020) 100238

U. Ceyhan, et al.

Nomenclature A Bo Ca f f f0 g h hs i j L n p q r0 S t u V V x y

λ Π ρ θ ζ

Droplet volume per unit depth Bond number Capillary number Film thickness Force vector Precursor film thickness Magnitude of gravitational acceleration Substrate topography function Step height Unit vector along x-coordinate Unit vector along y-coordinate Domain length Unit outward normal at the interface Pressure Volume flow per unit depth Initial radius of curvature Spreading parameter Time x-component of velocity Speed Droplet volume x-coordinate y-coordinate

Subscripts A c cg CL e lv m R r ret s sl sv

Advancing Critical Center of gravity Contact line Equilibrium Liquid and vapor Mesoscopic Receding Residual Retention Step Solid and liquid Solid and vapor

Superscripts l s v

Greek letters α η γ

Step steepness parameter Disjoining pressure Fluid density Contact angle Interface measured from horizontal

Liquid Scale Vapor

Other symbols

Substrate inclination angle Fluid dynamic viscosity Surface tension between interfaces

~

they both use averaging quantities for the effective interface energy. The existence of sub-micron scale concave surfaces, for example, alter the contact angles and hysteresis [15]. A simple way of visualization of hysteresis is the observation of droplet motion over tilted surfaces. Over an inclined flat substrate, the balance between gravitational and surface tension forces determines the shape of the droplet; while the former pushes the droplet down the incline, the latter pulls the droplet against the gravity due to a retention force. The droplet takes an asymmetric shape having contact angles different from the equilibrium contact angle θe on a flat substrate. When the critical inclination angle is achieved, the droplet moves over the substrate; the leading edge of the droplet advances with an advancing contact angle θA while its trailing edge recedes with a receding contact angle θR which is the dynamic component of the contact angle hysteresis originating from the droplet motion. The difference between these angles and Young angle depends on the Capillary number, Ca based on contact line speed defining the ratio of viscous forces to surface tension forces, and scales linearly with Ca1/3 which is given by the Cox-Voinov law [28,29]. The ratio of gravitational to surface tension forces defined by the Bond number, Bo, is a critical control variable on the motion and shape of the droplet over inclined surfaces. The competing force between gravity and interfacial forces causing retention due to contact angle hysteresis determines both the shape and the motion. Furmidge [21], Macdougall and Ockrent [30] experimentally show that there exists a critical inclination angle, αc, at which a droplet starts moving, i.e. whenever the gravitational force acting on the droplet is equal or greater than the contact line retention forces due to surface tensions, it moves. This condition is provided by the tangential force balance acting on a droplet over a flat substrate and given by

cos

Mark used for dimensional quantities

R

cos

A

=

gA

sin

c

= Bo sin

c.

(2)

In eq. (2), A is the droplet volume per unit depth for a two-dimensional droplet, ρ is the liquid density, g is the gravitational acceleration and γ is the surface tension between liquid and vapor. Different variations of eq. (2) are proposed to study a specific problem and are validated by others [31,32]. For small Bo sin α values, the two-dimensional droplets take almost circular arc shape; beyond a critical inclination angle, however, e.g. for fixed Bo, the droplet undergoes a wetting transition resulting with an elongated profile and completing this transition, it moves with a terminal speed [33–35]. Any model of droplet motion over substrates needs the treatment of the contact line singularity. The no-slip condition can be alleviated such as by using slip models [35–38], precursor film models. In the latter, the moving contact line is replaced with an apparent one and connected to a precursor film of constant thickness in front of the contact line at the mesoscopic length scale where the molecular interactions matter: there exists a disjoining pressure Π(f) [39], where f is the film thickness, with various models such as the ones derived from van der Waals forces [40,41] or diffuse interface models [42]. The existence of precursor film allows the macroscopic contact line to spread into a thin mesoscopic (or microscopic in some applications) film by which the stress and pressure singularities are alleviated. To see the difference between slip and precursor models, Savva and Kalliadasis [43], Sibley et al. [44] investigate these two models and show that modification of the model equations for precursor and slip model leads to equivalent quasi-static limit which is a sign of same dynamics for different models. Savva and Kalliadasis [35,45] work on two-dimensional droplets on heterogeneous surfaces by addressing both horizontal and inclined heterogeneous surfaces (including both chemical and periodic, singlewavelength and small-amplitude topographical heterogeneities), respectively. Using lubrication approximations, they model the droplet 2

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U. Ceyhan, et al.

motion with a slip boundary condition. They review the static and dynamic behavior of the droplet motion and use a singular perturbation method in order to evaluate position of the moving drops as well as the critical angle. Without imposing the hysteresis effect a priori, the equilibrium of droplets cannot be preserved over ideal surfaces, however, droplets can reach their equilibrium on the substrate having heterogeneities. Thiele et al. [34] revise the analysis of the evolution of a two and three-dimensional thin liquid films over tilted surfaces. In their model, they include a disjoining pressure which is formulated by a diffuse interface theory. They show that if the inclination angle increases, so does the speed of the contact line, and diffuse interface thickness increases and it provides the formation of a thinner drop and stronger oscillations. When the surfaces have topographical heterogeneities, the motion also depends on the defect parameters which may either cause the droplet to stick to it, or delay its motion. Park and Kumar [46] study on two-dimensional droplet, advancing over a surface having a single defect tilted at a certain angle measured from their level surface. Unlike Savva and Kalliadasis [35], who works on the behavior of droplets on tilted heterogeneous surfaces having periodic structures, Park and Kumar [46] focus on a single topographical defect of Gaussian shape over a flat tilted substrate to examine the influence of pinning at the defect by using lubrication equations utilizing the precursor-film model. They observe that droplet pinning occurs at the point which has a minimum slope on the defect surface. If retention force due to the hysteresis is bigger than sliding forces, droplet remains pinned at a nonzero sliding angle. At this configuration, advancing contact angle is maximum and retention forces are maximized. According to the experimental study of Kalinin et al. [47], pinningdepinning of a droplet is mostly determined by the slope of the topographical defect (the steepness of a step down of the rings with trapezoidal cross-sections), consistent with Gibbs inequality [48]. In general, as the slope of step increases, so does the apparent advancing angle. In other words, the strength of the pinning increases. Recently, Escobar et al. [49] observe the contact line pinning at the side-walls of the concentric rings with which they explain the zipping-depinning mode of sessile microdroplets. We study the motion of two-dimensional Wenzel-state droplets using lubrication approximation with a precursor film model. A typical rough surface consists of several troughs and peaks which can be modeled as step-ups and downs. Here, we define the droplets to be in the Wenzel-state to emphasize that the liquid wets the substrate and there is no air entrapment within the substrate structures though this state is used for relatively large contact angles; we do not make use of the Wenzel equation. Considering only one of the rough spots, the leading edge of the moving droplet first encounters with the positive slope side (with respect to motion direction), the apparent contact angle (measured from horizontal) decreases and the advancing contact line moves faster. It continues its motion slowing down towards the negative slope side. Depending whether retention forces dominate over the gravitational forces, the droplet either pins or depins. Assuming that there exist no sharp corners, because the negative slope side (stepdown) of the spot is responsible from this pinning and depinning process, we simplify the spot geometry to backward-facing step and first study on pinning-depinning transition of the advancing side of the droplet. The phase diagrams of this transition reveal that the critical condition is determined by a simple force balance around the contact line of the droplet; the difference from the flat substrate is the retention force due the existence of topography change. The existence of roughness around the receding contact line, however, alters this transition as it may also pin at one of these spots and may leave residual droplets behind the forward-facing steps. In the following sections, we first introduce our model problem, summarize the numerical technique to integrate the resulting equations and its validation (with details given in the Appendix), and discuss on

y f0

ζ g x

h(x) α

Fig. 1. Problem domain: droplet around an inclined step.

the pinning-depinning of droplets around the inclined steps. 2. Model problem In Fig. 1, we show a representative two-dimensional droplet around a step with a uniform precursor film of thickness f0 away from both contact lines. We fix the coordinate at the substrate and define its topography by h (x ) , the interface location is measured from y = 0 and h . The topography is dethe film thickness, f , is defined as f = fined as a step shown in Fig. 1 using a hyperbolic tangent function:

h (x ) =

hs 1 2

x

tanh

xs (3)

where hs is the step height, xs is the location at which the step has its most negative slope, is the steepness parameter; the smaller is, the steeper the step. As we will drop tildes shortly, in Fig. 1, we show the variables without tildes for convenience. We treat the fluid to be incompressible, Newtonian and non-volatile with uniform density ρ and dynamic viscosity η. The substrate is inclined with the horizontal at an angle α and the gravitational acceleration g = g(sinαi − cos αj) acts downwards. For a small slope interface, using the long-wave approximation, the interface , measured from y = 0 , satisfies

q =0 x

+

t

(4)

where q is the volume flow rate per unit depth passing across the film thickness f which is found by integrating the velocity across it. The xcomponent of the velocity satisfying the no-slip boundary condition at y = h (x ) and vanishing shear stress condition at the interface is

u=

p x

1 2

(y 2 + 2 (h

g sin

y)

2

h ).

(5)

Here, p is the pressure and its gradient to be determined from the normal force balance on the interface . Integrating (5) across the film from h to , the volume flow rate per unit depth q is found to be

q=

1 3

p x

g sin

(

h )3 .

(6)

The interfacial momentum balance in the normal direction determines the pressure difference at the interface as

pl

pv =

n

(7)

.

In eq. (7), and refer to the interface liquid and vapor pressures, respectively; γ is the surface tension between liquid and vapor interface, treated to be uniform throughout the study, n is the unit outward normal at the interface and is the disjoining pressure. We obtain the pressure distribution by integrating the y-momentum balance and is given by

pl

p = p v + g cos (

pv

y)

2

x2

.

(8)

The variation of the interface normal contribution to the pressure in 3

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U. Ceyhan, et al.

(8) is the fourth term on the right-hand side and we use the small slope approximation of the interface (which is consistent when θe is small) to

thickness f0 and equilibrium contact angle θe, the last two pairs result with slower convergence due to the increase in both the gradients of disjoining pressure and its energy density. We integrate eqs. (11−13) in space using a cubic finite element discretization and implicitly in time using a second order CrankNicolson scheme. We give the details of the numerical method in Appendix A and its validation in Appendix B.

2

treat the n as , to a leading order. We assume the vapor pressure x2 at the interface to be uniform and differentiate (8) with respect to x to rewrite (6) in terms of the interface , then we obtain the interface evolution equation to be

t

+

h )3

( x

g cos

3

sin

x

3

x3

3. Droplet motion around an inclined step

= 0.

x

As we stated earlier, to be able to analyze the motion of moving contact lines around rough surfaces or substrates having topographical structures, we simplify the substrate topography having a single backward facing step as shown in Fig. 1 to mimic a smooth roughness. We initialize the droplet motion by adding a circular arc droplet onto a precursor film thickness of f0 away from the step. Here, we should note that the initial circular arc approximation of the droplet profile is unsteady and it takes some time to reach an equilibrium shape. However, for fixed volume, the pinning-depinning transition is not affected by the initial profile's circular arc approximation other than the one obtained by the terminal profile of an initially circular arc sitting on flat substrate although it affects the time taken until the transition around the step occurs. The initial location of the droplet center is at x = 1.1 whereas the step is located at x = 2.6. We consider a partial wetting fluid and set the equilibrium contact angle to θe = 15°; this provides us with a slender droplet for which the long-wave approximation used in §2 is consistent. Considering a three-dimensional droplet on an inclined flat substrate shown in Fig. 2 (top), the retention force parallel to the substrate can be computed by integrating the surface tension forces at the triple line along the contour of the droplet's contact with the substrate:

(9)

We consider an initial two-dimensional partial wetting droplet being an arc of a circle meeting the substrate at equilibrium contact angle θe whose volume per unit depth is A = r02 ( e sin e cos e ) with r0 being the initial radius of curvature of the droplet and non-dimensionalize using the following length, velocity, time and pressure scales

e

f = f sf =

A (1 e

u = usu =

t = t st =

p = ps p =

1/2

A sin2 e sin e cos

x = x sx =

x,

(10a)

e

cos e ) 2 sin e cos e

1/2

f,

(10b)

(1 cos e )3 u, 3 sin3 e

3 sin3 e (1 cos e )3 (1

cos

(10c)

A sin2 e sin e cos

e e)

2(

sin

e

A sin4

1/2

t,

(10d)

e

e cos e )

fret =

1/2

p.

(10e)

e

s

sin e sin 1 cos

fret =

3 (f e

+ h) x3

lv (cos R

= 0.

x

g (x s ) 2

f (0, t ) f (L , t ) = = 0, x x

(12)

where f0 is the dimensionless precursor film thickness; we set f0 to 0.001 in all of the following simulations. The choice of precursor film thickness affects the contact line region. The thicker precursor films do not predict the imposed contact angles while the one we used, f0 = 0.001, predict the contact angle within an error of 1°. Going further below to f0 = 0.0005 does not vary the profile, but it has slower convergence compared with f0 = 0.001; therefore, we pick f0 = 0.001. For the disjoining pressure term, we use two-term disjoining pressure model [41] with powers m = 3, n = 2; its non-dimensional form is

=

1 e sin e f0 1 cos e

2

f0 f

3

f0 f

2

.

+

sl

sv ) nds .

(14)

lv (cos R

cos

(15)

A ) i.

cos

A)

= gA sin

c.

(16)

The term on the left-hand side of (16) is the retention force due to hysteresis and the one on the right side is the gravitational force on the droplet acting in the direction of motion. A similar equation is given by Furmidge [21], Macdougall and Ockrent [30]. Whenever α > αc, the gravitational forces overcome the retention forces due to hysteresis and the droplet starts sliding. On flat substrate, for a three-dimensional droplet symmetric with respect to x-axis and when ∮ CLnds = 0 or for a two-dimensional droplet, the lateral force contributions due to tensions γsl and γsv cancel for a chemically homogeneous substrate. That is not the case around the step (Fig. 2 (bottom)): there the lateral force of γsv − γsl at the receding contact line is bigger than the x-component of the force at the advancing contact line around the step which is (γsv − γsl) cos θs, A where θs, A is the angle of the step at the advancing contact line location measured from x-axis (see the magnified region at the bottom of Fig. 2). Summation of these two, where we replace γsv − γsl with γlv cos θe by Young condition, results with an additional term due to existence of a step, and the balance between these retention forces and gravitational force determines the pinning and depinning of a droplet around the step:

(11)

The Bond number in (11) is defined as Bo = . The boundary conditions away from both droplet and step (x = 0 and L, length of the domain) are

f (0, t ) = f (L, t ) = f0 ,

lv cos

For a flat substrate as shown in Fig. 2 (middle), the critical condition for the onset of two-dimensional droplet motion occurs when

(f + h ) x

f 3 Bo cos

(

In the case of two-dimensional droplet shown in Fig. 2(middle), (14) reduces to

s

In eq. (10), the variables x, f, u, t, p are non-dimensional and x , f , us, ts, ps are the corresponding scales. The dimensionless evolution of the film thickness is then governed by

f + t x

CL

lv [cos R

(13)

The choice of parameters m and n is based on the comparison of the results and the convergence of this choice with other possible powers such as m = 9, n = 3 and m = 4, n = 3. For fixed precursor film

cos

A

cos

e (1

cos s, A )] ~ gA sin

(17)

The advancing contact angle θA is the summation of the step angle θs, A and mesoscopic angle θm, A at the advancing contact line which are 4

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to the mid-point of the step at which the retention force due to it is maximum. It is clear from the inset that the contact line location approaches middle of the step as α → αc. A similar observation is made around a Gaussian shape defect [46].

y

x

n

x

n=-i

θR θA

x θs,A

n=-i

n=i

θm,A

θR θA

n

Fig. 2. (top)Three-dimensional droplet sliding along inclined flat substrate, n is the outward unit normal at the contact line which is tangent to the substrate; two-dimensional droplet sliding along inclined (middle) flat substrate, (bottom) step. The step size is exaggerated compared with the droplet size to clarify the contact line region around the step.

measured at the length scale of the step topography. In (17), we assume that the receding contact line is away from the step, so that θs, R → 0. In the limit as θs, A → 0, the substrate is flat and θm, A = θA; we recover (16) when α = αc. To show that the balance given in (17) determines the pinning and depinning transition, we plot, in Fig. 3, the phase diagrams for three different steps. To form the diagrams, we run a set of numerical experiments in which we vary the Bo and inclination angle α for fixed droplet volume of unity in each case. The dark shaded region is pinning region while the light one is the depinning region and the droplet follows a pinning-depinning transition determined by a curve shown with dashed lines in all cases that we obtain by the balance of retention forces and gravitational force given in (17). The existence of precursor film ahead of the contact lines changes the definition of the triple lines and the angles they make with the substrate. To guarantee that the balance is computed correctly, we construct the transition curve using the right-hand side of (17). This transition line shifts upwards as the step becomes steeper (from (a) to (c) in Fig. 3). The circular and triangular marks correspond respectively to pinned and depinned droplets around the step. For clarity, we also show some cases which are around the transition lines to emphasize that both varying the α for fixed Bo and varying the Bo for fixed α follow the transition. The retention force due to step is maximum whenever θs, A is maximum. This point corresponds to the middle of the step, namely the point at which its absolute slope is maximum. In Fig. 4(a), the variation of the contact line location with α, for fixed Bo, is shown. The advancing contact line pins around the step for α < αc and its location gets closer

Fig. 3. Phase diagrams of pinning-depinning transition of a two-dimensional droplet around a step of height hs = 0.05: (a) λ = 0.02, (b) λ = 0.01, (c) λ = 0.0075; the separators of the transitions shown as dashed black lines are obtained by plotting the force balance relation at the critical transition angle. The inclination angle α is in degrees. The insets show the step profile for the corresponding λ. 5

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existence increases the hysteresis. Up until the pinning occurs, the motion of the contact line is determined by the balance of viscous, capillary and retention forces. When we plot the advancing contact line locations, xA,CL, against the time, they collapse onto each other when time is scaled with Bo sin α as shown in Fig. 4(b). The shaded region refers to the pinning while the other is the advancing of the contact line. This plot simply shows that the contact line Capillary number (based on contact line speed defined as Ca = ηVCL/γ) scales linearly with the Bo sin α up to pinning (shaded area in Fig. 4(b)). A similar experimental observation can be seen in Fig. 2 of Podgorski et al. [50]. The variation of x-coordinate of the droplet's center of gravity follows a similar pattern compared with xA,CL except its location slowly varies after the advancing contact line is pinned because the receding contact line is still moving towards right. Though the phase diagrams are formed for three different step steepness parameter λ, varying λ and/or step height hs determines the slope. The step is defined by eq. (3) with its steepest point at xs. At this point the absolute slope is maximum with the value hs/(2λ). Varying only the step height hs alters the slope; e.g. halving the step height from hs = 0.05 to hs = 0.025 halves the steepest slope. A case in which we set Bo = 1.25, α = 20° with hs = 0.05 and λ = 0.01 pins at the step (see Fig. 3(b)). For this case, we halve only the step height and the droplet depins. The full set of runs for the step heights hs = 0.025 and λ = 0.01,0.0075 reveal that we obtain the same phase diagrams given in Fig. 3(a) and (b), respectively. It is the slope of the step that determines the pinning-depinning transition within the parameters studied. We should note here that if the height of the defect is sufficiently small, the droplet may not pin [47]. Varying the equilibrium contact angle, on the other hand, modifies the phase transitions as it determines the upper and lower bounds of the receding and advancing angles, respectively: θR < θe < θA. For the steepest step geometry, we form the transition curves for two different equilibrium contact angles below and above the θe = 15°: the dotted line in the dark shaded region of Fig. 3 (c) is the transition curve obtained for θe = 10° while the dotted line inside the light shaded region is the transition curve obtained for θe = 20°. The transition curve shifts upwards for a higher equilibrium contact angle as it is more difficult for the advancing contact line to move forward. The reverse is true for smaller contact angles; the transition curve is shifted downwards enlarging the depinning region as it is simpler for the advancing contact line to move.

Fig. 4. (a) Pinned contact line locations as α → αc. We plot for α = 10° to 13.95°, Bo = 1.25, θe = 15°, λ = 0.02. The circle on the step locates the maximum θs. The inset shows the contact line position for the critical case, the dashed line is the tangent line drawn from the minimum slope of the interface near the advancing contact line; (b) position of the advancing contact line near the step (left axis), x-coordinate of the center of gravity of the droplet (right axis); hs = 0.05, λ = 0.01, Bo = 0.25; ○: α = 10°, ∇: α = 20°, □: α = 30°, ☆: α = 40°, ◊: α = 50;.

The contribution of the step to the retention force due to the term cosθe(1 − cos θs, A) in (17) becomes more pronounced as the step becomes steeper. The changes in the receding contact angle with different steps is small compared with the changes at the advancing contact angle. This shows that the increase in the retention force with steeper step can be explained by a simple force balance around the advancing contact line. There, the magnitude of the upward retention force (in negative i-direction) is

fret , A

=

lv (cos A

cos

e cos s, A).

4. Effect of multiple forward and backward facing steps Over a rough surface, there exist step-ups and downs which alter the motion of the droplets. In this section, we mimic such a surface by combining the single step geometries to form substrates as shown in Fig. 5 with hatched regions around y = 0. In the analysis of the single step geometry, we assume that the receding contact line is away from the step. When the receding contact line encounters such a step, however, it alters the pinning-depinning transition because the mesoscopic angle of the receding contact line will be affected by the non-zero slope surfaces. In the case of a backward facing step, the substrate slope increases the mesoscopic angle of the receding contact line. The increase in the retention force alters the pinning-depinning transition of the droplet. If a droplet pins at such a surface, its pinning-depinning transition is determined by the balance of contact line forces and gravitational forces given by

(18)

As the step becomes steeper, both θA and θs, A increases in (18); however, the increase in θs, A is more pronounced compared with the increase in θA, as well as cosθs, A is multiplied with cosθe < 1, showing that the retention force increases with steeper step. This is simply observable by comparing the contributions of the terms to the retention force just before the depinning transition. We take three representative cases in which the droplets are pinned and they are located just before the pinning-depinning transition curves of three different λ values (see Fig. 3(a) for Bo = 1.25, α = 13.95°, Fig. 3(b) for Bo = 1.75, α = 24° and Fig. 3(c) for Bo = 1.75, α = 37.52 °). We respectively measure the receding and advancing contact angles by calculating the maximum and minimum slopes of the interface ζ (these are the apparent angles; the contact line locations are at the points where a tangent line to the interface meets the substrate at the corresponding angle locations). The measured receding contact angle contribution to the retention force due to cosθR remains nearly the same while advancing contact angle contribution increases as λ increases and retention force due the step

lv [cos R

cos

A

cos

e (cos s, R

cos s, A )] ~ gA sin .

(19)

Here θs, R is the substrate angle at the receding contact line and we recover (17) in the limit as θs, R → 0. The positive contribution to the retention force occurs around a step down where the pinning of the receding contact line is expected. While a step down increases the 6

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it is not only the step height which determines this formation but the slope of the steps. 5. Conclusion We consider the motion of a partially wetting droplet over inclined surfaces. When the surface over which a droplet moves is not clean, the hysteresis causes the droplet to pin/depin around chemical or topographical heterogeneities. As it is observed that the contact line pins at the negative slope side of a physical one, we simplify the substrate geometry to a topography having a single forward-facing step geometry. The model problem is for a two-dimensional slender droplet providing us with the use of lubrication theory; the evolution is governed by a 4th order non-linear diffusion equation using a precursor film model. To integrate the governing equations, we have developed a cubic finite element solver and validate it against well-known problems. The step geometry is determined by the step height and steepness parameter both determining its maximum absolute slope. The pinning of a droplet around a step simply shows that retention forces due surface energies dominate the gravitational forces and the balance between these determines the transition from a pinning to depinning regime. The phase diagrams obtained by numerical experiments clearly separates this transition: a simple force balance of gravitational force acting on the droplet volume and the surface tension forces around the contact line defines this separator above which the droplet depins from the step. The use of single forward facing step geometry helps explaining the increase in the retention force due to step by a simple force balance argument made around the pinned contact line. While, for fixed Bo and α, we show that it is the slope of the step that determines this transition, at the end, we show that the existence of multiple steps may impede the motion of droplets because the receding contact line may pin at the backward-facing steps or leave residual droplet behind

Fig. 5. Effect of substrate topography on the motion of droplet, (top)λ = 0.02 for both step-ups and downs, (middle) λ = 0.04 for step-downs and λ = 0.02 for step-ups, (bottom) λ = 0.04 for step-downs and λ = 0.01 for step-ups; Bo = 0.75, α = 40∘. Same line types correspond to the same time for comparison. The inset shows the residual droplet behind a forward-facing step.

0.2 (a)

0.1

mesoscopic angle of the moving contact lines, the reverse reduces it which helps the contact line motion. Decreasing the slope of step-downs compared with the step ups helps the motion of droplet over such a rough substrate. Fig. 5 compares such an example. We take a representative droplet from the depinning region of the phase diagram shown in Fig. 3(a), namely we set Bo = 0.75 and α = 40°. We, initially, place the droplet over the hills of the topography and let it slide. Different from the single step, the receding contact line alters the motion of the droplet. Making the backward-facing steps steeper (compare Fig. 5 (top) and (middle)) impedes the motion. For a steeper backward facing step as shown in Fig. 5 (top) compared with (middle), the receding contact line remains pinned longer at the steps. For a sufficiently steep forward-facing steps, however, the receding contact line may leave a residual droplet behind the step as the film touches the step corner before the contact line is able to rise over it. It happens, for example, when we set λ = 0.01 or below at the forwardfacing steps as shown in the inset of Fig. 5 (bottom). Though the computed volume of the residual droplets proves to be small compared with the unit droplet volume, we show that, for fixed maximum slope of the forward-facing step, the volume increases monotonically with step height. We change the substrate design having different heights from the aforementioned design over which residual droplet formation is observed. Fig. 6 shows the steps having heights 1.25hs and 1.5hs in (a), 1.75hs and 2hs in (b), 2.25hs and 2.5hs in (c) where hs = 0.05. For fixed slope, the residual droplet volume increases as the step height increases. We plot, in Fig. 6 (d), the variation of residual droplet volume, Vr , as function of the step height. For similar substrate designs corresponding to Fig. 5(b), we do not observe residual formation even at 2.5hs because

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0 10-3

6

(d)

4 2 0 0.06

Fig. 6. Effect of step height on the residual droplet formation, (a) residual droplets behind steps of heights hs=0.0625 and hs=0.075, (b) residual droplets behind steps of heights hs=0.0875 and hs=0.1, (c) residual droplets behind steps of heights hs=0.1125 and hs=0.125, (d) residual droplet volume as a function of step height. 7

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sufficiently steep forward-facing steps. We also show that, for fixed slope, the residual droplet volume increases with step height. The results contribute to the understanding of the pinning-depinning of contact lines around step-like heterogeneities. Our two-dimensional model problem can be extended to other two/three-dimensional models and would motivate further experiments and design of new substrates.

All the authors have read and approved the final manuscript. Declaration of Competing Interest The authors declare that they have no conflict of interest. Financial funding and support

Authors contributions

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

All the authors were involved in the preparation of the manuscript.

Appendix A. Weak formulation of the evolution of f and numerical solution procedure To integrate the evolution equation given in (11), we use cubic finite element discretization in space and second order discretization in time. The evolution equation is valid in a domain Ω : 0 ≤ x ≤ L. We approximate the test functions and f from the same Hilbert-Sobolov space ℋ1(Ω). We first convert the film evolution equation into a set of two coupled equations, both being second order, by letting 2f x2

w = 0,

f + t x sin

(A.1)

(f + h ) x

f 3 Bo cos sin e cos

1

3h

w x

e

x3

f x

f

= 0.

(A.2)

We obtain the weak form of the set of equations (A.1 − A.2) by first multiplying with corresponding test functions, namely w and f , respectively; then integrating over the domain Ω and weakening the differentiability requirement using the fact that the test functions are chosen to be zero at any Dirichlet boundary condition:

f w dx + x x

f fd t sin

f

wwdx = 0,

(A.3)

(f + h ) x

f 3 Bo cos

+

1

sin e cos

f x

f d x

3h

w x

e

x3

= 0.

(A.4) n

n+1

We integrate in time using a second-order Crank-Nicolson scheme by setting the θ = 1/2 on the standard θ-scheme for the time t ∈ (t , t Δt = tn+1 − tn being the time step between the old time tn and the new time tn+1:

f n+1 w dx + x x + (1 f n+1 t

sin + (1

fn w dx + x x

)

1

fn ¯

fd

sin e cos e

)

wn + 1 x 3

wn

3h

x

x3

w nwdx = 0,

(

3

( ( )

x3

n fn fn x

f n+1

(f n + h ) x

f¯ d x

(A.5)

(f n + 1 + h ) x

n + 1 f n+ 1

3h

f n Bo cos

)

w n + 1wdx

f n + 1 Bo cos

+

) and

x

sin

1

)

f¯ d x

sin e cos e

)

= 0.

(A.6) c

We use piecewise continuous cubic shape functions ϕ for the space discretization. We approximate f, w and corresponding test functions in isoparametric domain by

fn = f n+1 =

f jn + 1 f =

f jn c j,

c j,

wn =

wn+1 = fi

c i ,

w =

wjn

c j,

w nj + 1 jc , wi

c i .

(A.7)

Substituting approximations (A.7) into the weak form (A.5-A.6) for all possible test functions, we obtain a system of the form K(u)u = R to be solved for u, unknown vector comprising f and w, and a residual vector of the unknown variables due to nonlinearity of the film evolution equation. 8

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We treat the nonlinearity at every time step using the Newton's method given in Algorithm 1. Algorithm 1. Newton's algorithm. 1: while e > TOL do 2: um+1= um - T(um)−1r(um)

e = um+ 1

um

3: um=um+1 r In Algorithm 1, r is the residual vector and defined as r(u) = K(u)u − R; T is the tangent stiffness matrix and defined as T (u) = u . We set the −12 tolerance to be 10 in the Newton's iteration. Time step size and element sizes are determined by the independence of solutions from both Δt and Δx. We typically set Δt to 0.0005 and vary the distance between each space node from 1/600 to 1/1800 depending on the resolution. We use 7th order Gauss rule of integration to compute all the integrals. Appendix B. Numerical method validation: droplet motion on flat substrates We devote this section to the validation of our solver analyzing droplet spreading over a flat horizontal surface and the motion of droplets over a flat inclined one. B.1. Spreading droplets Tanner [51] and McHale et al. [52] predict the motion of two-dimensional viscous oil drops over flat substrates. To a first approximation, the maximum droplet height decreases with time t as t−1/7; to show this relation, we put a circular arc droplet onto a flat substrate making an initial angle of θ = 15° and let it spread till the contact angle reaches zero for high-energy surface. In Fig. 7 (top), we show the time history of maximum droplet height up to non-dimensional time of t = 100 and observe the same power-law relation after t ~ 1. Up to this time, the power-law relation does not hold because within this interval, the interface modifies itself to satisfy the evolution equation from an initially circular arc profile sitting over a flat precursor film of thickness f0; a similar observation is made by Schwartz and Eley [41] for the early time over which the initial droplet shape matters. We also check the dynamic variation of the advancing contact line while the droplet spreads. When the droplet advances over an existing precursor film, the contact line is an apparent one, we measure it by finding locations of the maximum and minimum slopes of the droplet interface, draw a tangent line from this point the substrate and compute both angles and contact line positions where this line meets the substrate. The advancing contact angle differs from its Young value and is given as a function of Ca by the Cox-Voinov model [28,29]: θA3 = θe3 − 9Ca ln (x/lmicro) with lmicro being a microscopic length-scale. This linear relationship between θA3 − θe3 and Ca, for the current spreading droplet is validated as shown in Fig. 7 (bottom). 1.6 1.4 1.2

7 -1

1 0.8 0.6

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10

-1

0

1

10

10

10

2

0.01

3 3 A e

0.008

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0.002

0

0

1

2

3

Ca

4

5 10

-4

Fig. 7. (top)Time variation of maximum droplet height: two-dimensional droplet spreading on a flat surface, see text for the initial conditions; (bottom)Speed of the advancing contact line of the spreading droplet. 9

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B.2. Droplet motion on inclined flat substrates When a droplet is driven by gravity over a flat inclined substrate, its shape does not remain circular at all times. Here, we show the droplet wetting transition over a flat inclined substrate. In all cases, we set θe to 10°, Bo to 0.75 and vary the inclination angle α from 0° to 50°, in 10° increments. As shown in Fig. 8, for small Bo sin α, droplet takes almost circular arc shape, surface tension forces determine the terminal shape. With increasing Bo sin α, however, the droplet elongates forming a tail shape towards the rear end; for all cases, the droplets move with constant speed after gaining their terminal shapes. Thiele et al. [34], Savva and Kalliadasis [35], Park and Kumar [46] observe similar transitions. Though the terminal shapes can be predicted with a steady model by changing the reference frame moving with the terminal contact line speed, we do not include it here as our focus in this paper is the motion around the step.

Fig. 8. Droplet wetting transition over an inclined flat substrate.

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