Dynamics of Contact Line Pinning and Depinning of Droplets

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Dynamics of Contact Line Pinning and Depinning of Droplets Evaporating on Micro-Ribs Ali Mazloomi Moqaddam, Dominique Derome, and Jan Carmeliet Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00409 • Publication Date (Web): 18 Apr 2018 Downloaded from http://pubs.acs.org on April 18, 2018

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Dynamics of Contact Line Pinning and Depinning of Droplets Evaporating on Micro-Ribs

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Ali Mazloomi Moqaddama,b, Dominique Deromeb and Jan Carmelieta,b

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a)

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b)

Chair of Building Physics, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zurich, Switzerland

Laboratory for Multiscale Studies in Building Physics Empa, Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

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Abstract

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The contact line dynamics of evaporating droplets deposited on a set of parallel micro-ribs is analyzed with the

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use of a recently developed entropic lattice Boltzmann model for two-phase flow. Upon deposition, part of the droplet penetrates into the space between ribs due to capillary action, while the remaining liquid of the droplet

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remains pinned on top of the micro-ribs. In a first stage, evaporation continues until the droplet undergoes a series of pinning-depinning events, showing alternatively constant contact radius and constant contact angle

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modes. While the droplet is pinned, evaporation results in a contact angle reduction while the contact radius

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remains constant. At a critical contact angle, the contact line depins and the contact radius reduces and the droplet rearranges to a larger apparent contact angle . This pinning-depinning behavior goes on until the liquid

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above the micro-ribs is evaporated. By computing the Gibbs free energy taking into account the interfacial energy, pressure terms as well as viscous dissipation due to drop internal flow, we found that the mechanism that

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causes the unpinning of the contact line results from an excess in Gibbs free energy. The spacing distance and the rib height play an important role in controlling the pinning-depinning cycling, the critical contact angle and

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the excess Gibbs free energy. However, we found that neither the critical contact angle nor the maximum excess

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Gibbs free energy depend on the rib width. We show that the different terms, i.e. pressure term, viscous dissipation and interfacial energy, contributing to the excess Gibbs free energy can be varied differently by

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varying different geometrical properties of the micro-ribs. It is demonstrated that, by varying the spacing distance between the ribs, the energy barrier is controlled by the interfacial energy while the contribution of the

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viscous dissipation is dominant if either rib height or width are changed. Main finding of this is study is that, for micro-rib patterned surfaces, the energy barrier required for the contact line to depin can be enlarged by

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increasing the spacing or the rib height, which can be important for practical applications.

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1. Introduction

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Wetting and evaporation of sessile droplets are phenomena ubiquitous in daily life and present in many

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engineering applications such as substrate patterning [1], inkjet printing [2], spray cooling [3], electronic chip fabrication [4]. These applications have sparked a renewed interest in understanding the underlying phenomena,

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but one less studied aspect is the role of the surface texture on which an evaporating droplet lies. Although an apparently simple process, droplet evaporation is accompanied by a series of interesting phenomena studied

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since long [5]. Due to non-uniform evaporation, a flow inside the drop is generated [6] [7] and this advective

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flow brings liquid to the triple line where evaporation is higher for droplets on a hydrophilic surface. In general, a drop deposited on a surface evaporates in of two distinct evaporation modes: constant contact angle (CCA)

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mode and constant contact radius (CCR) mode. During evaporation on textured surfaces or chemically heterogeneous surfaces, a combination of CCR and CCA modes namely “pinning-depinning” occurs. In the

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“pinning” mode, the contact line is pinned as evaporation evolves, retaining a constant contact radius while the apparent contact angle and drop height decrease, until reaching a critical contact angle, where the triple contact

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line slips or jumps to a new equilibrium condition. In the new position, the drop has a smaller contact radius and

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obtains a larger apparent contact angle and height. This cycle continues until full evaporation of the liquid drop on the surface. In general, the depinning mode is much faster than the pinning mode [6].

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The “pinning-depinning” mode can be considered as a transition between CCA and CCR. This transition has

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been extensively observed experimentally for non-flat surfaces with geometrical morphologies [8] [9] [10] [11] [12]. For example, using micro- and nano-structured non-patterned hydrophobic surfaces, the effect of surface

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structure on the evaporation characteristics of water droplets was investigated by Lee et al. [12]. Shin and his coworkers also experimentally examined the natural evaporation of a water droplet on submicron non-patterned

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and patterned post array hydrophobic surfaces [11]. However, most efforts have been mainly focused on experimental studies of droplet evaporating on patterned surfaces decorated by post arrays of few ten micrometer

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[6] [13] [14] [15]. For instance, Xu et. al [13] studied different evaporation modes by investigating the droplet

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profile during evaporation of droplets on micro-pillar-patterned superhydrophobic surfaces. The micro-pillar

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used in [13] was made to have the same diameter (5 𝜇𝑚) and height (25 𝜇𝑚) but varying spaces between pillars

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(5 − 50 𝜇𝑚).The dynamics of contact line depinning during droplet evaporation was also studied both theoretically and experimentally for hydrophilic and hydrophobics surfaces designed by micropillars of a few ten

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micrometer of height and diameter [16] [17] [6], and in [18] [19] showing that the wetting conditions close to the triple contact line impact the apparent contact angle. In [16], the co-occurrence of pinning and moving of a

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contact line around a micropillar is described. Different contact angles such as equilibrium, receding and

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advancing contact angles were studied in [17] and a model proposed that bridges the connection among these contact angles. Moreover, the evaporation process of a sessile drop of water on a regular pillar-like soft patterned

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substrate was experimentally investigated by Yu-Chen et. al. [14]. The pillar size used in [14] was in the order of

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10 𝜇𝑚 and it was shown that for the soft micropillar patterned substrate the receding contact angle is smaller and the evaporation rate is faster in contrast with the solid surfaces. Recently, the wetting and evaporation behavior

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of water droplets containing bacteria on substrates decorated by pillars of a few ten micrometer size was studied in [15]. It was shown that the pattern of bacterial distribution after drying is affected by the pillar height together

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with the concentration of suspended particle in the water droplet. Very recently, water evaporation in a wide

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range of droplet diameters and wall temperatures on both smooth and corrugated surfaces was studied experimentally by Misyura [20]. The structured surface employed in [20] was made as sine waves with an

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amplitude and wave length of around 20 and 80 𝜇𝑚, respectively. Using experimental observations, Misyura demonstrated that both static contact angle and evaporation rate are affected by droplet radius for the corrugated surfaces in comparison with the smooth ones.

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To note, pinning-depinning has also been quite well studied in the evaporation of colloidal droplets on flat surfaces, where a set of ring-shaped stains is formed due to contact line self-pinning induced by the micro-

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nanoparticles deposition [21] [22] [23] [24]. In all these studies, the dynamics of pinning-depinning is analyzed

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using one of two approaches. The first approach is based on computing the critical pinning force required to be overcome for contact line depinning (see [25] and references therein). In the second approach, the dynamics of

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the contact line depinning is analyzed using Gibbs free energy calculations to determine the energy barrier required to release the pinning contact line [6] [26] [27]. Thermodynamically, the Gibbs-free-energy-based

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scheme is more consistent than the pinning-force-based approach [17]. However, both approaches can be criticized, as limitations in experimental measurements might lead to a lack of some information which results

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from the internal flow of the evaporated droplet. Recently, the dynamics of contact line depinning during droplet

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evaporation is investigated in more detail both theoretically and experimentally based on Gibbs free energy calculations. A proposed theoretical model estimates the receding contact angle at which the transition from

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CCR to CCA occurs [13].

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However, to the best of our knowledge, all the studies based on Gibbs free energy are restricted to the interfacial

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Gibbs free energy term of the Gibbs free energy equation and do not take into account the remaining terms such as the pressure term and particularly the Gibbs free energy loss due to viscous dissipation induced by internal

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liquid flow. Nevertheless, an adequate numerical simulation method like entropic lattice Boltzmann can allow to


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accurately capture the pinning-depinning behavior and to compute the Gibbs free energy of system with all terms involved.

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In this study, using tri-dimensional numerical simulations, we aim to investigate the dynamics of the contact line

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depinning for an evaporating drop deposited on a surface decorated by a set of parallel micro-ribs. Micro-ribs are structures, which resemble the surface structure of a cut wood surface, where the sliced cell walls represent

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the ribs and the cellular voids the space in between the ribs. The impact of liquid drops on wood surfaces has been a topic of a few experimental studies in literature [28] [29] [30]. However to the best of our knowledge, in

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comparison to evaporation on micro-pillar arrays [6] [13,14] [15], there is no effort in literature to study liquid drop evaporation on rib structures in terms of both distance and size of the ribs as performed in this study.

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Further, the computation of the Gibbs free energy of an evaporating droplet includes all the terms involved in

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this paper, namely the interfacial energy, pressure term and viscous dissipation. The effects of the geometry of

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the surface, namely the rib height, rib width and spacing between the ribs on the critical contact angle and energy barriers is studied in details.

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Evaporation is considered here as a slow diffusive process where the temperature change due to evaporation is very small. Thus, we assume that the evaporation process is an isothermal process. As a result, the evaporation is

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modelled isothermally by imposing a constant vapor flux on the side and top boundaries of the computational

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domain. Although Marangoni flow may occur during evaporation due to a gradient in surface tension, which stems from a gradient in temperature along the liquid-gas interface, as the simulations are isothermal, no surface

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tension gradient will be observed in the simulation. We start the analysis once the droplet has penetrated into the ribs, since we focus mainly on the pinning-depinning process, and less on the initial wetting behavior.

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The remainder of this paper is structured as following: first, we briefly explain our numerical model and present the simulation setup. Next, the simulation results along with a detailed discussion on the dynamics of the contact

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line depinning during drop evaporation on micro-ribs are presented. Finally, we summarize the paper.

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2. Methodology

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A. Numerical method

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The entropic lattice Boltzmann for two-phase flow proposed by Mazloomi et. al [31] is used to simulate the

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drying process of a droplet deposited on a series of micro-ribs. The method is discussed in details in [32] and a summary of the method is given here. The entropic lattice Boltzmann equation for a system of liquid and vapor

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separated by an interface reads as: 𝑒𝑞 𝑒𝑞 𝑒𝑞 𝑓𝑖 (𝒙 + 𝒗𝑖 𝛿𝑡, 𝑡 + 𝛿𝑡) = 𝑓𝑖 (𝒙, 𝑡) + 𝛼𝛽[𝑓𝑖 (𝜌, 𝒖) − 𝑓𝑖 (𝒙, 𝑡)] + [𝑓𝑖 (𝜌, 𝒖 + 𝛿𝒖) − 𝑓𝑖 (𝜌, 𝒖)]

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(1)

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where 𝑓𝑖 (𝒙, 𝑡) are the discrete populations and 𝒗𝑖 (𝑖 = 1, … , 𝑁) denote the discrete velocities describing the underlying lattice structure. The 𝐷3𝑄27 lattice (𝑁 = 27) is used for our thri-dimensional simulations. The

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parameter  (0 < 𝛽 < 1) is fixed by the kinematic viscosity, 𝜈, through 𝜈 = 𝑐𝑠2 𝛿𝑡 [

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the lattice speed of sound; lattice units 𝛿𝑥 = 𝛿𝑡 = 1 are used. The equilibrium population 𝑓𝑖

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1 2𝛽

1

𝛿𝑥

2

√3𝛿𝑡

− ] where the 𝑐𝑠 = 𝑒𝑞

is

is the minimizer of

𝑓𝑖

the discrete entropy function 𝐻 = ∑𝑖 𝑓𝑖 ln( ), under the constraints of local mass and momentum conservations, 𝑊𝑖

𝑒𝑞

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{𝜌, 𝜌𝒖} = {1, 𝒗𝑖 }{𝑓𝑖 }, where 𝑊𝑖 are the lattice weights. The stabilizer parameter 𝛼 defines the maximal over-

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relaxation, which is computed at each time step for each computational node from the entropy estimate equation

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[33].

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In Eq. (1), the two-phase effects as a result of intermolecular forces, are present through the velocity increment

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𝛿𝒖 = 𝛿𝑡, with the force 𝑭 = 𝑭𝑓−𝑓 + 𝑭𝑓−𝑠 and 𝑭𝑓−𝑓 and 𝑭𝑓−𝑠 representing the fluid-fluid and fluid-solid

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interactions, respectively. Phase separation occurs by defining the fluid-fluid interaction as 𝑭𝑓−𝑓 = 𝛁 ∙

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(𝜌𝑐𝑠2 𝑰 − 𝑷) using the Korteweg’s stress 𝑷 as,

𝑭

𝜌


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𝜅 𝑷 = (𝑝 − 𝜅𝜌∇2 𝜌 − |∇𝜌|2 ) 𝑰 + 𝜅(∇𝜌) ⊗ (∇𝜌) 2

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(2)

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where 𝜅 is the coefficient that controls the surface tension, 𝑰 is the unit tensor and 𝑝 denotes the non-ideal equation of state [34]. The Carnahan-Starling (C-S) equation of state is used for this study [35]. The introduction of cohesive interaction through the velocity increment in Eq. 1, leads to surface tensional forces separating the

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liquid and vapor by an interface, which maintains the liquid and vapor in equilibrium state. The wettability

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condition is modeled by taking into account the fluid-solid interaction 𝑭𝑓−𝑠 as follows: 𝑭𝑓−𝑠 (𝒙, 𝑡) = 𝜅𝑤 𝜌(𝒙, 𝑡) ∑ 𝑤𝑖 𝑠(𝒙 + 𝒗𝑖 𝛿𝑡)𝒗𝑖 .

(3)

𝑖

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where the strength of the fluid-solid interaction is reflected by 𝜅𝑤 . Different contact angles are modeled by

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adjusting 𝜅𝑤 , which may range from negative to positive values corresponding to hydrophilic and hydrophobic

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surfaces, respectively. The indicator function 𝑠(𝒙 + 𝒗𝑖 𝛿𝑡)𝒗𝑖 in Eq. (3) is equal to one for solid nodes and zero

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otherwise and 𝑤𝑖 are the weight coefficients [32].

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B. Simulation setup and boundary conditions

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The simulation setup is initialized with a spherical droplet of radius 𝑅0 = 1.3 𝑚𝑚 (corresponding to 100 lattice nodes) deposited on top of a series of parallel ribs on a base, as schematically represented in Fig. 1. The liquid

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drop and its surrounding vapor are initialized with a zero velocity without imposed gravity. Since the droplet size is smaller than the capillary length for a water droplet, gravitational effects are negligible. The computational

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domain size, set in function of the size of the droplet, is 5𝑅0 × 5𝑅0 × 4𝑅0 for all simulations reported here, unless otherwise stated. The resulting grid is checked for sensitivity by varying the grid resolution. The ribs

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have a height ℎ varying from 65 to 650 𝜇𝑚, and a width 𝑏 in a range from 65 to 326 𝜇𝑚. Ribs are at a spacing 𝑤 ranging from 39 to 234 𝜇𝑚. In all simulations, the liquid to vapor density ratio is fixed to 𝜌𝑙 ⁄𝜌𝑣 ≅ 100

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(𝜌𝑙 /𝜌𝑐 = 3.05, 𝜌𝑣 /𝜌𝑐 = 0.0302 where 𝜌𝑐 is the critical density computed at the critical temperature, 𝑇𝑐 , and the critical pressure, 𝑝𝑐 , from the C-S equation of state used in this study [35]) at 𝑇⁄𝑇𝑐 = 0.6715 in accordance with the coexistence curve of the C-S equation of state [35]. Our previous studies showed that a density ratio of

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around 100 is sufficient to correctly capture the dynamics of two-phase problems under consideration [36] [32] [37] [38] [33]. Using the C-S equation of state at the aforementioned temperature gives an interfacial surface

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tension of 𝜎/𝑝𝑐 𝑅0 = 0.0315as a dimensionless number. The wetting contact angle is fixed to 67.6𝑜 using a fluid-solid interaction strength 𝜅𝑤 = 0.0035 in Eq. (3). The viscosity is fixed to 𝜈 = 1⁄6 in the lattice unit. The

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viscosity of the simulated liquid is one order of magnitude larger than water at the same temperature.

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Figure 1: Schematic representation of an evaporating drop of radius 𝑅0 deposited on arrays of microribs of width 𝑏, height ℎ and the spacing distance 𝑤. In all simulations the drop radius 𝑅0 is fixed to 1.3 𝑚𝑚 (corresponding to a 100 lattice nodes) while 𝑤, ℎ and 𝑏 are varied in a range between 39 to 234 𝜇𝑚, 65 to 650 𝜇𝑚 and 65 to 325 𝜇𝑚 respectively.

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As mentioned above, we assume that the drop evaporates diffusively without temperature changes and thus evaporation is modeled by imposing a constant vapor flux, more precisely a constant vapor phase velocity at the

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side and top outer boundaries. Imposing a constant outflow on the borders results in a reduction of the vapor

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pressure inside the domain. For the system to come back to its equilibrium, some of the high density liquid phase evaporates and brings the vapor density to its equilibrium value. It should be noticed that, since the temperature

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is constant and the interfacial surface tension controlling the liquid to vapor density ratio is related to the temperature, the density ratio between the liquid and vapor remains constant during the evaporation process. We

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do not impose a uniform evaporation flux at the liquid-vapor interface of the droplet, but allow that the drop evaporates diffusively by imposing a constant vapor flux on the top and side outer boundaries of the calculation

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domain, which are sufficiently far away from the droplet surface. The needed size of the computational domain

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to ensure a diffusive evaporation process was determined after performing simulations of different computational domain sizes.

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Numerically, the vapor flux at the boundaries is modeled by replacing all the populations on the boundary nodes with the equilibrium populations corresponding to a prescribed outflow velocity and vapor density. For the

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boundary nodes, the density of vapor is computed based on the known populations at the boundaries derived from mass and momentum conservation. For example, for the top boundary, the vapor density can be easily

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computed in respect to the known populations on the boundary nodes by solving the mass conservation equation

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𝜌𝑣 = ∑𝑖 𝑓𝑖 in conjunction with the moment conservation equation 𝜌𝑣 𝑢𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 = ∑𝑖 𝑓𝑖 𝒗𝑖 . We also apply a wall boundary condition as explained in detail in Ref. [32] at the bottom boundary and all wall surfaces.

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By prescribing a vapor pressure on the boundaries, which are far enough from the droplet, we can capture the

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slow diffusive evaporation process allowing non-isothermal effects to be neglected. The same approach was used in other works such as [39].

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C. Numerical validation

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The entropic lattice Boltzmann method for two phase flow used in this study has been previously validated and verified against a large number of experimental and theoretical predictions [36] [32] [37] [38] [33] [40].

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However, to show the validity of the method for the problem under consideration, numerical results are first

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compared to the 𝐷2 -law for the evaporation of a static droplet suspended in vapor. The classical 𝐷2 -law says that, during droplet evaporation, the surface area, represented by square of the droplet diameter, decreases


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linearly with the time [41]. For a droplet of the diameter D, the droplet mass and mass loss rate are 𝑚𝑑 =

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and 𝑚̇𝑑 =

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reads as:

𝜌𝑙 𝐷π 4

2

𝜌𝑙 π𝐷3 6

2

𝑑(𝐷 )/𝑑𝑡, respectively. By assuming 𝑚̇𝑑 /𝐷=constant, the 𝐷 -law in a dimensionless form 𝐷 ∗ 2 = 1 − Κ𝑡 ∗

(4) 2𝑡𝜎

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where the non-dimensional diameter 𝐷 ∗ and time 𝑡 ∗ are defined as 𝐷/𝐷0 and 𝑡 ∗ =

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denotes a constant value equal to Κ =

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rate 𝑚̇𝑑 is measured from the simulations by computing the slope of mass evolution versus time. In Fig. 2, the square of the droplet diameter as a function of the evaporation time is plotted for both simulation results and

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those obtained from Eq. (4). A good agreement between the numerical results and theoretical predictions is found.

2𝜇𝑙 𝑚̇𝑑

𝜋𝜎𝐷02 𝜌𝑙

𝜇𝐷0

, respectively and Κ

with the dynamic viscosity 𝜇𝑙 and the liquid density 𝜌𝑙 . The mass

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Figure 2: 𝑫𝟐 -law: comparison between numerical simulations (open squares) and theoretical prediction (solid line).

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3. Results and discussions

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A. Droplet behavior and volume variation on a flat and a slightly rough surface

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As a reference we first document and analyze the evaporation of a liquid droplet deposited on a flat surface and on a smooth surface with asperities. The contour lines of a liquid drop during evaporation on a flat surface is

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shown in Fig. 3a. During evaporation, the contact radius (gray line) smoothly decreases while the apparent contact angle (red line) remains constant confirming the CCA (constant contact angle) mode. Fig 3c displays the

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droplet profiles during evaporation on a slightly rough surface due to asperities of 2 × 5 (width×height) lattices at a 50 lattice spacing. The contact line is initially pinned at the first asperities, while the apparent contact angle and concurrently droplet height decreases leading to a CCR (constant contact radius) mode. Upon reaching a

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critical contact angle, the droplet depins and the drop slides over the surface reducing its wetting contact radius

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until it pins again at the next asperity. This pinning-depinning process is repeated until the droplet reaches the last pairs of asperities on the surface.


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Figure 3: a) Simulation results of an evaporating liquid drop on a flat surface where the apparent contact angle

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remains constant during evaporation, while the drop contact diameter decreases (CCA mode). b) Normalized droplet contact radius (left axis) as well as drop contact angle (right axis). c) Evolution of an evaporating droplet

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in CCR mode on a rough surface. d) Normalized droplet contact radius and apparent contact angle on a rough surface versus normalized time.

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B. Dynamic behavior of evaporating drops on micro-ribs I.

Volume variation

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We now consider an evaporating droplet deposited on a series of parallel micro-ribs. Three main phases can be

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observed in Fig. 4: capillary suction, pinning and depinning during droplet evaporation from Fig. 4a to 4c. Fig. 4d illustrates the second depinning event. Fig 4e gives the time evolution of the total volume (solid line), the

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volume of the droplet residing on top of the ribs (dash line) and the volume penetrated into the space between the

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ribs (dash-dot line) for two different spacing distances, 𝑤/𝑅0 .

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In the first phase, the space between the parallel ribs is invaded by the liquid droplet due to capillary suction (Fig. 4a). The capillary process results in a fast decrease of the droplet volume residing above the ribs (Fig. 4e;

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dash line). At the end of the capillary suction period, the part of the droplet residing on the ribs remains pinned at a rib edge (Fig 4b). It should be noted that, for the entire range of spacing distances studied, we found that the

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space between ribs is always invaded and wetted by the droplets, indicating that the pinning-depinning process during evaporation is studied for droplets in the Wenzel state. Thereafter, the total volume decreases linearly

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with time due to the constant evaporation flux imposed on the top and side boundaries (Fig. 4e). As the

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evaporation proceeds, the pinned contact angle and the droplet height decrease while the contact radius remains constant. The first depinning mode (Fig. 4c) occurs after reaching a critical contact angle followed by a second

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pinning of the droplet (Fig 4d). The depinning and pinning sequence is repeated until the above ribs volume is totally vanished and liquid remains only in between the ribs.


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Figure 4: Selected snapshots of an evaporating droplet depostited on a series of parallel micro-ribs: a) capillary suction phase, b) pinning mode, c) the first slip or depinning event, d) the second slip event. In (a), (b), (c) and (d), the top panel shows the density iso-surface in a three-dimensional view, the middle panel represents the density conture in the middle droplet cross-plane and in the bottom panel the velocity field inside the drop and around the triple contact line in the same plane is visualized. In (e) the total liquid volume (solid line), liquid volume of the droplet residing on the ribs (dashed line) and liquid volume penetrated into micro-ribs (dash-dot line) are plotted versus time for two different rib widths (in both cases the height and width of ribs are kept constant to ℎ/𝑅0 = 0.5 and 𝑏/R 0 = 0.05).

257 258

We also visualize the velocity vectors for the middle plane of the droplet with the velocity magnitude contour in

259 260

the background (third row in Fig. 4). The internal flow in the droplet takes place from the evaporating liquidvapor surface towards the ribs underneath the droplet (Fig 4 (a)). Initially there is a downward flow towards and

261

into the space between the ribs driven by capillary suction. At the triple contact line, in addition to the downward

262 263

flow in the ribs, liquid flows towards the triple line due to high evaporative flux at this interface. We recall that when the droplet contact angle is less than 90° the evaporative flux is the largest at the triple line and smallest at

264 265

the apex of the droplet [42]. As evaporation proceeds accompanied with pinning-depinning sequence, the vapor velocity distribution at the droplet surface becomes strongly non-uniform leading to a high evaporative flux near


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Langmuir

266 267

the triple contact line (Fig. 4(c) and (d)). The high evaporative flux is replenished by an internal liquid flow in the droplet towards the triple line. We observed that, after each depinning event, the liquid in the rib space flows

268

downwards while the vapor in the other ribs flows upward leading to a large circulation around the triple point

269 270

(Fig. 4(c) and (d)). We conclude that the singularity of the evaporative flow at the triple line of the droplet interface highly influences the depinning of the contact line, since it drives the internal liquid flow in the droplet

271 272

and in the inter-rib spaces. We remark that the internal flow in the droplet will also play an important role in the dissipating energy due to viscosity and thus should be taken into account in the Gibbs free energy calculations,

273

which will be explained in section III.

274

II.

Contact angle and droplet contact radius

275

As shown in Fig.4, for an evaporating drop deposited on micro-ribs, the drop has an elliptic shape dictated by the

276 277

surface structure. The contact line profile of the drop placed on top of the ribs for different evaporating times is shown in Fig. 5a. It is evident from this Fig. 5a that the liquid drop keeps its elliptic shape as evaporation

278 279

proceeds. The characteristic time evolution of the minor (𝑟) and major (𝑅) semi-axis of the elliptical droplet during evaporation is plotted in Fig. 5b. As expected, both minor and major semi-axis reduces gradually as the

280 281

evaporation time elapses. Since the drop pins and depins along the minor semi-axis, a small fluctuation in the minor semi-axis magnitude can be observed from Fig. 5b (gray line) which corresponds to each pinning-

282 283 284

depinning event. It is interesting that the aspect ratio (𝑟/𝑅) between the minor and major semi-axis remains almost constant during evaporation as shown in Fig. 5c. This means that one can use one of this two as the characteristic length to calculate the excess Gibss free energy per unit length (see further). Here in this study we

285 286 287

will use the minor semi-axis, 𝑟, for computing the excess Gibbs free energy per unit length, which will be presented in the next section. From now, we will use “contact radius” instead of “minor semi-axis” in agreement with other studies in literature. We also remark that the contact angle during evaporation will measured in a

288

plane normal to the blades (along the minor semi-axis). This is the plane where contact line pinning and

289 290

depinning occurs. The contact angle is then the angle between the contact line and a horizontal line at the top of the ribs.

291 292

As shown on Fig. 5d, the critical contact angle over all pinning-depinning events reduces as evaporation proceeds. A similar behavior was observed by experiment for an evaporating droplet containing nanoparticles on

293

a silicon wafer surface [43]. In our case, we observed that as evaporation proceeds, less liquid remains on top of

294 295

the ribs generating nearly an elliptical cap which becomes more and more flat resulting in smaller pinned contact angles with time (see Fig. 4: top panel). The critical contact angle over all pinning-depinning events is averaged

296

and named, (𝜃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 )𝑎𝑣𝑔 . The first pinning event lasts for a longer time compared to the consecutive events

297

since when the drop is deposited and relaxed, the apparent contact angle at first pinning is around 78.5𝑜 and thus

298 299

a substantial evaporation time is required to reach the critical contact angle of 36.5𝑜 at which the first depinning occurs. The long first pinning period results in a substantial volume loss inducing a large reduction of the drop

300

height which results the apparent contact angle 𝜃0,1 at pinning right after the first slipping becomes much lower

301 302 303

than the Young equilibrium contact angle on the flat solid (67.6𝑜 ). As the evaporation evolves, the drop height constantly reduces leading to a decrease with time in the initial contact angle of the following pinning-depinning cycles as shown in Fig. 5d. Another approach is based on the maximum pinning force, where the detachment of

304 305

the triple contact line occurs at the critical contact angle, when the depinning force exceeds the maximum pinning force provided by the micro-ribs [44]. A more complete approach is however an energy approach

306 307

considering all terms in Gibbs free energy equation including pressure, viscous dissipation and interfacial energy terms which has been followed in this study.

308


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309 310 311 312 313 314 315 316 317 318

Figure 5: (a) Top view of the contact line profile of the liquid drop pinned on top of the ribs for diffferent times. (b) Charactristic time evolution of the minor (left axis) and major semi-axis (right axis) of the ellips contact area. (c) The aspect ratio between the minor and the major semi-axis of ellips contact area during evaporation. (d) Variation of the apparent contact angle of the pinned contact line during evaporation process of the liquid drop. The micro-ribs have a spacing distance of 𝑤/𝑅0 =0.06, height of ℎ/𝑅0 = 0.5 and 𝑏/R 0 = 0.05. 𝜃0,𝑖 in (d) shows the apparent contact angle at the beginning of each pinning event 𝑖. After pinning the 𝜃0,𝑖 is decreased to a critical value at which the drop decomes depinned. The critical contact angle is averaged over all slipping modes and it is shown by (𝜃𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 )𝑎𝑣𝑔 .

319 320

The results are further compared to the pinning time model for an evaporating droplet proposed by Shanahan as follows [45] [46]: 𝑡𝑝𝑖𝑛,𝑖 = 𝑡0,𝑖 −

𝜋𝑟 3 (𝜃0,𝑖 − 𝜃𝑐𝑟𝑖𝑡,𝑖 ) . 2 (1 + 𝑐𝑜𝑠𝜃0,𝑖 ) 𝑑𝑉⁄𝑑𝑡

(5)

321

where 𝑟 is the contact radius and 𝑑𝑉 ⁄𝑑𝑡 represents the droplet volume change rate due to evaporation.

322

For each pining-depinning event (i), we measure the initial contact angle (𝜃0,𝑖 ) and critical contact angle 𝜃𝑐𝑟𝑖𝑡,𝑖

323

from the simulations and then the pinning time computed from Eq. (5) is compared with that obtained directly

324

from the simulations.

325

Table 1 presents the pinning times for both numerical simulations and those obtained by Eq. (5) together with the

326 327

initial contact angle and the critical contact angle for all pinning-depinning events. The pinning times computed from simulations compare very well to those from Eq. (5) as listed in Table 1.

328 329


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330 331 332

Langmuir

Table 1: Pinning time, initial and critical contact angles for all pinning and depinning events during evaporation of a liquid drop deposited on micro-ribs of spacing distance 𝑤/𝑅0 =0.06, height of ℎ/𝑅0 = 0.5 and rib width of 𝑏/𝑅0 = 0.05. (Fig. 5b) 𝑖

𝑡 ∗ 𝑝𝑖𝑛,𝑖 𝑡ℎ𝑒𝑜𝑟𝑦

𝑡 ∗ 𝑝𝑖𝑛,𝑖 𝑠𝑖𝑚𝑢𝑙𝑎𝑡𝑖𝑜𝑛

𝜃0,𝑖 (0 )

𝜃𝑐𝑟𝑖𝑡,𝑖 (0 )

0

70.7

72.4

78.5

36.5

1

75.1

78.0

39.6

34.8

2

81.3

83.0

37.8

32.8

3

86.3

87.5

36.1

29.2

4

90.8

90.9

34.0

25.1

5

94.8

92.6

28.5

17.3

333 334 335

III.

Excess Gibbs free energy and average critical contact angle variation

336

The Gibbs free energy for our system i.e. an evaporating liquid droplet deposited on a set of parallel micro-ribs is

337 338

composed of the Gibbs free energy of the phases as well as the interfacial Gibbs free energy induced by the surface tension and the fluid-solid interactions.. Here, for a diffusive evaporation process, the Gibbs free energy

339 340

changes of the vapor and solid phases are assumed negligible compared to those of the liquid phase. We also remind that the LBM used here in this study does not consider thermal effects and isothermal conditions are

341

assumed. Therefore, the Gibbs free energy of the evaporating droplet reads as: 𝐺𝑠𝑦𝑠𝑡𝑒𝑚 = (𝑈 + 𝑃𝑉 − 𝑇𝑆)𝐿 + 𝐴𝑑𝑟𝑜𝑝 𝜎𝐿𝑉 + 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 𝜎𝑆𝐿 − 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 𝜎𝑆𝑉 ,

(6)

342

where G, U, P, V, T, S are the Gibbs free energy, internal energy, pressure , volume, temperature and the entropy

343

of the liquid drop, respectively, as the subscript L stands for the liquid phase. 𝐴𝑑𝑟𝑜𝑝 is the surface area of the

344

drop residing above the ribs and 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 is the contact area between the droplet and the substrate. 𝜎𝐿𝑉 , 𝜎𝑆𝐿 and

345

𝜎𝑆𝑉 are the interfacial tensions between liquid/vapor, solid/liquid and solid/vapor respectively.

346

Since the pressures inside the drop and in the liquid between the ribs are different, the 𝑃𝑉 term is divided into

347

two separate terms (𝑃𝑉𝑑𝑟𝑜𝑝 , 𝑃𝑉𝑟𝑖𝑏 ). We also express the temperature and the entropy term in Eq. (6) in terms of

348

the Gibbs free energy loss due to viscous dissipation, Φ [36]. Φ=

349 350

𝜇𝑙 𝑡 ∫ ∫ (∇𝑢 + ∇𝑢 𝑇 )2 𝑑𝑉𝑑𝑡. 2 0 𝑉

where 𝜇𝑙 is the dynamic viscosity of the liquid phase and 𝑉 denotes the liquid volume. Assuming Young’s equation valid: 𝜎𝑆𝑉 − 𝜎𝑆𝐿 = 𝜎𝐿𝑉 𝑐𝑜𝑠𝜃𝑌

351

(8)

where the 𝜃𝑌 is the Young’s equilibrium contact angle, and combining with Eq. (6) , one obtains: 𝐺𝑠𝑦𝑠𝑡𝑒𝑚 = 𝑈 + 𝑃𝑉𝑑𝑟𝑜𝑝 + 𝑃𝑉𝑟𝑖𝑏 + Φ + 𝐴𝑑𝑟𝑜𝑝 𝜎𝐿𝑉 −𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 𝜎𝐿𝑉 𝑐𝑜𝑠𝜃𝑌 ,

352

(7)

(9)

The excess Gibbs free energy is then defined as: 𝛿𝐺 = 𝐺𝑠𝑦𝑠𝑡𝑒𝑚 (𝑡) − 𝐺𝑠𝑦𝑠𝑡𝑒𝑚 (𝑡 = 𝑡0,𝑖 )

(10)

353

where 𝐺𝑠𝑦𝑠𝑡𝑒𝑚 (𝑡) is the total Gibbs free energy at the instant t and 𝐺𝑠𝑦𝑠𝑡𝑒𝑚 (𝑡 = 𝑡0,𝑖 ) is the total Gibbs free

354

energy at the moment when a new stick phase begins. At 𝑡 = 𝑡0,𝑖 the initial contact angle 𝜃 is 𝜃 = 𝜃0,𝑖 where 𝑖 is

355

the order of the pinning cycle (see Fig. 5b). In the calculation of the excess Gibbs free energy, the change of the

356

internal energy (𝑈) can be assumed to be zero as the drop evaporation is modeled as an isothermal process.


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357 358 359 360 361 362 363

Figure 6: Evolution of excess Gibbs free energy per unit length of triple line (contact radius 𝒓) normalized by the initial Gibbs free enrgy during evaporation proccess of a droplet deposited on a micro-ribs with 𝑤/𝑅0 = 0.06, ℎ/𝑅0 = 0.5 and 𝑏/R 0 = 0.05, taking into account all terms (black) or only the interfacial terms (blue). The maximum excess Gibbs free energy or the energy barrier required for the drop to be depinned is averaged over all depinning events except the first, depinning denoted by (𝛿𝐺𝑚𝑎𝑥 /𝐺0 )𝑎𝑣𝑔 ∗ (𝑅0 /𝑟).

364 365

Numerical simulations enable us to compute all the terms given in Eq. (9). Therefore, unlike other studies, the excess Gibbs free energy during drop evaporation is computed by taking into account not only the interfacial

366

energies but also the pressure terms as well as the viscous dissipation. In Fig. 6, the excess Gibbs free energy per

367

unit length of the triple line 𝑟 (defined as the minor semi-axis of the elliptical droplet generated on top of the

368

ribs) is plotted as a function of time for ribs with w/R 0 = 0.06, h/R 0 = 0.5 and b/R 0 = 0.05. The excess Gibbs

369

free energy per unit length, (𝛿𝐺/𝑟), is normalized by 𝐺0 /𝑅0 where 𝐺0 represents the total Gibbs free energy at

370 371

𝑡 = 0 and 𝑅0 stands for the initial drop radius. When the excess Gibbs free energy attains its maximum value, there is sufficient energy available to overcome the energy barrier and the triple line jumps to the next position or

372 373

stick phase where 𝛿𝐺 becomes zero again and the process is repeated.. As seen from Fig. 6, after the first depinning event, the maximum excess Gibbs free energy or the energy barrier per unit length of triple line is

374 375

almost constant for the other pinning-depinning motions. Here, we make an average over all these depinning events leading to the average maximum excess Gibbs free energy per unit length of triple line (𝛿𝐺𝑚𝑎𝑥 ⁄𝐺0 )𝑎𝑣𝑔 ∗

376

(𝑅0 ⁄𝑟), called for short the maximum excess Gibbs free energy.

377

The Gibbs free energy equation for droplets is commonly evaluated considering only interfacial energy terms,

378 379

i.e. the last two terms in Eq. (9). To analyze the influence of the pressure and viscous dissipation terms, the excess Gibbs free energy is calculated including all terms and also taking into account only the interfacial terms.

380

Fig. 6 compares the excess Gibbs free energies for these two cases (case all terms: black symbols, case only

381 382

interfacial terms: blue symbols). Note that the initial Gibbs free energy, 𝐺0 , is different for the two cases. Indeed, in the case with all terms the pressure term is included while in the other case (only interfacial term) is not

383

presented. In the first pinning event, the two cases differ a lot, due to the important influence of pressure and

384 385

viscous dissipation terms attributed to important internal flow. It is however interesting to note that after the first depinning event, the excess Gibbs free energy magnitude for both cases (all versus only interfacial terms) are

386 387

almost the same meaning that the pressure and viscous dissipation terms do not play a significant role during pinning-depinning cycles as evaporation proceeds. This can be attributed to the fact that the pressure and viscous

388 389

dissipation terms, which can be mainly attributed to the internal flow, remain quite constant during the different pinning-depinning events (with exception to the first depinning event). As a result the change in pressure and

390

viscous dissipation, as calculated in the excess Gibbs free energy, are very small compared to the change in the

391

interfacial terms.

The results shown in Fig. 6 confirm the validity, as done in previous studies, to only


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Langmuir

392 393

considering the interfacial terms for the calculation of energy barrier required for depinning contact line during pining-depinning motions.

394

To further investigate the effect of the geometrical properties of the ribs such as spacing distance 𝑤, rib height ℎ

395

and rib width 𝑏 on the dynamics of the contact line depinning, we performed further simulations by varying 𝑤, ℎ

396 397

and 𝑏. For each case, one geometrical characteristic was altered, while the other two remain constant.

398 399 400 401 402

Figure 7: Variation of (a) the average critical contact angle at droplet depinning as well as (b) the average maximum excess Gibbs free energy as a function of the spacing distance 𝑤. For all simulations here, the rib height and width are kept equal to ℎ/𝑅0 = 0.5 and 𝑏/R 0 = 0.05, respectively. The dashed line is fitted to the simualtion results (symbols) as a guide to the eye.

403

Figs. 7, 9 and 11 show the simulation results for the average critical contact angle and the maximum excess

404

Gibbs free energy versus the dimensionless spacing distance w, rib height h and rib width b, respectively. In Fig.

405

7, the dimensionless spacing distance 𝑤/𝑅0 is varied in a range between 0.03 to 0.18, while keeping the rib

406

height and width constant (ℎ/𝑅0 = 0.5, 𝑏/𝑅0 = 0.05). Fig. 7a shows that the critical contact angle decreases

407

with increasing spacing distance, while as seen from Fig.7b the maximum excess Gibbs free energy increases

408 409

with spacing distance. This means that the energy barrier for a droplet to depin and move over a capillary filled by liquid is larger for wider than for a smaller spacing distance. The dashed line is added to guide the eye. Bear

410 411

in mind that for all simulations shown in Fig. 7, the droplet size was fixed to 𝑅0 = 1.3 𝑚𝑚 (or 100 lattice nodes) and thus by increasing the spacing distance the droplet will span over less ribs resulting in less pinning-depinning

412

events. However, for the range of 𝑤/𝑅0 values studied here, two pinning-depinning events occur at least.

413

To analyze the effect of the different terms involved in the calculation of the excess Gibbs free energy, these

414

terms i.e. the change in pressure term (PV), viscous dissipation and interfacial energy, were determined.

415

Rewriting Eq. (10) we get: 𝛿𝐺 = 𝛿𝑃𝑉 + 𝛿Φ + 𝛿𝐴𝜎

(11)

416

where 𝛿𝑃𝑉 = 𝑃𝑉(𝑡) − 𝑃𝑉(𝑡 = 𝑡0,𝑖 ), 𝛿Φ = Φ(𝑡) − Φ(𝑡 = 𝑡0,𝑖 ), 𝛿𝐴𝜎 = 𝐴𝜎(𝑡) − 𝐴𝜎(𝑡 = 𝑡0,𝑖 ) and 𝐴𝜎 is the

417

interfacial energy 𝐴𝜎 = 𝐴𝑑𝑟𝑜𝑝 𝜎𝐿𝑉 −𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 𝜎𝐿𝑉 𝑐𝑜𝑠𝜃𝑌 . Equation 11 enables us to analyze the impact of a change

418

of a geometrical property of the ribs on the three different energy terms, or how by changing the geometry we

419

can for example increase/decrease the energy barrier for depinning.

420

The contribution of the pressure (PV) term (diamond symbols), the viscous dissipation (square symbols) and the

421

interfacial energy (triangular symbols) to the maximum excess Gibbs free energy, or energy barrier as function

422 423

of the spacing distance is shown in Fig. 7b. As seen from Fig. 7b, the increase in energy barrier by increasing the spacing distance is mainly caused due to an increase of the interfacial energy. When increasing the spacing

424

distance while keeping the initial droplet volume fixed, the liquid droplet during spreading will wet a smaller

425

number of ribs resulting in a decrease in the solid-fluid contact surface area 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 , while the fluid-fluid


Langmuir 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

426

surface area 𝐴𝑑𝑟𝑜𝑝 roughly remains constant. This can be clearly seen from Fig. 8 representing the wetting

427

contact area 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 and the drop surface 𝐴𝑑𝑟𝑜𝑝 versus time for different values of spacing distance 𝑤/𝑅0 . The

428

wetting contact area 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 decreases with spacing distance, while the drop surface 𝐴𝑑𝑟𝑜𝑝 is less dependent on

429

the spacing distance. As a result, the term 𝐴𝜎 = 𝐴𝑑𝑟𝑜𝑝 𝜎𝐿𝑉 −𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 𝜎𝐿𝑉 𝑐𝑜𝑠𝜃𝑌 increases with spacing distance,

430

resulting in an increase of the energy barrier or the maximum excess Gibbs free energy. Our simulations further

431 432

demonstrate that the PV and viscous dissipation terms have less effect on the change of the energy barrier as the spacing distance varies.

433 434 435 436 437

Figure 8: Evolution of (a) the solid-fluid contact area, 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 , together with (b) the fluid-fluid interfacial area, 𝐴𝑑𝑟𝑜𝑝 , during evaporation of the liquid droplet deposited on a rib-structured surface for different spacing distance 𝑤. In both (a) and (b) the surface area has been normalized by 𝐴0 i.e. the initial spherical drop surface area.

438 439 440

This means that the internal flow leading to viscous dissipation and the pressure (PV) terms has a smaller contribution to the energy barrier. It should be noted that each symbol shown in Fig. 7b is an average of the

441

maximum change of a corresponding term over all pinning-depinning events for an individual spacing distance.

442 443 444 445 446

Figure 9: Variation of (a) the average critical contact angle and (b) the average maximum excess Gibbs free energy as a function of the height of microribs . In all cases in (a) and (b) the spacing distance and rib width are kept constant to 𝑤/𝑅0 = 0.06 and 𝑏/R 0 = 0.05, respectively. The dashed line is fitted to the simualtion results (symbols) as a guide to the eye.

447

In Fig. 9, the results for the average critical contact angle and the maximum excess Gibbs free energy are

448

represented for different heights of the micro-ribs varying in a range from ℎ/𝑅0 = 0.05 to h/R 0 = 0.5 . The drop

449

size was fixed to 𝑅0 = 1.3 𝑚𝑚 and the spacing distance and rib width remained constant to w/R 0 = 0.06 and


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Langmuir

450 451 452

b/R 0 = 0.05, respectively. Fig. 9a shows that the critical contact angle decreases with increasing height of the ribs while as seen from Fig. 9b, the required energy to allow the drop to depin becomes larger by enlarging the height of the micro-ribs. This can be further analyzed by studying the contribution of different terms to the

453

maximum excess Gibbs free energy as plotted in Fig. 9b. The increase of the energy barrier by enlarging the rib

454 455

height can be attributed to an increase of viscous dissipation and surface energy and to a less extent to the pressure (PV) term. We further determine the solid-fluid contract area and the surface area of the deposited drop

456 457

on the ribs during evaporation for different rib heights (Fig. 10). As it is seen from Fig. 10, by increasing the rib height the wetting contact area increases while the drop surface area only slightly changes. The increase in

458

wetting contact area results in first increase of viscous dissipation due to increase of boundary layer a location

459

where the most viscous dissipation occurs and additionally increase of the interfacial energy (𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 increases

460

while the 𝐴𝑑𝑟𝑜𝑝 remains nearly constant by increasing rib height, Fig. 10) as shown in Fig. 9b. However, from

461

simulation results shown in Fig. 9b, we observe that the PV term slightly changes by varying the rib height and

462

thus has less contribution to the value of energy barrier, which is similar to the pervious case shown in Fig. 7b.

463

464 465 466 467 468

Figure 10: Evolution of (a) the solid-fluid contact area, 𝐴𝑤𝑒𝑡𝑡𝑖𝑛𝑔 , together with (b) the fluid-fluid interfacial area, 𝐴𝑑𝑟𝑜𝑝 , during evaporation of the liquid droplet deposited on the rib-structured surface for different rib heights ℎ. In both (a) and (b) the surface area has been normalized by 𝐴0 i.e. the initial spherical drop surface area.

469

470 471 472 473

Figure 11: Variation of (a) the average critical contact angle and (b) the average maximum excess Gibbs free energy as a function of the rib width. The spacing distance and rib height are fixed to 𝑤/𝑅0 = 0.06 and ℎ/𝑅0 = 0.5.


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474

To complete our geometrical study of the rib patterned surfaces on the pinning-depinning mode, the critical

475

contact angle and the maximum excess Gibbs free energy were measured by varying the rib width, 𝑏, in a range from b⁄R 0 = 0.05 to 0.25 while the distance and the rib height are kept constant to w/R 0 = 0.06 and h/R 0 =

476 477 478 479 480

0.5, respectively. The results are shown in Fig. 11. For all simulations reported in Fig. 11, the drop size was fixed to 1.3 𝑚𝑚 (100 lattice nodes). We chose a range for variation of the rib width (b⁄R 0 = 0.05 to 0.25 ) such that two pinning-depinning modes are recorded at least. As evident from Fig. 11, it was found that neither the average critical contact angle nor the maximum excess Gibbs free energy depend on the rib width. However, for nano-

481 482

droplets deposited on nano-textured surfaces the variation of the width of the geometrical properties may significantly affect the contact line dynamics and consequently the energy barrier. It was shown in [47] [48]

483 484

using molecular dynamics simulations that for nano-droplets on chemically heterogeneous solid substrates the characteristic width of the inhomogeneity can have a significance impact on the behavior of the contact line

485

motion during evaporation. The constant fitted dashed lines in Fig. 11a and 11.b, were measured by averaging

486 487

over all simulation results (symbols). We again measured the contribution of the different terms in the excess Gibbs free energy as plotted in Fig. 11b. We observe that the viscous dissipation predominantly contributes to

488 489

the maximum excess Gibbs free energy calculations, while the contribution of the pressure (PV) and interfacial energy terms is lower. It is also seen from Fig.11b. that the variation of the viscous dissipation is almost constant

490 491

as the size of the rib width changes resulting in the total energy barrier being almost constant. From these results, we conclude that the internal flow in the droplet is not significantly affected during evaporation by changing the

492

width of the ribs in the range under consideration. As a result the viscous dissipation as the main contributor of

493

the maximum excess Gibbs free energy remains constant.

494

From the simulation results shown in Fig. 7, 9 and 11, one can conclude that for an evaporating droplet pinned

495 496

on a series of parallel ribs, the energy barrier required for depinning the contact line is mainly dominated by either interfacial energy or the viscous dissipation and the contribution of the pressurebterm in the maximum

497 498

excess Gibbs free energy calculations is almost negligible. Moreover, by varying either spacing distance, rib height or rib width the changes in viscous dissipation are rather small.

499 500

4. Conclusions

501 502

In this study, using numerical simulations based on the entropic lattice Boltzmann method for two-phase flows, we studied the dynamics of the contact line pinning-depinning during evaporating drops on a set of parallel

503 504

micro-ribs. This type of surfaces, with ribs of a few tens of micrometer are different to common textured surfaces patterned by pillars. The dynamics of the contact line depinning was further investigated by varying the spacing

505 506

distance, the surface height and the rib width in a wide range from few ten to hundreds of micrometer. It was observed that upon deposition of the drop, first the space between ribs is invaded by the liquid drop due to the

507

effect of capillary suction. This process is followed by a pinning-depinning behavior, showing alternatively CCR

508 509

(constant contact radius) and CCA (constant contact angle) modes. The droplet depins at a critical contact angle, which is controlled by the maximum excess Gibbs free energy per unit length, when there is sufficient energy to

510 511

overcome the energy barrier resulting in a move of the triple contact line to a new pinning condition. Numerical simulations enabled us to correctly calculate the excess Gibbs free energy by taking into account not only the

512 513

interfacial energy terms studied previously in literature but also the pressure term as well as the viscous dissipation induced by the internal flow in the droplet. Unlike the rib width, both the spacing distance between

514

the ribs and the height of the ribs play a significant role in the control of the pinning/depinning cycles, the critical

515 516

contact angle and the maximum excess Gibbs free energy. Evaluating the contribution of the different terms to the energy barrier, i.e. the pressure, viscous dissipation as well as the interfacial energy terms, we surprisingly

517 518

found that the contribution of different terms can be different for different geometrical properties. By varying the spacing distance between the ribs the energy barrier is predominantly controlled by the interfacial energy while

519

the viscous dissipation is dominant in the case the rib height and width are altered.


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520 521

However to get further insight into the dynamics of the contact line depinning for drops evaporating on ribs, further work is still needed to investigate the effect of rib shape and also to explore the behavior of contact line

522

depinning on porous architectures without base plate which will open a new window for future research.

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5. Acknowledgment

525

This work was supported by the Swiss National Science Foundation (project No. 200021_175793). Prof. Iliya V.

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Karlin and Dr. Shyam S. Chikatamarla acknowledge the support from the computational resources at the Swiss National Super Computing Center CSCS. The important contribution of the PhD work of Soyoun Son is also

528

acknowledged.

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11 Shin, Dong Hwan and Lee, Seong Hyuk and Choi, Chang Kyoung and Retterer, Scott. The evaporation and wetting dynamics of sessile water droplets on submicron-scale patterned silicon hydrophobic surfaces. Journal of Micromechanics and Microengineering, 20 (2010), 055021. 12 Lee, Chi Young and Zhang, Bong June and Park, Jiyeon and Kim, Kwang J. Water droplet evaporation on Cu-based hydrophobic surfaces with nano-and micro-structures. International Journal of Heat and Mass Transfer, 2012 (2012), 2151--2159. 13 Xu, Wei and Leeladhar, Rajesh and Kang, Yong Tae and Choi, Chang-Hwan. Evaporation kinetics of sessile water droplets on micropillared superhydrophobic surfaces. Langmuir, 29 (2013), 6032-6041. 14 Chuang, Yu-Chen and Chu, Che-Kang and Lin, Shih-Yao and Chen, Li-Jen. Evaporation of water droplets on soft patterned surfaces. Soft matter, 10 (2014), 3394--3403. 15 Susarrey-Arce, A and Marin, A and Massey, A and Oknianska, A and D. Pattern formation by staphylococcus epidermidis via droplet evaporation on micropillars arrays at a surface. Langmuir, 32 (2016), 7159--7169. 16 Lu, Zhengmao and Preston, Daniel J and Antao, Dion S and Zhu, Yangying and Wang, Evelyn N. Coexistence of pinning and moving on a contact line. Langmuir, 33 (2017), 8970--8975. 17 Wu, Shuai and Ma, Ming. A contact angle hysteresis model based on the fractal structure of contact line. Journal of colloid and interface science, 505 (2017), 995--1000. 18 Bormashenko, E and Musin, A and Whyman, G and Zinigrad, M. Wetting transitions and depinning of the triple line. Langmuir, 28 (2012), 3460--3464. 19 Bahadur, Vaibhav and Garimella, Suresh V. Preventing the Cassie- Wenzel transition using surfaces with noncommunicating roughness elements. Langmuir, 25 (4815--4820 2009). 20 Misyura, SY. Contact angle and droplet heat transfer during evaporation on structured and smooth surfaces of heated wall. Applied Surface Science, 414 (2017), 188--196. 21 Deegan, Robert D and Bakajin, Olgica and Dupont, Todd F and Huber, Greg Huber. Capillary flow as the cause of ring stains from dried liquid drops. Nature, 389 (1997), 827. 22 Deegan, Robert D. Pattern formation in drying drops. Physical review E, 61 (2000), 475. 23 Shao, Fen Fen and Neild, Adrian and Alan, Tuncay. Controlled particle self-assembly in an evaporating droplet. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 398 (2012), 64--68. 24 Lin, Shih-Yao and Yang, Kai-Chieh and Chen, Li-Jen. Effect of Surface Hydrophobicity on Critical Pinning Concentration of Nanoparticles To Trigger the Coffee Ring Formation during the Evaporation Process of Sessile Drops of Nanofluids. The Journal of Physical Chemistry C, 119 (2015), 3050--3059.


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25 Orejon, Daniel and Sefiane, Khellil and Shanahan, Martin ER. Stick--slip of evaporating droplets: substrate hydrophobicity and nanoparticle concentration. Langmuir, 27 (2011), 12834--12843. 26 Shanahan, Martin ER. Simple theory of" stick-slip" wetting hysteresis. Langmuir, 11 (1995), 1041-1043. 27 Bormashenko, Edward and Musin, Albina and Zinigrad, Michael. Evaporation of droplets on strongly and weakly pinning surfaces and dynamics of the triple line. Colloids and Surfaces A: Physicochemical and Engineering Aspects, 385 (2011), 235--240. 28 Chen, Ping-ping and Wang, Xi-shi. Experimental study of water drop impact on wood surfaces. International Journal of Heat and Mass Transfer, 54 (2011), 4143--4147. 29 Lan, Meijuan and Wang, Xishi and Chen, Pingping and Zhao, Xiangdi. Effects of surface tension and wood surface roughness on impact splash of a pure and multi-component water drop. Case Studies in Thermal Engineering, 8 (2016), 218--225. 30 Huang, Xianjia and Chen, Pingping and Lan, Meijuan and Wang, Xishi and Liao, Guangxuan. Experimental study of water drops with additive impact on wood surfaces. Procedia Engineering, 62 (2013), 852--858. 31 Ali, Mazloomi Moqaddam and Chikatamarla, Shyam S and Karlin, Iliya V. Entropic lattice Boltzmann method for multiphase flows. Physical review letters, 114 (2015), 174502. 32 Ali, Mazloomi Moqaddam and Chikatamarla, Shyam S and Karlin, Iliya V. Entropic lattice Boltzmann method for multiphase flows: Fluid-solid interfaces. Physical Review E, 92 (2015), 023308. 33 Ali, Mazloomi Moqaddam. Entropic Lattice Boltzmann Method for Two-Phase Flows. 2016. 34 Slemrod, Marshall. Dynamic phase transitions in a van der Waals fluid. Journal of differential equations, 52 (1984), 1--23. 35 Yuan, Peng and Schaefer, Laura. Equations of state in a lattice Boltzmann model. Physics of Fluids, 18 (2006), 042101. 36 Ali, Mazloomi Moqaddam and Chikatamarla, Shyam S and Karlin, Iliya V. Drops bouncing off macrotextured superhydrophobic surfaces. Journal of Fluid Mechanics, 824 (2017), 866--885. 37 Ali, Mazloomi Moqaddam and Chikatamarla, Shyam S and Karlin, Iliya V. Simulation of binary droplet collisions with the entropic lattice Boltzmann method. Physics of Fluids, 28 (2016), 022106. 38 Ali, Mazloomi Moqaddam and Chikatamarla, Shyam S and Karlin, Iliya V. Simulation of droplets collisions using two-phase entropic lattice Boltzmann method. Journal of Statistical Physics, 161 (2015), 1420--1433. 39 Joshi, Abhijit S and Sun, Ying. Wetting dynamics and particle deposition for an evaporating colloidal drop: A lattice Boltzmann study. Physical Review E, 82 (2010), 041401.


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40 Chantelot, Pierre and Moqaddam, Ali Mazloomi and Gauthier, Ana. Water ring-bouncing on repellent singularities. Soft Matter (2018). 41 Godsave, GAE. Studies of the combustion of drops in a fuel spray—the burning of single drops of fuel. Symposium (International) on Combustion (1953), 818--830. 42 Stauber, Jutta M and Wilson, Stephen K and Duffy, Brian R and Sefiane, Khellil. On the lifetimes of evaporating droplets with related initial and receding contact angles. Physics of Fluids, 27 (2015), 122101. 43 Oksuz, Melik and Erbil, H Yildirim. Comments on the Energy Barrier Calculations during “Stick--Slip” Behavior of Evaporating Droplets Containing Nanoparticles. The Journal of Physical Chemistry C, 118 (2014), 9228--9338. 44 Wang, FengChao and Wu, HengAn. Molecular origin of contact line stick-slip motion during droplet evaporation. Scientific reports (2015), 17521. 45 Shanahan, Martin ER. Simple theory of" stick-slip" wetting hysteresis. Langmuir, 11 (1995), 1041-1043. 46 Shanahan, MER and Sefiane, K. Kinetics of triple line motion during evaporation. Contact angle, wettability and adhesion, 6 (2009), 19--31. 47 Wang, Feng-Chao and Wu, Heng-An. Pinning and depinning mechanism of the contact line during evaporation of nano-droplets sessile on textured surfaces. Soft Matter, 9 (2013), 5703--5709. 48 Zhang, Jianguo and Müller-Plathe, Florian and Leroy, Frédéric. Pinning of the contact line during evaporation on heterogeneous surfaces: slowdown or temporary immobilization? Insights from a nanoscale study. Langmuir, 31 (2015), 7544--7552. 49 Erbil, H Yildirim and McHale, Glen and Newton, MI. Drop evaporation on solid surfaces: constant contact angle mode. Langmuir, 18 (2002), 2636--2641. 50 Erbil, H Yildirim and Dogan, Muzeyyen. Determination of Diffusion Coefficient- Vapor Pressure Product of Some Liquids from Hanging Drop Evaporation. Langmuir, 16 (2000), 9267--9273. 51 Erbil, H Yildirim. Determination of the peripheral contact angle of sessile drops on solids from the rate of evaporation. Journal of adhesion science and technology, 13 (1999), 1405--1413. 52 Larson, Ronald G. Re-Shaping the Coffee Ring. Angewandte Chemie International Edition, 51 (2012), 2546--2548. 53 Chen, Xuemei and Ma, Ruiyuan and Li, Jintao and Hao, Chonglei and Guo, Wei and Luk, Bing Lam and Li, Shuai Cheng and Yao, Shuhuai and Wang, Zuankai. Evaporation of droplets on superhydrophobic surfaces: Surface roughness and small droplet size effects. Physical review letters, 109 (2012), 116101.


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54 Cates, ME and Stratford, K and Adhikari, R and Stansell, P and Desplat, JC and Pagonabarraga, I and Wagner, AJ. Simulating colloid hydrodynamics with lattice Boltzmann methods. Journal of Physics: Condensed Matter, 16 (2004), S3903. 531 532 533


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Graphical TOC Entry

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For Table of Content Only 342x178mm (300 x 300 DPI)


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Schematic representation of an evaporating drop of radius R_0 deposited on arrays of micro-ribs of width b, height h and the spacing distance w. In all simulations the drop radius R_0 is fixed to 1.3 mm (corresponding to a 100 lattice nodes) while w, h and b are varied in a range between 39 to 234 µm, 65 to 650 µm and 65 to 325 µm respectively. 211x187mm (300 x 300 DPI)


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D^2-law: comparison between numerical simulations (open squares) and theoretical prediction (solid line). 352x215mm (300 x 300 DPI)


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a) Simulation results of an evaporating liquid drop on a flat surface where the contact angleapparent contact angle remains constant during evaporation, while the drop contact diameter decreases (CCA mode). b) Normalized droplet contact radius (left axis) as well as drop contact angle (right axis). c) Evolution of an evaporating droplet in CCR mode on a rough surface. d) Normalized droplet contact radius and contact angleapparent contact angle on a rough surface versus normalized time. 127x76mm (300 x 300 DPI)


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Selected snapshots of an evaporating droplet depostited on a series of parallel micro-ribs: a) capillary suction phase, b) pinning mode, c) the first slip or depinning event, d) the second slip event. In (a), (b), (c) and (d), the top panel shows the density iso-surface in a three-dimensional view, the middle panel represents the density conture in the middle droplet cross-plane and in the bottom panel the velocity field inside the drop and around the triple contact line in the same plane is visualized. In (e) the total liquid volume (solid line), liquid volume of the droplet residing on the ribs (dashed line) and liquid volume penetrated into micro-ribs (dash-dot line) are plotted versus time for two different rib widths (in both cases the height and width of ribs are kept constant to h/R_0=0.5 and b/R_0=0.05). 204x211mm (300 x 300 DPI)


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(a) Top view of the contact line profile of the liquid drop pinned on top of the ribs for diffferent times. (b) Charactristic time evolution of the minor (left axis) and major semi-axis (right axis) of the ellips contact area. (c) The aspect ratio between the minor and the major semi-axis of ellips contact area during evaporation. (d) Variation of the contact angleapparent contact angle of the pinned contact line during evaporation process of the liquid drop. The micro-ribs have a spacing distance of w/R_0=0.06, height of h/R_0=0.5 and b/R_0=0.05. θ_(0,i) in (d) shows the contact angleapparent contact angle at the beginning of each pinning event i. After pinning the θ_(0,i) is decreased to a critical value at which the drop decomes depinned. The critical contact angle is averaged over all slipping modes and it is shown by (θ_critical )_avg. 127x89mm (300 x 300 DPI)


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Evolution of excess Gibbs free energy per unit length of triple line (contact radius r) normalized by the initial Gibbs free enrgy during evaporation proccess of a droplet deposited on a micro-ribs with w/R_0=0.06, h/R_0=0.5 and b/R_0=0.05, taking into account all terms (black) or only the interfacial terms (blue). The maximum excess Gibbs free energy or the energy barrier required for the drop to be depinned is averaged over all depinning events except the first, depinning denoted by 〖(δG_max/G_0)〗_avg*(R_0/r). 114x75mm (300 x 300 DPI)


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Variation of (a) the average critical contact angle at droplet depinning as well as (b) the average maximum excess Gibbs free energy as a function of the spacing distance w. For all simulations here, the rib height and width are kept equal to h/R_0=0.5 and b/R_0=0.05, respectively. The dashed line is fitted to the simualtion results (symbols) as a guide to the eye. 127x53mm (300 x 300 DPI)


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Evolution of (a) the solid-fluid contact area, A_wetting, together with (b) the fluid-fluid interfacial area, A_drop, during evaporation of the liquid droplet deposited on a rib-structured surface for different spacing distance w. In both (a) and (b) the surface area has been normalized by A_0 i.e. the initial spherical drop surface area. 127x45mm (300 x 300 DPI)


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Variation of (a) the average critical contact angle and (b) the average maximum excess Gibbs free energy as a function of the height of microribs . In all cases in (a) and (b) the spacing distance and rib width are kept constant to w/R_0=0.06 and b/R_0=0.05, respectively. The dashed line is fitted to the simualtion results (symbols) as a guide to the eye. 127x50mm (300 x 300 DPI)


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Evolution of (a) the solid-fluid contact area, A_wetting, together with (b) the fluid-fluid interfacial area, A_drop, during evaporation of the liquid droplet deposited on the rib-structured surface for different rib heights h. In both (a) and (b) the surface area has been normalized by A_0 i.e. the initial spherical drop surface area. 127x49mm (300 x 300 DPI)


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Variation of (a) the average critical contact angle and (b) the average maximum excess Gibbs free energy as a function of the rib width. The spacing distance and rib height are fixed to w/R_0=0.06 and h/R_0=0.5. 127x54mm (300 x 300 DPI)


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Dynamics of Contact Line Pinning and Depinning of Droplets

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