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On the onset of motion of sliding drops Cite this: Soft Matter, 2014, 10, 3325
Ciro Semprebona and Martin Brinkmann*ab In this article we numerically investigate the onset of motion of liquid drops in contact with a plane and homogeneous substrate with contact angle hysteresis. The drops are driven by a body force F ¼ rgV, where r is the density of the liquid, g is the acceleration of gravity, and V is the volume of the drop. We compare two protocols to vary the bond number Bo ¼ lv/lc by changes of either the drop size lv ¼ V1/3 or the capillary length lc ¼ (g/rg)1/2 where g is the interfacial tension, revealing that the transition between pinned and steady moving states can be either continuous or discontinuous. In a certain range
Received 19th July 2013 Accepted 3rd February 2014
both pinned and moving states can be found for a given bond number Bo, depending on the history of the control parameters g and V. Our calculations are extended to arbitrary combinations of static advancing and receding contact angles and provide a comprehensive picture of the depinning transition
DOI: 10.1039/c3sm51959g
induced by a quasi-static variation of the control parameters. Finally, we demonstrate that the particular
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form of the contact line mobility in our model has an impact on the interfacial shape of steady moving drops.
1
Introduction
Rain drops running down a window pane can be easily trapped by impurities like dust grains or other surface heterogeneities, while sticking drops can be observed even on freshly cleaned glass surfaces. These two behaviours are ultimately related to surface heterogeneities at the micron and submicron scale, which can be sufficient to pin the three phase contact line. The depinning mechanism of drops under the action of gravity and their subsequent motion has been particularly important to dene an optimum drop size in pesticide sprays.1,2 Systematic measurements of sliding drops help provide a simple way to characterise the homogeneity of surfaces.4,5 Most of the experimental procedures to induce the onset of sliding motion rely on the quasi-static variation of the body force, for example achieved by tilting a surface or increasing the rotation velocity of a spinning plate. An extended review on the subject can be found in an article by Santos et al.6 However, despite the remarkable amount of studies, the description of the transition between pinned and moving states is still incomplete. In particular, it has been experimentally shown that different experimental protocols lead to different depinning conditions,7 while a more quantitative analysis is sill lacking. In this work we attempt to give a comprehensive picture of the evolution of drop shapes and of the onset of motion, driven by the quasi-static variation of the body force. In particular, we are interested in the order of the depinning
a
Department Dynamics of Complex Fluids, Max-Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 G¨ottingen, Germany
b Experimental Physics, Saarland University, 66123 Saarbr¨ ucken, Germany. E-mail: [email protected]
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transition, which will turn out to depend on the protocol adopted to vary the body force. Consider a sessile drop in contact with a plane substrate subjected to a horizontal body force F, as sketched in Fig. 1. In a typical experiment, we have F ¼ rgV, where g is the acceleration of gravity, r is the mass density of the liquid, and V is the drop volume. In the present work we assume for simplicity, unless otherwise stated, that the component of the body force in the vertical direction is negligible. This situation corresponds to drops on a vertical window pane, or drops on a spinning plate sufficiently small and far from the axis of rotation.
Profile of the drop (top) and contour of the contact line (bottom) of a steady moving drop shortly after depinning, driven by a body force F acting horizontally. H, W, and L indicate the height, width and length of the drop, respectively.
Fig. 1
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In the presence of hysteresis the contact line of the drop will locally advance over the substrate only if the apparent contact angle is larger than the advancing angle qa, and recede only if it is smaller than the receding angle qr. Apparent contact angles q measured on a static drop prole always fall into the hysteresis interval, dened as the range between these two values. A quantitative estimate for the depinning condition can be obtained considering a mechanical equilibrium of the body force F ¼ exrgV acting on the bulk of the liquid and the horizontal components of the interfacial tension at the contact line, þ rgV ¼ g dsex $n cos q; (1)
The paper is organised as follows: rst, we introduce the physical model and its numerical implementation in Section 2. In the following Section 3 we analyse the quasi-static evolution of the drop morphologies for a specic combination of advancing and receding angles. Here, we consider two paradigmatic protocols under various initial conditions, and discuss the transition to moving states. In Section 4 we extend our results to substrates with arbitrary combinations of advancing and receding angles and discuss the implications for tilted plate experiments. Finally, in Section 5 we analyse the impact of the driving force and some simple laws for the contact line mobility on the morphologies of steady moving drops.
G
where G denotes the contour of the contact line parameterised by the arc-length s and g denotes the interfacial energy of the liquid interface. The vector n is the normal to the contact line in the plane of the substrate while ex is the downhill direction, cf. also the sketch in Fig. 1. A relationship between the static advancing and static receding contact angles and the magnitude of the body force at which sliding occurs can be established under certain simplifying assumptions regarding the shape of the contact line, and the distribution of local contact angles. Based on experimental observations, Furmidge2 proposed a simple model for drop shapes where the contact line contour G exhibits straight sides aligned with the body force. This situation, illustrated in Fig. 1, appears in tilting plate experiments. In the curved front and rear part of G, the local contact angles at depinning are equal to the advancing and receding angles, respectively. Under these assumptions, the RHS of eqn (1) can be integrated and reduced to rgV ¼ gkW(cos qr cos qa)
(2)
where W denotes the extension of the drop perpendicular to the force F. While the Furmidge model corresponds to k ¼ 1, experimental measurements4 and numerical calculations6,8–10 show that the empirical factor k in eqn (2) deviates signicantly for various systems. The reason is that the balance expressed in eqn (1) crucially depends on the actual shape of the drop contour. Practically, eqn (2) requires a reasonable estimate for the drop width W to correctly predict the onset of motion. In this work, rather than imposing a priori simplied shapes to the contact line contour, we quasi-statically drive the drop to the depinning transition. The drop morphology, and in particular its width and elongation, is then uniquely determined by the initial condition and the protocol adopted to change the set of control parameters. Recently it has been challenged by Krasovitski et al.3 that the advancing and receding angles correspond to the maximum and minimum angles measured at the onset of motion. This is the case if the hysteresis is induced by a heterogeneity of the substrate with a typical wavelength comparable to the drop size. Practically however, the hysteresis is related to a surface heterogeneity at a much smaller scale, and it can be regarded as a material parameter. In this case the maximum and minimum angles equal the advancing and receding angles, as observed in most of the experiments.
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2 Physical model and numerical implementation Throughout this work we consider ideally plane, rigid and chemically inert substrates, with isotropic contact angle hysteresis. The macroscopic wetting properties are described in terms of the static advancing contact angle, qa, and the static receding contact angle, qr. If the drop velocity is small and, at the same time, the drop is not too at, the shape of the interface is governed by the interplay of interfacial stresses and gravity. This approximation is good if the modied capillary number Ca* h hUH3/gW3 is small compared to one. Here, h is the dynamic viscosity of the liquid, g is the interfacial tension, U is the drop velocity, while H denotes the height of the drop, cf. also the sketch in Fig. 1. If Ca* 1, the injected work during drop movement will be dissipated in the direct vicinity of the contact line. In this case, it is reasonable to assume a separation of length and time scales for the interfacial deformation, between the macroscopic global shape of the drop and the microscopic distortion of the contact line. As a consequence, for a given contour G of the contact line, we assume that the interface of the drop instantaneously adopts the shape S that minimises the sum of energy of the free interface and the gravitational contribution E{S} ¼ gAlv rgVhxi.
(3)
Here, Alv denotes the area of the liquid vapour interface, r is the mass density of the liquid, g is the acceleration,† and hxi is the x-coordinate of the liquid's centre of mass given by ð 1 d3 r ex $r : (4) hxih V U As we assume the liquid to be non-volatile on the time-scale of the experiment, we apply the constraint of a xed volume V to the drop's body U. The local velocity u of the three phase contact line is zero if q falls into the hysteresis interval. Otherwise, we suppose that u is described by a power law of the local contact angle q: † An acceleration g ¼ u2r parallel to the surface can be realised for a vertical orientation of the substrate or on a spinning plate, where u is the angular frequency of rotation and r is the distance of the centre of mass to the rotation axis.
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u¼
8 u 0 b b > > < bq b1 q qa
for q $ qa
u0 b q qr b b1 bqr
for q # qr
a
> > :
;
(5)
The exponent b in eqn (5) mimics the leading order scaling of certain models proposed for the contact line motion. For example, b ¼ 2 corresponds to the molecular kinetic theory according to Blake11 while b ¼ 3 corresponds to the continuum hydrodynamic model of Voinov12 and Cox.13 The linearized relationship around the static values of the contact angles qa and qr reduces to a friction law, taking b ¼ 1. Models for the contact line dynamics including slippage are necessary to overcome the paradox rst described by Huh and Scriven,14 for which the stresses and viscous dissipation diverge at the contact line in the presence of no slip boundary condition. In our case the prefactor u0 describes a typical velocity scale related to the combination of uids and solid materials. Note that our results about the quasistatic evolution are general and not bound to the specic form of eqn (5). Instead results regarding transient states and steady moving drops are bound to the model details. For simplicity we dened in eqn (5) a prefactor such that the linearised mobility is identical for all exponents b both at q ¼ qa and q ¼ qr. A different choice for the normalization could be motivated by additional information on the specic system under investigation. Note, that the model presented here is not appropriate whenever a separation of length scales is not possible. In view of the numerical implementation of our model it is useful to non-dimensionalise all relevant physical quantities. In general, one is free to chose, besides a mass scale M0, a length scale L0 and a time scale T0. Two length scales naturally arise if we consider static drop shapes under the inuence of gravity: the capillary length lc h (g/rg)1/2 and the linear dimension of a droplet lv h V1/3. The ratio of both length scales denes the magnitude of the bond number Bo h lv2/lc2. The particular choice L0 h lc is useful in experiments where the acceleration g is xed while the volume V is changing. If, instead, g is changing while V is constant, the choice L0 h lv is more appropriate. Alternatively, if both length scales are changing during the experiment, L0 could be either dened as lv(t0) or lc(t0) at the start of the experiment t ¼ t0. Given the length scale L0, a natural denition of the timescale is T0 h L0/u0. We compute drop shapes by minimising the interfacial and gravitational energy in eqn (3) under the constraint of a xed liquid volume V. The liquid–vapor interface S is represented by a mesh of nodes connected by the edges of triangular facets. Nodes belonging to the contact line are constrained to move in the plane of the substrate. In this representation, any quantity like the energy in eqn (3) or the enclosed volume is a function of 3N Cartesian coordinates and can be numerically evaluated and minimised under local and global constraints by standard optimisation algorithms. We use the public domain soware Surface Evolver15 to set up the interface, maintain the triangulation, and to minimise the energy (3). In order to improve the accuracy, the mesh is gradually rened close to the contact line, close to the symmetry plane and in parts where the interface is
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Fig. 2 (a) Snapshot of a sliding drop in our numerical computations illustrating the different levels of the mesh refinement. (b) Functional dependence of the model for the contact line velocity for different exponents b. (c) Magnification of the region close to the contact line to illustrate the implementation of the dynamic law. The local contact angle is computed from the surface normal of the triangles sharing an edge with the contact line (shown in green).
strongly curved. The smallest edge element is chosen such that the contact line is typically composed of 200 elements of which only half are simulated exploiting the mirror symmetry of the drop. In a typical simulation run, the interface of half of the drop, as shown in Fig. 2a, is represented by roughly 500 nodes and 1500 edges. The algorithm to simulate the temporal evolution of the droplet shape contains two steps: in the rst step the energy of the drop is minimised given the contour Gn of the contact line at time tn. In the second step, aer convergence of the interfacial shape, the local contact angle q at each edge of Gn is evaluated. If qr # q # qa holds for both the edges connected to a node of Gn, the contact line around the node is in a pinned state and this node is not moved. Whenever the local contact angle of at least one of the neighbouring edges falls outside the hysteresis interval, i.e., q > qa or q < qr, the node is allowed to move. To perform the motion of the contact line between time tn and tn+1 h tn + Dt, a virtual displacement Dr ¼ nuDt is rst assigned to each edge of Gn, where n is the co-normal vector of the contact line in the plane of the substrate. The local velocity of the contact line, u, is a function of q according to the dynamic law in eqn (5). As shown in the sketch in Fig. 2c, the new contact line contour Gn+1 is then obtained by moving each node according to a weighted average of the virtual displacements assigned to the neighbouring edges. At this point, an updated interfacial shape Sn+1 for a new contact line contour Gn+1 has to be computed. The drop shape is evolved in time by a cyclic repetition of the two steps. To reduce the computational time we use an adaptive
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time step chosen such that the maximum displacement is 10% of the length of the shortest element of the contact line. To check the accuracy of the algorithm we numerically integrated the stresses along the contour of the contact line projected along the direction of the body force. According to the nondimensional form of eqn (1), the local contact angle has to satisfy the identity þ ds cos q ex $n ¼ Bo; (6) Published on 03 February 2014. Downloaded by Université Laval on 26/05/2014 21:23:37.
G
where we made use of the non-dimensional arc length parameter s h S/lv. Eqn (6) has been veried within 1% of accuracy in our simulation runs. A typical simulation run is performed as follows: rst the drops are introduced under an initial condition chosen according to the protocols described in Section 3. Then the bond number is quasi-statically increased by changing one of the two length scales. At every iteration we check if the contact line is entirely pinned or if some parts have been displaced. If all the elements of the contact line reach a pinned state we stop the iterations and save the conguration. If aer N steps some parts of the contact line are still being displaced the drop is assumed to be in a steady moving state and we perform N more steps averaging all quantities of interest. We choose N ¼ 300, as a compromise between accuracy and computational time. A similar algorithm has been proposed by Santos and White6,16 to simulate the quasi-static evolution of drops in tilt plate experiments. To test our implementation we considered the same system investigated by Santos and White6 accounting for both the normal and tangential components of the body force. In the quasi static regime both implementations are equivalent, and our results conform with the results in ref. 6. In addition our algorithm allows us to investigate the shape of steady moving drops for whenever the energy dissipation occurs in the vicinity of the contact line, implying that the free liquid interface is close to a state of minimum energy. However, if the viscous dissipation in the liquid bulk cannot be neglected, the full Navier Stokes equations need to be numerically solved, which is a signicantly more complex task.17,18
3 Two depinning protocols In the following we investigate the quasi-static evolution of drop shapes under two different protocols to slowly increase the bond number Bo h lv2/lc2. In protocol (I), the volume V is increased in small steps while the capillary length lc is kept constant. Employing the capillary length as a unit length L0 h lc, the volume of the drop in the initial state is set to V ¼ lv3 ¼ 103L03. Distortions of the interface induced by the body force are negligible because the linear dimension of the drop, lv, satises lv lc. In this case the initial value of the contact angle qin of the drop is approximately constant along its perimeter and taken to be qin ¼ qa. Above a certain value of the bond number, the drop does not anymore relax into a static shape and enters a dynamic, sliding state. In the complementary protocol (II), we consider a small drop of constant liquid volume V and slowly increase the bond
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number Bo by decreasing the capillary length lc. In this situation, it is more appropriate to use the length scale L0 h lv related to the drop volume as a reference length. We start with a value lc ¼ 10L0 and consider a variety of initial contact angles qin in the interval of the hysteresis. In the special case where qin ¼ qa the initial drop shape is the same as in protocol (I) but with a linear dimension scaled up by a factor of 10. As for protocol (I), we nd that the drop will not relax anymore into a static shape if a certain value of the bond number is exceeded. Examples of the main results of our numerical investigations are illustrated in Fig. 3 for particular values qa ¼ 120 and qr ¼ 60 . Each panel in the rst row of Fig. 3 displays a set of drop proles projected to the symmetry plane. The corresponding contact line contours are shown in the second row. Variations of the local angle q as a function of polar angle f are displayed in the third row. Here, the centre of the area in contact with the liquid is chosen as the origin of the polar coordinate system. The rst column in Fig. 3 refers to protocol (I) while the second and third column refer to protocol (II) for initial contact angles qin ¼ qa and qin ¼ (qa + qr)/2, respectively. In protocol (I) major parts of the contact line are moving already at small bond numbers Bo 1 as the drop has to accommodate the increasing liquid volume. Consequently, the drop grows both in the direction of the body force and in the direction perpendicular to it, while the trailing edge remains immobilised. As expected, the local contact angle q equals the advancing contact angle qa in the parts of the contact line which have advanced. Following the contact line from the front to the back of the drop, q decreases in the pinned part to a minimum located in the rear apex. For larger bond numbers the local contact angle in the rear apex reaches the value qr and the trailing edge also depins. The condition that the leading and the trailing edge are depinned does not imply that the entire drop is depinned, i.e., it still relaxes into a static shape aer a small increment of Bo. Finally, depinning of the drop occurs at a bond number Bo ¼ Bo*, which is of the order unity. For the present choice of advancing and receding contact angles, the contact line contour at depinning does not include the initial contour of the drop. As a consequence it is not possible to experimentally reproduce the depinning induced by protocol (I) just by injecting the liquid from a hole in the substrate, as the contact line would pin at the edges of the hole before the onset of motion. Once the drop has reached a steady moving state, the contact line contour displays two parallel straight sides aligned with the direction of the driving body force. The leading and trailing edges are curved and both contours are close to a circular arc, with different radius of curvature. This observation has been already reported by Furmidge in ref. 2. The last pinned drop shape detected for protocol (I) at Bo ¼ Bo* also coincides with the shape of the rst steady moving drop in the limit of zero velocity as Bo approaches Bo* from above. The latter shape is represented by a black dashed line in Fig. 3a–c, which indicates a continuous transition from pinned to moving drops. Apart from the fact that the length scale of the drop, lv, does not change during the increase of the bond number, the shape of the last pinned drop in protocol (II) for an initial contact angle qin ¼ qa is similar to the one in protocol (I). The
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Upper, middle, and lower panels show sequences of drop profiles, contact line contours, and distributions of contact angles, respectively, during a quasi-static increase of the bond number Bo according to protocols (I) and (II), see the main text. Data are rescaled by the respective characteristic length that is kept fixed. Dashed lines represent the last detected profile before depinning. The substrate is characterised by qa ¼ 120 and qr ¼ 60 . Panels (a), (b), and (c): drop size lv is increased while the capillary length lc is constant for Bo ¼ 0.01, 0.16, 0.49, 0.72. Panels (d), (e), and (f): lc is decreased while lv is constant for Bo ¼ 0.01, 0.16, 0.36, 0.64, 1, 1.14. The initial shape is a spherical cap with qin ¼ 120 . Panels (g), (h), (i): lc is decreased at a constant lv for Bo ¼ 0.01, 0.36, 0.72, 1.42 and qin ¼ (qa + qr)/2 ¼ 90 . Fig. 3
differences in the evolution of the drop shapes between protocols (I) and (II), however, become more pronounced when qin < qa. The drop proles and their corresponding contact line contours for the case qin ¼ (qa + qr)/2 shown in Fig. 3 reveal that the contact line is displaced in equal proportions in the front and back of the drop while the overall shape of the contact line contour stays close to a circle. Details of the contours and proles, indicate qualitative agreement with the observations of Berejnov and Thorne.19 During the depinning transition in protocol (II), the drop ‘hatches’ through the narrow sector of the moving contact line contour in the front. In order to further investigate the morphological changes during the depinning processes for protocols (I) and (II), and to highlight their main characteristics, we compare in Fig. 4a the evolution of the base eccentricity, dened as the ratio between the parallel and perpendicular length, 3 h L/W, as function of Bo for both protocols. Quasi-static branches are represented by continuous lines, thick for protocol (I) and thin for protocol (II). As expected, the most rapid increase of the base eccentricity 3 occurs for protocol (I) at a small Bo. Close to the point of depinning at Bo ¼ Bo*, the rate at which 3 grows with Bo is reduced. During protocol (II) for qin ¼ qa ¼ 120 , the base eccentricity 3 increases linearly with Bo. For the initial contact
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Fig. 4 Comparison of the evolution of the eccentricity 3 h L/W of the
contact line as a function of bond number Bo for protocols (I) (thick black line) and (II) (thin lines). The thick red line indicates steady moving drops for b ¼ 3. See the main text for details.
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angle qin ¼ (qa + qr)/2 ¼ 90 both contact line displacements occur almost simultaneously and lead to only a slight variation of 3. For an initial contact angle qin ¼ 72 , the trailing edge displaces rst, inducing a linear decrease of 3 with Bo. The bond number at the depinning transition for protocol (II), Bos, satises Bos $ Bo* and is monotonously increasing with the decrease of the initial contact angle qin. The most striking difference between protocols (I) and (II) can be observed during the transition from static to steady moving drops: As Bo approaches Bo* from below, the shape of static drops on the branch corresponding to protocol (I) continuously evolve toward the shape of steady moving drops at Bo $ Bo*. This behaviour is in contrast to the observation made during protocol (II): here, the shape of the drop undergoes a discontinuous change between the last static shape at Bo ¼ Bos and the drop shapes on the steady moving branch at Bo $ Bos. The pinned lateral parts of the contact line form a bottle neck for the free interface of the drop and a certain excess of driving force is required to overcome the energy barrier. For protocol (II) it is possible to obtain the quasi-static evolution by imposing a global constraint to x the lateral position of the center of mass in eqn (4). The Lagrange multiplier corresponding to this global constraint plays the same role as the prefactor rgV in the total energy in eqn (3). In contrast to the unconstrained ensemble, the acceleration is adjusted such that the lateral component of the center of mass position, hxi, is kept at a xed value. Integration of the Lagrange multiplier, and the equivalent body force, yields the work dissipated during the movement as a function of the displacement. Both methods, xing the body force F ¼ rgV or the lateral position of the center of mass, hxi, yield equivalent results for protocol (II) if vBo/vhxi $ 0. Fixing the value of hxi instead of Bo allows us to observe solution branches of drop shapes with v Bo/vhxi < 0. A similar method has been applied in several problems regarding the stability of liquid interfaces in contact with substrates with complex geometries20–23. In the present case, the lateral position of the center of mass hxi is slowly dragged out of its initial position. At the same time the contact line G is allowed to relax according to the contact line mobility in eqn (5). Within the numerical accuracy, the evolution of the drop base eccentricity 3 is identical to the quasistatic evolution described above. In addition, it is possible to observe branches in the ensemble where the body force is xed and the center of mass position is free to move. These branches are shown as thin dashed lines in Fig. 4. The corresponding interfacial conguration represents drop shapes aer the depinning. Interestingly, all unstable branches are attracted to the conguration of the last pinned shape at critical depinning Bo* according to protocol (I). Once the centre of mass hxi has been displaced sufficiently far from its initial position, any information about the initial state is lost and the free interface will adopt the shape at critical depinning. In view of the discontinuous transition found for protocol (II), it is not surprising to nd regions of bond numbers where both pinned and steady moving drop shapes exist. In particular, it is likely that the lower bound to Bos is determined by the depinning threshold Bo* of protocol (I). The upper bound to Bos
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has been determined by Dimitrakopoulos24 using an optimisation algorithm that converges to the drop shape with the highest resistance to depinning. From our results, and the investigations of Dimitrakopoulos, we suggest that depinning events are discontinuous when the depinning threshold is larger than Bo*. A steady moving branch for the case b ¼ 3 is represented with a thick red dashed line in Fig. 4 and will be described in more detail in Section 5. To resolve the temporal evolution of the drop shapes during the depinning transition for protocol (II), we restart our simulations from the closest conguration saved before the depinning threshold Bos. The bond number is then slightly increased above Bos and the simulation run is continued without further increase of the bond number until the drop has displaced twice its own length. Fig. 5a shows the temporal evolution of the drop base eccentricity 3 during the discontinuous depinning, for some of the cases considered in Fig. 4. Panels (b), (c) and (d) of Fig. 5 illustrate a sequence of out of equilibrium proles, contact line contours, and contact angle distributions for the
(a) Temporal evolution of the base eccentricity of the contact line, 3, in protocol (II) after depinning for initial contact angles qin ¼ 60 , qin ¼ 72 , qin ¼ 90 , and qin ¼ 108 . Inset: relationship between the transition time Dt and the initial condition qin. The reference time t0 refers to the inflexion point. Colored dots on the curve for qin ¼ 72 refer to the profiles, contours and angle distribution shown in panels (b), (c) and (d) respectively. The dashed curves refer to the last pinned morphology detected (t ¼ N). Fig. 5
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case of qin ¼ 72 . The corresponding morphologies are marked by colored points in panel (a). Two steps can be distinguished in the transition: initially the contact line slowly creeps forward in the central part of the leading edge while the sides remain pinned. Once the drop has passed through the bottle neck it quickly approaches the characteristic morphology of a steady moving drop. The total transition time can be thought as the sum of the time necessary to complete both steps. The rst transition time is related to the magnitude of the bond number increment when the simulations are restarted. The second transition time is related to the excess of driving force across the transition and diverges as Bo approaches the critical value Bo* of the continuous depinning transition for protocol (I). The inset of Fig. 5a illustrate how this transition time Dt depends on the initial condition qin for protocol (II).
4 Depinning threshold for arbitrary hysteresis In the previous section we analysed the depinning process applying two different protocols to increase the bond number. So far, we restricted our analysis to the specic values qa ¼ 120 and qr ¼ 60 . To check if the same qualitative results hold also for different combination of angles, we performed additional calculations by either xing qr ¼ 60 and varying qa or xing qa ¼ 120 and varying qr. The results referred to the quasi-static evolution with protocol (I) displayed in Fig. 6 conform to the descriptions in Section 3. Interestingly, the quasi-static evolution of the base eccentricity 3 is a monotonous function of the bond number Bo only when the difference between qa and qr is sufficiently large. As expected, the smaller the value of the depinning threshold Bo* for protocol (I) is, the smaller the contact angle hysteresis. In this case, the base eccentricity of the drop exhibits a maximum at some intermediate value Bo < Bo*. The effect is also clearly visible from the contour plots in Fig. 7a. The same calculations have been repeated for protocol (II), and the same behaviour has been observed considering both the case qin ¼ qa and the case qin ¼ (qa + qr)/2. In particular we observed that although the two protocols are qualitatively different, the depinning threshold Bos for protocol (II) in the case qin ¼ qa is very close to the critical value Bo* for protocol (I), and they can be confused if Dq < 10 . Naively, one would expect that the bond number Bo* at critical depinning grows monotonously with contact angle hysteresis Dq h qa qr, regardless of how the static contact angles qa and qr are varied. Inspection of Fig. 6a shows that this statement holds only in a restricted number of cases, i.e., if qa is xed and qr is decreased. Instead, if qr is xed and qa is increased, Bo* displays a maximum, see Fig. 6b. The non-monotonous variation of the depinning threshold Bo* can be qualitatively explained simply considering drops whose width W is set by the diameter of the equivalent spherical cup set by the advancing angle qa. According to eqn (2) the retention force is proportional to cos qa cos qr and to the width W of the drop. While keeping xed the value of qr, a larger value of qa induces not only a larger retention force per unit
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Quasi-static evolution according to protocol (I) of the aspect ratio of the contact line for (a) fixed qa ¼ 120 and various qr, and (b) fixed qr ¼ 60 and various qa. Fig. 6
length, but also a smaller width W of the contact line contour. As a consequence, at a certain value of the advancing angle, qmax , the two effects must balance each other, and the threshold a bond number Bo* shows a maximum. To complete our analysis, we will from now on focus on protocol (I) and determine the values of the bond number Bo* at critical depinning for all combinations of contact angle values qa and qr in the range of contact angles between 20 and 160 . To this end, we performed 435 independent calculations for qa and qr varied in steps of 5 . The results are presented in Fig. 7, where the values of Bo* have been interpolated with a multinomial function in cos qa and cos qr and encoded with a color scale. To precisely determine the depinning threshold Bo* for each simulation run, we linearly interpolate the relationship between the bond number Bo and velocity u for steady moving drops in the range 0.05 < u/u0 < 0.3. An example is shown in the inset of Fig. 7a. Isolines in the main plot of Fig. 7a identify regions of constant depinning threshold Bo* and suggest the presence of a global maximum located around qa z 118 and qr z 0 where Bo* z 1.65. Oen the depinning threshold is calculated in the limit of systems with small contact angle hysteresis, assuming that the drop shape is approximately symmetric, with only small distortions induced by gravity. This approach can lead to analytical relationships as outlined by Qu´ er´ e in ref. 25. A useful benchmark can be obtained by analysing the base eccentricity 3 in our systematic calculations, here presented as a color plot in Fig. 7b. Interestingly, 3 is mainly proportional to the hysteresis Dq, with only a mild dependence on the Soft Matter, 2014, 10, 3325–3334 | 3331
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the order of one, the attening is small and the relationship between the sliding angle as and the threshold bond number Boa can be approximated by Boa z sin asBo*
(7)
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where Bo* is the critical bond number on a vertical wall (a ¼ 90 ). A suitable interpolation for Bo* in the range [20 , 160 ] is given by eqn (8) and (9) in the appendix.
5 Steady moving drops
Fig. 7 (a) Color map of the bond number Bo* at critical depinning for
protocol (I) in terms of advancing and receding contact angles in range qa, qr ˛ [20 , 160 ]. Data are obtained by interpolation combinations of qr < qa in steps of 5 . (b) Aspect ratio 3 ¼ L/W of drop footprint at the depinning threshold Bo* corresponding to data shown in panel (a).
the for the the
specic values qa and qr. For example considering an hysteresis interval of Dq ¼ 20 , the base eccentricity 3 does not exceed the value z1.2. The results presented until now refer to systems where the body force F acts tangentially to the substrate, i.e., F$ez ¼ 0 with respect to the coordinate system, as shown in Fig. 1. However, the standard experimental set up is the inclined plate, and it is worthy to consider the case where the substrates are tilted by an arbitrary angle a < 90 with respect to the horizontal orientation. Hence, we performed a set of simulations allowing for two non-zero components of the body force F ¼ rgV(ez cos a ex sin a) in the minimisation of the interfacial and gravitational energy. From the simplied model in eqn (2) it is expected that the dominant scaling of the threshold bond number on a tilted plate, Boa, is simply described by the factor sin a. Our calculations show that in the limit of small inclination angles a 1 and large bond numbers Bo [ 1, the liquid interface of a pinned drop is considerably attened. In this situation the differences between protocols (I) and (II) are enhanced and the scaling deviates. Restricting our focus to bond numbers Bo of
3332 | Soft Matter, 2014, 10, 3325–3334
In Section 3 we described the transition from static to steady moving drops induced by an increase of the bond number. Once a drop reaches the steady moving state it has lost its memory of the shapes during the transient period. According to our dynamic contact angle model in eqn (5) the normal component of the local contact line velocity in any point of the contact line is determined by the local apparent contact angle. The terminal velocity of the whole drop at a given driving force, however, is also linked to the global shape of the drop. We expect that the shape of the drop beyond the depinning threshold will depend on the particular form of the contact line mobility law in eqn (5), i.e., on the exponent b. In our model, we assume that the free interface is at any time close to an energy minimum of the interfacial and gravitational energy. Then, the magnitude of the driving force, xed by Bo, and the dynamic law for the contact line in eqn (5) uniquely determine the shape of a steady moving drop. In other words, a simultaneous rescaling of lv and lc such that the value of Bo stays the same, does not affect the drop deformation and the distribution of contact angles. Because the contact line velocity is related only to the local contact angle and not to the drop size, the velocity of a steadily moving drop will depend on the bond number Bo, the advancing and receding contact angle qa and qr, and the parameters b and u0 of the contact line mobility in eqn (5). Steady moving morphologies calculated for values qa ¼ 120 and qr ¼ 60 are displayed in Fig. 8. Panel (a) of Fig. 8 summarises the aspect ratio 3 of the contact line contour as a function of the bond number Bo for a number of integer exponents b. Curves converge in the linear regime close to the depinning threshold Bo*, while for larger bond numbers Bo $ Bo* the steeper the increase of 3 is, the larger the exponent b. Similarly, the dynamic advancing and receding contact angles change more rapidly for a larger b. The receding angle saturates when approaching qr ¼ 0 , see the inset of Fig. 8a. Panels (b), (c) and (d) of Fig. 8 illustrate drop proles, contact line contours and distribution of contact angles of steady moving drops for b ¼ 3 and various bond numbers Bo. Similarly, panels (e), (f), and (g) of Fig. 8 display the drop prole, contact line contours and distribution of contact angles for Bo ¼ 1.6 and various integer exponents b. Values of Bo and 3 corresponding to the drop morphologies shown in panels (b) to (g) of Fig. 8 are marked by lled symbols in panel (a). As
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formation of a corner singularity in the tail of the drop is a multiscale problem where the viscous dissipation in the bulk cannot be neglected.29,30 However, it is interesting to note that a similar singularity appears for sufficiently high drop velocities, as reported also in other studies under assumptions similar to our model.31
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6
Fig. 8 (a) Base eccentricity 3 of the drop as a function of the bond number Bo for different exponents b in the dynamic contact angle model in eqn (5). Advancing and receding contact angles are qa ¼ 120 and qr ¼ 60 , and the corresponding depinning threshold is Bos* ¼ 1.1. The droplet profile in side view contact line contour, and the polar diagram of the contact angle distribution for b ¼ 1 a sequence of bond numbers Bo ¼ 1.2, 1.3, 1.4, 1.5, 1.6 in (b), (c), and (e). Same plots as before, but for a bond number Bo ¼ 1.6 and exponents b ¼ 1, 2, 3 in (c), (f), (g).
expected, larger bond numbers cause the shape of a steady moving drop to be more elongated and narrower. Differences in the exponent b appear to affect primarily the length L of the contact line contour and not its width W. From the distribution of local contact angles one can observe that the maximum and minimum angles are attained only at the symmetry plane in the front and the rear part of the contact line, respectively. On the parallel lateral segments, the contact line is pinned and the angles fall in the range of the hysteresis. Interestingly the main reason for the increase of the aspect ratio of the drop contour is the formation of a pointed tip in the rear. This phenomenon has been experimentally observed on the motion of viscous drops26,27 and provides a useful benchmark to interpret macroscopic features in terms of microscopic processes. In particular Stone and Limat28 obtained a self-similar solution for the thin lm equation in three dimensions, with a conical shape. Clearly our model cannot reproduce the conical shape, being in contrast to our assumption of separation of time and length scale. In fact it has been claried that the
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Conclusion
In this work we numerically investigated the onset of motion of a drop on a rigid and inert substrate in the presence of contact angle hysteresis. To quantify the impact of the contact angle hysteresis on the evolution of drop shapes before depinning, we applied two protocols to quasi-statically increase the bond number. For simplicity, we assumed gravity to act parallel to the substrate. In protocol (I), where the drop volume is increased at xed acceleration, the transition from static to steady moving drops is continuous. The corresponding bond number Bo* is a function of both the static advancing contact angle qa and the static receding contact angle qr. During a quasi-static decrease we observe repinning of the steady moving drops once Bo reaches Bo* from above. For protocol (II), where the acceleration is varied at a constant volume, we obtained larger depinning thresholds Bos > Bo* for the same values of qa and qr. For protocol (II), the depinning transition is discontinuous, and the initial contact angle also affects the depinning threshold. This indicates the existence of a region where both steady moving and static drop shapes could be found for the same parameter values. A similar phenomenon has been recently described for 2D viscous drops on a periodically varying chemical pattern, controlled by the chemical contrast and the slip length.32 We also extended our calculations to arbitrary combinations of static advancing and receding contact angles, providing a map for the threshold values of the critical bond number for the continuous transition. Limited to the cases where the drop attening is small, the present map allows to determine the sliding angles also on inclined substrates. Besides the depinning threshold, we investigated steady moving states assuming a simple dynamic law in eqn (5) which relates the contact line velocity u to the local contact angle q. It is possible to modify the dynamic law to account for the effect of regular patterns on the substrate. For example the presence of different advancing modes33 may lead to an anisotropic contact line mobility. The present approach can also be extended to simulate the motion of drops directly accounting for wettability and/or topographic patterns, with periodicity comparable to the drop size.34
Appendix The bond number at critical depinning can be interpolated using a multinomial expansion in x ¼ cos qa and y ¼ cos qr: Bo* ¼ A(x) + B(x)y + C(x)y2
(8)
where A(x), B(x) and C(x) are sixth degree polynomials
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8 < AðxÞ ¼ 0:01417 1:827x 1:352x2 0:4879x3 0:9299x4 1:786x5 1:360x6 BðxÞ ¼ 1:830 þ 1:788x þ 1:292x2 þ 1:305x3 þ 2:253x4 þ 3:190x5 þ 1:419x6 : CðxÞ ¼ 0:4262 0:6571x 0:5434x2 0:7523x3 1:452x4 1:236x5 0:2059x6
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Eqn (9) reproduces within an accuracy of 1% the simulation data in the range qa,r ˛ [20 , 160 ].
Acknowledgements We thank Prof. Jacco Snoeijer and Daniel Herde for stimulating and helpful discussions. Funding from DFG within the Grant no. HE 2016/14-2 Piko is gratefully acknowledged.
References 1 J. J. Bikerman, J. Colloid Sci., 1950, 349–359. 2 C. Furmidge, J. Colloid Sci., 1962, 17, 309–324. 3 B. Krasovitski and A. Marmur, Langmuir, 2005, 21, 3881– 3885. 4 C. W. Extrand and Y. Kumagai, J. Colloid Interface Sci., 1995, 170, 515–521. 5 I. ElSherbini and M. Jacobi, J. Colloid Interface Sci., 2004, 273, 556–565. 6 M. J. Santos, S. Velasco and J. White, Langmuir, 2012, 28, 11819–11826. 7 E. Pierce, F. J. Carmona and A. Amirfazli, Colloids Surf., A, 2008, 323, 73–82. 8 R. Brown, F. Orr, L. Scriven and Others, J. Colloid Interface Sci., 1980, 73, 76–87. 9 A. Lawal and R. Brown, J. Colloid Interface Sci., 1982, 89, 332– 345. 10 E. B. Dussan V. and R. T. Chow, J. Fluid Mech., 1983, 137, 1. 11 T. D. Blake, A. Clarke, J. De Coninck and M. J. De Ruijter, Langmuir, 1997, 13, 21642166. 12 O. V. Voinov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza, 1976, 5, 76–84. 13 R. Cox, J. Fluid Mech., 1986, 168, 169194. 14 C. Huh and L. Scriven, J. Colloid Interface Sci., 1971, 35, 85– 101. 15 K. Brakke, Philos. Trans. R. Soc., A, 1996, 354, 2143–2157. 16 M. Santos and J. White, Langmuir, 2011, 27, 14868–14875.
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17 L. Schwartz, D. Roux and J. Cooperwhite, Phys. D, 2005, 209, 236–244. 18 B. M. Mognetti, H. Kusumaatmaja, J. M. Yeomans and P. J. Rossky, Faraday Discuss., 2010, 146, 153. 19 V. Berejnov and R. Thorne, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2007, 75, 1–6. 20 M. Brinkmann and R. Lipowsky, J. Appl. Physiol., 2002, 92, 4296. 21 H. B. Eral, J. de Ruiter, R. de Ruiter, J. M. Oh, C. Semprebon, M. Brinkmann and F. Mugele, So Matter, 2001, 7, 5138. 22 R. de Ruiter, J. de Ruiter, H. B. Eral, C. Semprebon, M. Brinkmann and F. Mugele, Langmuir, 2012, 28, 13300– 13306. 23 D. Ferraro, C. Semprebon, T. T´ oth, E. Locatelli, M. Pierno, G. Mistura and M. Brinkmann, Langmuir, 2012, 28, 13919– 13923. 24 P. Dimitrakopoulos and J. J. L. Higdon, J. Fluid Mech., 1999, 395, 181–209. 25 D. Qu´ er´ e, M. J. Azzopardi and L. Delattre, Langmuir, 1998, 7463, 2213–2216. 26 T. Podgorski, J. M. Flesselles and L. Limat, Phys. Rev. Lett., 2001, 87, 1–4. 27 N. Le Grand, A. Daerr and L. Limat, J. Fluid Mech., 2005, 541, 293. 28 L. Limat and H. Stone, Europhys. Lett., 2004, 65, 365–371. 29 J. H. Snoeijer, E. Rio, N. Le Grand and L. Limat, Phys. Fluids, 2005, 17, 072101. 30 J. H. Snoeijer and B. Andreotti, Ann. Rev. Fluid Mech., 2013, 45, 269–292. 31 M. Ben Amar, L. J. Cummings and Y. Pomeau, Phys. Fluids, 2003, 15, 2949. 32 D. Herde, U. Thiele, S. Herminghaus and M. Brinkmann, Europhys. Lett., 2012, 100, 16002. 33 C. Semprebon, S. Herminghaus and M. Brinkmann, So Matter, 2012, 8, 6301–6309. 34 S. Varagnolo, D. Ferraro, et al., Phys. Rev. Lett., 2013, 111, 066101.
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PAPER
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On the onset of motion of sliding drops Cite this: Soft Matter, 2014, 10, 3325
Ciro Semprebona and Martin Brinkmann*ab In this article we numerically investigate the onset of motion of liquid drops in contact with a plane and homogeneous substrate with contact angle hysteresis. The drops are driven by a body force F ¼ rgV, where r is the density of the liquid, g is the acceleration of gravity, and V is the volume of the drop. We compare two protocols to vary the bond number Bo ¼ lv/lc by changes of either the drop size lv ¼ V1/3 or the capillary length lc ¼ (g/rg)1/2 where g is the interfacial tension, revealing that the transition between pinned and steady moving states can be either continuous or discontinuous. In a certain range
Received 19th July 2013 Accepted 3rd February 2014
both pinned and moving states can be found for a given bond number Bo, depending on the history of the control parameters g and V. Our calculations are extended to arbitrary combinations of static advancing and receding contact angles and provide a comprehensive picture of the depinning transition
DOI: 10.1039/c3sm51959g
induced by a quasi-static variation of the control parameters. Finally, we demonstrate that the particular
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form of the contact line mobility in our model has an impact on the interfacial shape of steady moving drops.
1
Introduction
Rain drops running down a window pane can be easily trapped by impurities like dust grains or other surface heterogeneities, while sticking drops can be observed even on freshly cleaned glass surfaces. These two behaviours are ultimately related to surface heterogeneities at the micron and submicron scale, which can be sufficient to pin the three phase contact line. The depinning mechanism of drops under the action of gravity and their subsequent motion has been particularly important to dene an optimum drop size in pesticide sprays.1,2 Systematic measurements of sliding drops help provide a simple way to characterise the homogeneity of surfaces.4,5 Most of the experimental procedures to induce the onset of sliding motion rely on the quasi-static variation of the body force, for example achieved by tilting a surface or increasing the rotation velocity of a spinning plate. An extended review on the subject can be found in an article by Santos et al.6 However, despite the remarkable amount of studies, the description of the transition between pinned and moving states is still incomplete. In particular, it has been experimentally shown that different experimental protocols lead to different depinning conditions,7 while a more quantitative analysis is sill lacking. In this work we attempt to give a comprehensive picture of the evolution of drop shapes and of the onset of motion, driven by the quasi-static variation of the body force. In particular, we are interested in the order of the depinning
a
Department Dynamics of Complex Fluids, Max-Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 G¨ottingen, Germany
b Experimental Physics, Saarland University, 66123 Saarbr¨ ucken, Germany. E-mail: [email protected]
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transition, which will turn out to depend on the protocol adopted to vary the body force. Consider a sessile drop in contact with a plane substrate subjected to a horizontal body force F, as sketched in Fig. 1. In a typical experiment, we have F ¼ rgV, where g is the acceleration of gravity, r is the mass density of the liquid, and V is the drop volume. In the present work we assume for simplicity, unless otherwise stated, that the component of the body force in the vertical direction is negligible. This situation corresponds to drops on a vertical window pane, or drops on a spinning plate sufficiently small and far from the axis of rotation.
Profile of the drop (top) and contour of the contact line (bottom) of a steady moving drop shortly after depinning, driven by a body force F acting horizontally. H, W, and L indicate the height, width and length of the drop, respectively.
Fig. 1
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In the presence of hysteresis the contact line of the drop will locally advance over the substrate only if the apparent contact angle is larger than the advancing angle qa, and recede only if it is smaller than the receding angle qr. Apparent contact angles q measured on a static drop prole always fall into the hysteresis interval, dened as the range between these two values. A quantitative estimate for the depinning condition can be obtained considering a mechanical equilibrium of the body force F ¼ exrgV acting on the bulk of the liquid and the horizontal components of the interfacial tension at the contact line, þ rgV ¼ g dsex $n cos q; (1)
The paper is organised as follows: rst, we introduce the physical model and its numerical implementation in Section 2. In the following Section 3 we analyse the quasi-static evolution of the drop morphologies for a specic combination of advancing and receding angles. Here, we consider two paradigmatic protocols under various initial conditions, and discuss the transition to moving states. In Section 4 we extend our results to substrates with arbitrary combinations of advancing and receding angles and discuss the implications for tilted plate experiments. Finally, in Section 5 we analyse the impact of the driving force and some simple laws for the contact line mobility on the morphologies of steady moving drops.
G
where G denotes the contour of the contact line parameterised by the arc-length s and g denotes the interfacial energy of the liquid interface. The vector n is the normal to the contact line in the plane of the substrate while ex is the downhill direction, cf. also the sketch in Fig. 1. A relationship between the static advancing and static receding contact angles and the magnitude of the body force at which sliding occurs can be established under certain simplifying assumptions regarding the shape of the contact line, and the distribution of local contact angles. Based on experimental observations, Furmidge2 proposed a simple model for drop shapes where the contact line contour G exhibits straight sides aligned with the body force. This situation, illustrated in Fig. 1, appears in tilting plate experiments. In the curved front and rear part of G, the local contact angles at depinning are equal to the advancing and receding angles, respectively. Under these assumptions, the RHS of eqn (1) can be integrated and reduced to rgV ¼ gkW(cos qr cos qa)
(2)
where W denotes the extension of the drop perpendicular to the force F. While the Furmidge model corresponds to k ¼ 1, experimental measurements4 and numerical calculations6,8–10 show that the empirical factor k in eqn (2) deviates signicantly for various systems. The reason is that the balance expressed in eqn (1) crucially depends on the actual shape of the drop contour. Practically, eqn (2) requires a reasonable estimate for the drop width W to correctly predict the onset of motion. In this work, rather than imposing a priori simplied shapes to the contact line contour, we quasi-statically drive the drop to the depinning transition. The drop morphology, and in particular its width and elongation, is then uniquely determined by the initial condition and the protocol adopted to change the set of control parameters. Recently it has been challenged by Krasovitski et al.3 that the advancing and receding angles correspond to the maximum and minimum angles measured at the onset of motion. This is the case if the hysteresis is induced by a heterogeneity of the substrate with a typical wavelength comparable to the drop size. Practically however, the hysteresis is related to a surface heterogeneity at a much smaller scale, and it can be regarded as a material parameter. In this case the maximum and minimum angles equal the advancing and receding angles, as observed in most of the experiments.
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2 Physical model and numerical implementation Throughout this work we consider ideally plane, rigid and chemically inert substrates, with isotropic contact angle hysteresis. The macroscopic wetting properties are described in terms of the static advancing contact angle, qa, and the static receding contact angle, qr. If the drop velocity is small and, at the same time, the drop is not too at, the shape of the interface is governed by the interplay of interfacial stresses and gravity. This approximation is good if the modied capillary number Ca* h hUH3/gW3 is small compared to one. Here, h is the dynamic viscosity of the liquid, g is the interfacial tension, U is the drop velocity, while H denotes the height of the drop, cf. also the sketch in Fig. 1. If Ca* 1, the injected work during drop movement will be dissipated in the direct vicinity of the contact line. In this case, it is reasonable to assume a separation of length and time scales for the interfacial deformation, between the macroscopic global shape of the drop and the microscopic distortion of the contact line. As a consequence, for a given contour G of the contact line, we assume that the interface of the drop instantaneously adopts the shape S that minimises the sum of energy of the free interface and the gravitational contribution E{S} ¼ gAlv rgVhxi.
(3)
Here, Alv denotes the area of the liquid vapour interface, r is the mass density of the liquid, g is the acceleration,† and hxi is the x-coordinate of the liquid's centre of mass given by ð 1 d3 r ex $r : (4) hxih V U As we assume the liquid to be non-volatile on the time-scale of the experiment, we apply the constraint of a xed volume V to the drop's body U. The local velocity u of the three phase contact line is zero if q falls into the hysteresis interval. Otherwise, we suppose that u is described by a power law of the local contact angle q: † An acceleration g ¼ u2r parallel to the surface can be realised for a vertical orientation of the substrate or on a spinning plate, where u is the angular frequency of rotation and r is the distance of the centre of mass to the rotation axis.
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u¼
8 u 0 b b > > < bq b1 q qa
for q $ qa
u0 b q qr b b1 bqr
for q # qr
a
> > :
;
(5)
The exponent b in eqn (5) mimics the leading order scaling of certain models proposed for the contact line motion. For example, b ¼ 2 corresponds to the molecular kinetic theory according to Blake11 while b ¼ 3 corresponds to the continuum hydrodynamic model of Voinov12 and Cox.13 The linearized relationship around the static values of the contact angles qa and qr reduces to a friction law, taking b ¼ 1. Models for the contact line dynamics including slippage are necessary to overcome the paradox rst described by Huh and Scriven,14 for which the stresses and viscous dissipation diverge at the contact line in the presence of no slip boundary condition. In our case the prefactor u0 describes a typical velocity scale related to the combination of uids and solid materials. Note that our results about the quasistatic evolution are general and not bound to the specic form of eqn (5). Instead results regarding transient states and steady moving drops are bound to the model details. For simplicity we dened in eqn (5) a prefactor such that the linearised mobility is identical for all exponents b both at q ¼ qa and q ¼ qr. A different choice for the normalization could be motivated by additional information on the specic system under investigation. Note, that the model presented here is not appropriate whenever a separation of length scales is not possible. In view of the numerical implementation of our model it is useful to non-dimensionalise all relevant physical quantities. In general, one is free to chose, besides a mass scale M0, a length scale L0 and a time scale T0. Two length scales naturally arise if we consider static drop shapes under the inuence of gravity: the capillary length lc h (g/rg)1/2 and the linear dimension of a droplet lv h V1/3. The ratio of both length scales denes the magnitude of the bond number Bo h lv2/lc2. The particular choice L0 h lc is useful in experiments where the acceleration g is xed while the volume V is changing. If, instead, g is changing while V is constant, the choice L0 h lv is more appropriate. Alternatively, if both length scales are changing during the experiment, L0 could be either dened as lv(t0) or lc(t0) at the start of the experiment t ¼ t0. Given the length scale L0, a natural denition of the timescale is T0 h L0/u0. We compute drop shapes by minimising the interfacial and gravitational energy in eqn (3) under the constraint of a xed liquid volume V. The liquid–vapor interface S is represented by a mesh of nodes connected by the edges of triangular facets. Nodes belonging to the contact line are constrained to move in the plane of the substrate. In this representation, any quantity like the energy in eqn (3) or the enclosed volume is a function of 3N Cartesian coordinates and can be numerically evaluated and minimised under local and global constraints by standard optimisation algorithms. We use the public domain soware Surface Evolver15 to set up the interface, maintain the triangulation, and to minimise the energy (3). In order to improve the accuracy, the mesh is gradually rened close to the contact line, close to the symmetry plane and in parts where the interface is
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Fig. 2 (a) Snapshot of a sliding drop in our numerical computations illustrating the different levels of the mesh refinement. (b) Functional dependence of the model for the contact line velocity for different exponents b. (c) Magnification of the region close to the contact line to illustrate the implementation of the dynamic law. The local contact angle is computed from the surface normal of the triangles sharing an edge with the contact line (shown in green).
strongly curved. The smallest edge element is chosen such that the contact line is typically composed of 200 elements of which only half are simulated exploiting the mirror symmetry of the drop. In a typical simulation run, the interface of half of the drop, as shown in Fig. 2a, is represented by roughly 500 nodes and 1500 edges. The algorithm to simulate the temporal evolution of the droplet shape contains two steps: in the rst step the energy of the drop is minimised given the contour Gn of the contact line at time tn. In the second step, aer convergence of the interfacial shape, the local contact angle q at each edge of Gn is evaluated. If qr # q # qa holds for both the edges connected to a node of Gn, the contact line around the node is in a pinned state and this node is not moved. Whenever the local contact angle of at least one of the neighbouring edges falls outside the hysteresis interval, i.e., q > qa or q < qr, the node is allowed to move. To perform the motion of the contact line between time tn and tn+1 h tn + Dt, a virtual displacement Dr ¼ nuDt is rst assigned to each edge of Gn, where n is the co-normal vector of the contact line in the plane of the substrate. The local velocity of the contact line, u, is a function of q according to the dynamic law in eqn (5). As shown in the sketch in Fig. 2c, the new contact line contour Gn+1 is then obtained by moving each node according to a weighted average of the virtual displacements assigned to the neighbouring edges. At this point, an updated interfacial shape Sn+1 for a new contact line contour Gn+1 has to be computed. The drop shape is evolved in time by a cyclic repetition of the two steps. To reduce the computational time we use an adaptive
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time step chosen such that the maximum displacement is 10% of the length of the shortest element of the contact line. To check the accuracy of the algorithm we numerically integrated the stresses along the contour of the contact line projected along the direction of the body force. According to the nondimensional form of eqn (1), the local contact angle has to satisfy the identity þ ds cos q ex $n ¼ Bo; (6) Published on 03 February 2014. Downloaded by Université Laval on 26/05/2014 21:23:37.
G
where we made use of the non-dimensional arc length parameter s h S/lv. Eqn (6) has been veried within 1% of accuracy in our simulation runs. A typical simulation run is performed as follows: rst the drops are introduced under an initial condition chosen according to the protocols described in Section 3. Then the bond number is quasi-statically increased by changing one of the two length scales. At every iteration we check if the contact line is entirely pinned or if some parts have been displaced. If all the elements of the contact line reach a pinned state we stop the iterations and save the conguration. If aer N steps some parts of the contact line are still being displaced the drop is assumed to be in a steady moving state and we perform N more steps averaging all quantities of interest. We choose N ¼ 300, as a compromise between accuracy and computational time. A similar algorithm has been proposed by Santos and White6,16 to simulate the quasi-static evolution of drops in tilt plate experiments. To test our implementation we considered the same system investigated by Santos and White6 accounting for both the normal and tangential components of the body force. In the quasi static regime both implementations are equivalent, and our results conform with the results in ref. 6. In addition our algorithm allows us to investigate the shape of steady moving drops for whenever the energy dissipation occurs in the vicinity of the contact line, implying that the free liquid interface is close to a state of minimum energy. However, if the viscous dissipation in the liquid bulk cannot be neglected, the full Navier Stokes equations need to be numerically solved, which is a signicantly more complex task.17,18
3 Two depinning protocols In the following we investigate the quasi-static evolution of drop shapes under two different protocols to slowly increase the bond number Bo h lv2/lc2. In protocol (I), the volume V is increased in small steps while the capillary length lc is kept constant. Employing the capillary length as a unit length L0 h lc, the volume of the drop in the initial state is set to V ¼ lv3 ¼ 103L03. Distortions of the interface induced by the body force are negligible because the linear dimension of the drop, lv, satises lv lc. In this case the initial value of the contact angle qin of the drop is approximately constant along its perimeter and taken to be qin ¼ qa. Above a certain value of the bond number, the drop does not anymore relax into a static shape and enters a dynamic, sliding state. In the complementary protocol (II), we consider a small drop of constant liquid volume V and slowly increase the bond
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number Bo by decreasing the capillary length lc. In this situation, it is more appropriate to use the length scale L0 h lv related to the drop volume as a reference length. We start with a value lc ¼ 10L0 and consider a variety of initial contact angles qin in the interval of the hysteresis. In the special case where qin ¼ qa the initial drop shape is the same as in protocol (I) but with a linear dimension scaled up by a factor of 10. As for protocol (I), we nd that the drop will not relax anymore into a static shape if a certain value of the bond number is exceeded. Examples of the main results of our numerical investigations are illustrated in Fig. 3 for particular values qa ¼ 120 and qr ¼ 60 . Each panel in the rst row of Fig. 3 displays a set of drop proles projected to the symmetry plane. The corresponding contact line contours are shown in the second row. Variations of the local angle q as a function of polar angle f are displayed in the third row. Here, the centre of the area in contact with the liquid is chosen as the origin of the polar coordinate system. The rst column in Fig. 3 refers to protocol (I) while the second and third column refer to protocol (II) for initial contact angles qin ¼ qa and qin ¼ (qa + qr)/2, respectively. In protocol (I) major parts of the contact line are moving already at small bond numbers Bo 1 as the drop has to accommodate the increasing liquid volume. Consequently, the drop grows both in the direction of the body force and in the direction perpendicular to it, while the trailing edge remains immobilised. As expected, the local contact angle q equals the advancing contact angle qa in the parts of the contact line which have advanced. Following the contact line from the front to the back of the drop, q decreases in the pinned part to a minimum located in the rear apex. For larger bond numbers the local contact angle in the rear apex reaches the value qr and the trailing edge also depins. The condition that the leading and the trailing edge are depinned does not imply that the entire drop is depinned, i.e., it still relaxes into a static shape aer a small increment of Bo. Finally, depinning of the drop occurs at a bond number Bo ¼ Bo*, which is of the order unity. For the present choice of advancing and receding contact angles, the contact line contour at depinning does not include the initial contour of the drop. As a consequence it is not possible to experimentally reproduce the depinning induced by protocol (I) just by injecting the liquid from a hole in the substrate, as the contact line would pin at the edges of the hole before the onset of motion. Once the drop has reached a steady moving state, the contact line contour displays two parallel straight sides aligned with the direction of the driving body force. The leading and trailing edges are curved and both contours are close to a circular arc, with different radius of curvature. This observation has been already reported by Furmidge in ref. 2. The last pinned drop shape detected for protocol (I) at Bo ¼ Bo* also coincides with the shape of the rst steady moving drop in the limit of zero velocity as Bo approaches Bo* from above. The latter shape is represented by a black dashed line in Fig. 3a–c, which indicates a continuous transition from pinned to moving drops. Apart from the fact that the length scale of the drop, lv, does not change during the increase of the bond number, the shape of the last pinned drop in protocol (II) for an initial contact angle qin ¼ qa is similar to the one in protocol (I). The
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Upper, middle, and lower panels show sequences of drop profiles, contact line contours, and distributions of contact angles, respectively, during a quasi-static increase of the bond number Bo according to protocols (I) and (II), see the main text. Data are rescaled by the respective characteristic length that is kept fixed. Dashed lines represent the last detected profile before depinning. The substrate is characterised by qa ¼ 120 and qr ¼ 60 . Panels (a), (b), and (c): drop size lv is increased while the capillary length lc is constant for Bo ¼ 0.01, 0.16, 0.49, 0.72. Panels (d), (e), and (f): lc is decreased while lv is constant for Bo ¼ 0.01, 0.16, 0.36, 0.64, 1, 1.14. The initial shape is a spherical cap with qin ¼ 120 . Panels (g), (h), (i): lc is decreased at a constant lv for Bo ¼ 0.01, 0.36, 0.72, 1.42 and qin ¼ (qa + qr)/2 ¼ 90 . Fig. 3
differences in the evolution of the drop shapes between protocols (I) and (II), however, become more pronounced when qin < qa. The drop proles and their corresponding contact line contours for the case qin ¼ (qa + qr)/2 shown in Fig. 3 reveal that the contact line is displaced in equal proportions in the front and back of the drop while the overall shape of the contact line contour stays close to a circle. Details of the contours and proles, indicate qualitative agreement with the observations of Berejnov and Thorne.19 During the depinning transition in protocol (II), the drop ‘hatches’ through the narrow sector of the moving contact line contour in the front. In order to further investigate the morphological changes during the depinning processes for protocols (I) and (II), and to highlight their main characteristics, we compare in Fig. 4a the evolution of the base eccentricity, dened as the ratio between the parallel and perpendicular length, 3 h L/W, as function of Bo for both protocols. Quasi-static branches are represented by continuous lines, thick for protocol (I) and thin for protocol (II). As expected, the most rapid increase of the base eccentricity 3 occurs for protocol (I) at a small Bo. Close to the point of depinning at Bo ¼ Bo*, the rate at which 3 grows with Bo is reduced. During protocol (II) for qin ¼ qa ¼ 120 , the base eccentricity 3 increases linearly with Bo. For the initial contact
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Fig. 4 Comparison of the evolution of the eccentricity 3 h L/W of the
contact line as a function of bond number Bo for protocols (I) (thick black line) and (II) (thin lines). The thick red line indicates steady moving drops for b ¼ 3. See the main text for details.
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angle qin ¼ (qa + qr)/2 ¼ 90 both contact line displacements occur almost simultaneously and lead to only a slight variation of 3. For an initial contact angle qin ¼ 72 , the trailing edge displaces rst, inducing a linear decrease of 3 with Bo. The bond number at the depinning transition for protocol (II), Bos, satises Bos $ Bo* and is monotonously increasing with the decrease of the initial contact angle qin. The most striking difference between protocols (I) and (II) can be observed during the transition from static to steady moving drops: As Bo approaches Bo* from below, the shape of static drops on the branch corresponding to protocol (I) continuously evolve toward the shape of steady moving drops at Bo $ Bo*. This behaviour is in contrast to the observation made during protocol (II): here, the shape of the drop undergoes a discontinuous change between the last static shape at Bo ¼ Bos and the drop shapes on the steady moving branch at Bo $ Bos. The pinned lateral parts of the contact line form a bottle neck for the free interface of the drop and a certain excess of driving force is required to overcome the energy barrier. For protocol (II) it is possible to obtain the quasi-static evolution by imposing a global constraint to x the lateral position of the center of mass in eqn (4). The Lagrange multiplier corresponding to this global constraint plays the same role as the prefactor rgV in the total energy in eqn (3). In contrast to the unconstrained ensemble, the acceleration is adjusted such that the lateral component of the center of mass position, hxi, is kept at a xed value. Integration of the Lagrange multiplier, and the equivalent body force, yields the work dissipated during the movement as a function of the displacement. Both methods, xing the body force F ¼ rgV or the lateral position of the center of mass, hxi, yield equivalent results for protocol (II) if vBo/vhxi $ 0. Fixing the value of hxi instead of Bo allows us to observe solution branches of drop shapes with v Bo/vhxi < 0. A similar method has been applied in several problems regarding the stability of liquid interfaces in contact with substrates with complex geometries20–23. In the present case, the lateral position of the center of mass hxi is slowly dragged out of its initial position. At the same time the contact line G is allowed to relax according to the contact line mobility in eqn (5). Within the numerical accuracy, the evolution of the drop base eccentricity 3 is identical to the quasistatic evolution described above. In addition, it is possible to observe branches in the ensemble where the body force is xed and the center of mass position is free to move. These branches are shown as thin dashed lines in Fig. 4. The corresponding interfacial conguration represents drop shapes aer the depinning. Interestingly, all unstable branches are attracted to the conguration of the last pinned shape at critical depinning Bo* according to protocol (I). Once the centre of mass hxi has been displaced sufficiently far from its initial position, any information about the initial state is lost and the free interface will adopt the shape at critical depinning. In view of the discontinuous transition found for protocol (II), it is not surprising to nd regions of bond numbers where both pinned and steady moving drop shapes exist. In particular, it is likely that the lower bound to Bos is determined by the depinning threshold Bo* of protocol (I). The upper bound to Bos
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has been determined by Dimitrakopoulos24 using an optimisation algorithm that converges to the drop shape with the highest resistance to depinning. From our results, and the investigations of Dimitrakopoulos, we suggest that depinning events are discontinuous when the depinning threshold is larger than Bo*. A steady moving branch for the case b ¼ 3 is represented with a thick red dashed line in Fig. 4 and will be described in more detail in Section 5. To resolve the temporal evolution of the drop shapes during the depinning transition for protocol (II), we restart our simulations from the closest conguration saved before the depinning threshold Bos. The bond number is then slightly increased above Bos and the simulation run is continued without further increase of the bond number until the drop has displaced twice its own length. Fig. 5a shows the temporal evolution of the drop base eccentricity 3 during the discontinuous depinning, for some of the cases considered in Fig. 4. Panels (b), (c) and (d) of Fig. 5 illustrate a sequence of out of equilibrium proles, contact line contours, and contact angle distributions for the
(a) Temporal evolution of the base eccentricity of the contact line, 3, in protocol (II) after depinning for initial contact angles qin ¼ 60 , qin ¼ 72 , qin ¼ 90 , and qin ¼ 108 . Inset: relationship between the transition time Dt and the initial condition qin. The reference time t0 refers to the inflexion point. Colored dots on the curve for qin ¼ 72 refer to the profiles, contours and angle distribution shown in panels (b), (c) and (d) respectively. The dashed curves refer to the last pinned morphology detected (t ¼ N). Fig. 5
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case of qin ¼ 72 . The corresponding morphologies are marked by colored points in panel (a). Two steps can be distinguished in the transition: initially the contact line slowly creeps forward in the central part of the leading edge while the sides remain pinned. Once the drop has passed through the bottle neck it quickly approaches the characteristic morphology of a steady moving drop. The total transition time can be thought as the sum of the time necessary to complete both steps. The rst transition time is related to the magnitude of the bond number increment when the simulations are restarted. The second transition time is related to the excess of driving force across the transition and diverges as Bo approaches the critical value Bo* of the continuous depinning transition for protocol (I). The inset of Fig. 5a illustrate how this transition time Dt depends on the initial condition qin for protocol (II).
4 Depinning threshold for arbitrary hysteresis In the previous section we analysed the depinning process applying two different protocols to increase the bond number. So far, we restricted our analysis to the specic values qa ¼ 120 and qr ¼ 60 . To check if the same qualitative results hold also for different combination of angles, we performed additional calculations by either xing qr ¼ 60 and varying qa or xing qa ¼ 120 and varying qr. The results referred to the quasi-static evolution with protocol (I) displayed in Fig. 6 conform to the descriptions in Section 3. Interestingly, the quasi-static evolution of the base eccentricity 3 is a monotonous function of the bond number Bo only when the difference between qa and qr is sufficiently large. As expected, the smaller the value of the depinning threshold Bo* for protocol (I) is, the smaller the contact angle hysteresis. In this case, the base eccentricity of the drop exhibits a maximum at some intermediate value Bo < Bo*. The effect is also clearly visible from the contour plots in Fig. 7a. The same calculations have been repeated for protocol (II), and the same behaviour has been observed considering both the case qin ¼ qa and the case qin ¼ (qa + qr)/2. In particular we observed that although the two protocols are qualitatively different, the depinning threshold Bos for protocol (II) in the case qin ¼ qa is very close to the critical value Bo* for protocol (I), and they can be confused if Dq < 10 . Naively, one would expect that the bond number Bo* at critical depinning grows monotonously with contact angle hysteresis Dq h qa qr, regardless of how the static contact angles qa and qr are varied. Inspection of Fig. 6a shows that this statement holds only in a restricted number of cases, i.e., if qa is xed and qr is decreased. Instead, if qr is xed and qa is increased, Bo* displays a maximum, see Fig. 6b. The non-monotonous variation of the depinning threshold Bo* can be qualitatively explained simply considering drops whose width W is set by the diameter of the equivalent spherical cup set by the advancing angle qa. According to eqn (2) the retention force is proportional to cos qa cos qr and to the width W of the drop. While keeping xed the value of qr, a larger value of qa induces not only a larger retention force per unit
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Quasi-static evolution according to protocol (I) of the aspect ratio of the contact line for (a) fixed qa ¼ 120 and various qr, and (b) fixed qr ¼ 60 and various qa. Fig. 6
length, but also a smaller width W of the contact line contour. As a consequence, at a certain value of the advancing angle, qmax , the two effects must balance each other, and the threshold a bond number Bo* shows a maximum. To complete our analysis, we will from now on focus on protocol (I) and determine the values of the bond number Bo* at critical depinning for all combinations of contact angle values qa and qr in the range of contact angles between 20 and 160 . To this end, we performed 435 independent calculations for qa and qr varied in steps of 5 . The results are presented in Fig. 7, where the values of Bo* have been interpolated with a multinomial function in cos qa and cos qr and encoded with a color scale. To precisely determine the depinning threshold Bo* for each simulation run, we linearly interpolate the relationship between the bond number Bo and velocity u for steady moving drops in the range 0.05 < u/u0 < 0.3. An example is shown in the inset of Fig. 7a. Isolines in the main plot of Fig. 7a identify regions of constant depinning threshold Bo* and suggest the presence of a global maximum located around qa z 118 and qr z 0 where Bo* z 1.65. Oen the depinning threshold is calculated in the limit of systems with small contact angle hysteresis, assuming that the drop shape is approximately symmetric, with only small distortions induced by gravity. This approach can lead to analytical relationships as outlined by Qu´ er´ e in ref. 25. A useful benchmark can be obtained by analysing the base eccentricity 3 in our systematic calculations, here presented as a color plot in Fig. 7b. Interestingly, 3 is mainly proportional to the hysteresis Dq, with only a mild dependence on the Soft Matter, 2014, 10, 3325–3334 | 3331
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the order of one, the attening is small and the relationship between the sliding angle as and the threshold bond number Boa can be approximated by Boa z sin asBo*
(7)
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where Bo* is the critical bond number on a vertical wall (a ¼ 90 ). A suitable interpolation for Bo* in the range [20 , 160 ] is given by eqn (8) and (9) in the appendix.
5 Steady moving drops
Fig. 7 (a) Color map of the bond number Bo* at critical depinning for
protocol (I) in terms of advancing and receding contact angles in range qa, qr ˛ [20 , 160 ]. Data are obtained by interpolation combinations of qr < qa in steps of 5 . (b) Aspect ratio 3 ¼ L/W of drop footprint at the depinning threshold Bo* corresponding to data shown in panel (a).
the for the the
specic values qa and qr. For example considering an hysteresis interval of Dq ¼ 20 , the base eccentricity 3 does not exceed the value z1.2. The results presented until now refer to systems where the body force F acts tangentially to the substrate, i.e., F$ez ¼ 0 with respect to the coordinate system, as shown in Fig. 1. However, the standard experimental set up is the inclined plate, and it is worthy to consider the case where the substrates are tilted by an arbitrary angle a < 90 with respect to the horizontal orientation. Hence, we performed a set of simulations allowing for two non-zero components of the body force F ¼ rgV(ez cos a ex sin a) in the minimisation of the interfacial and gravitational energy. From the simplied model in eqn (2) it is expected that the dominant scaling of the threshold bond number on a tilted plate, Boa, is simply described by the factor sin a. Our calculations show that in the limit of small inclination angles a 1 and large bond numbers Bo [ 1, the liquid interface of a pinned drop is considerably attened. In this situation the differences between protocols (I) and (II) are enhanced and the scaling deviates. Restricting our focus to bond numbers Bo of
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In Section 3 we described the transition from static to steady moving drops induced by an increase of the bond number. Once a drop reaches the steady moving state it has lost its memory of the shapes during the transient period. According to our dynamic contact angle model in eqn (5) the normal component of the local contact line velocity in any point of the contact line is determined by the local apparent contact angle. The terminal velocity of the whole drop at a given driving force, however, is also linked to the global shape of the drop. We expect that the shape of the drop beyond the depinning threshold will depend on the particular form of the contact line mobility law in eqn (5), i.e., on the exponent b. In our model, we assume that the free interface is at any time close to an energy minimum of the interfacial and gravitational energy. Then, the magnitude of the driving force, xed by Bo, and the dynamic law for the contact line in eqn (5) uniquely determine the shape of a steady moving drop. In other words, a simultaneous rescaling of lv and lc such that the value of Bo stays the same, does not affect the drop deformation and the distribution of contact angles. Because the contact line velocity is related only to the local contact angle and not to the drop size, the velocity of a steadily moving drop will depend on the bond number Bo, the advancing and receding contact angle qa and qr, and the parameters b and u0 of the contact line mobility in eqn (5). Steady moving morphologies calculated for values qa ¼ 120 and qr ¼ 60 are displayed in Fig. 8. Panel (a) of Fig. 8 summarises the aspect ratio 3 of the contact line contour as a function of the bond number Bo for a number of integer exponents b. Curves converge in the linear regime close to the depinning threshold Bo*, while for larger bond numbers Bo $ Bo* the steeper the increase of 3 is, the larger the exponent b. Similarly, the dynamic advancing and receding contact angles change more rapidly for a larger b. The receding angle saturates when approaching qr ¼ 0 , see the inset of Fig. 8a. Panels (b), (c) and (d) of Fig. 8 illustrate drop proles, contact line contours and distribution of contact angles of steady moving drops for b ¼ 3 and various bond numbers Bo. Similarly, panels (e), (f), and (g) of Fig. 8 display the drop prole, contact line contours and distribution of contact angles for Bo ¼ 1.6 and various integer exponents b. Values of Bo and 3 corresponding to the drop morphologies shown in panels (b) to (g) of Fig. 8 are marked by lled symbols in panel (a). As
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formation of a corner singularity in the tail of the drop is a multiscale problem where the viscous dissipation in the bulk cannot be neglected.29,30 However, it is interesting to note that a similar singularity appears for sufficiently high drop velocities, as reported also in other studies under assumptions similar to our model.31
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6
Fig. 8 (a) Base eccentricity 3 of the drop as a function of the bond number Bo for different exponents b in the dynamic contact angle model in eqn (5). Advancing and receding contact angles are qa ¼ 120 and qr ¼ 60 , and the corresponding depinning threshold is Bos* ¼ 1.1. The droplet profile in side view contact line contour, and the polar diagram of the contact angle distribution for b ¼ 1 a sequence of bond numbers Bo ¼ 1.2, 1.3, 1.4, 1.5, 1.6 in (b), (c), and (e). Same plots as before, but for a bond number Bo ¼ 1.6 and exponents b ¼ 1, 2, 3 in (c), (f), (g).
expected, larger bond numbers cause the shape of a steady moving drop to be more elongated and narrower. Differences in the exponent b appear to affect primarily the length L of the contact line contour and not its width W. From the distribution of local contact angles one can observe that the maximum and minimum angles are attained only at the symmetry plane in the front and the rear part of the contact line, respectively. On the parallel lateral segments, the contact line is pinned and the angles fall in the range of the hysteresis. Interestingly the main reason for the increase of the aspect ratio of the drop contour is the formation of a pointed tip in the rear. This phenomenon has been experimentally observed on the motion of viscous drops26,27 and provides a useful benchmark to interpret macroscopic features in terms of microscopic processes. In particular Stone and Limat28 obtained a self-similar solution for the thin lm equation in three dimensions, with a conical shape. Clearly our model cannot reproduce the conical shape, being in contrast to our assumption of separation of time and length scale. In fact it has been claried that the
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Conclusion
In this work we numerically investigated the onset of motion of a drop on a rigid and inert substrate in the presence of contact angle hysteresis. To quantify the impact of the contact angle hysteresis on the evolution of drop shapes before depinning, we applied two protocols to quasi-statically increase the bond number. For simplicity, we assumed gravity to act parallel to the substrate. In protocol (I), where the drop volume is increased at xed acceleration, the transition from static to steady moving drops is continuous. The corresponding bond number Bo* is a function of both the static advancing contact angle qa and the static receding contact angle qr. During a quasi-static decrease we observe repinning of the steady moving drops once Bo reaches Bo* from above. For protocol (II), where the acceleration is varied at a constant volume, we obtained larger depinning thresholds Bos > Bo* for the same values of qa and qr. For protocol (II), the depinning transition is discontinuous, and the initial contact angle also affects the depinning threshold. This indicates the existence of a region where both steady moving and static drop shapes could be found for the same parameter values. A similar phenomenon has been recently described for 2D viscous drops on a periodically varying chemical pattern, controlled by the chemical contrast and the slip length.32 We also extended our calculations to arbitrary combinations of static advancing and receding contact angles, providing a map for the threshold values of the critical bond number for the continuous transition. Limited to the cases where the drop attening is small, the present map allows to determine the sliding angles also on inclined substrates. Besides the depinning threshold, we investigated steady moving states assuming a simple dynamic law in eqn (5) which relates the contact line velocity u to the local contact angle q. It is possible to modify the dynamic law to account for the effect of regular patterns on the substrate. For example the presence of different advancing modes33 may lead to an anisotropic contact line mobility. The present approach can also be extended to simulate the motion of drops directly accounting for wettability and/or topographic patterns, with periodicity comparable to the drop size.34
Appendix The bond number at critical depinning can be interpolated using a multinomial expansion in x ¼ cos qa and y ¼ cos qr: Bo* ¼ A(x) + B(x)y + C(x)y2
(8)
where A(x), B(x) and C(x) are sixth degree polynomials
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8 < AðxÞ ¼ 0:01417 1:827x 1:352x2 0:4879x3 0:9299x4 1:786x5 1:360x6 BðxÞ ¼ 1:830 þ 1:788x þ 1:292x2 þ 1:305x3 þ 2:253x4 þ 3:190x5 þ 1:419x6 : CðxÞ ¼ 0:4262 0:6571x 0:5434x2 0:7523x3 1:452x4 1:236x5 0:2059x6
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Eqn (9) reproduces within an accuracy of 1% the simulation data in the range qa,r ˛ [20 , 160 ].
Acknowledgements We thank Prof. Jacco Snoeijer and Daniel Herde for stimulating and helpful discussions. Funding from DFG within the Grant no. HE 2016/14-2 Piko is gratefully acknowledged.
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