The debate on the dependence of apparent contact angles on drop contact area or three-phase contact line: A review

A sessile drop is an isolated drop which has been deposited on a solid substrate where the wetted area is limited by the three-phase contact line and characterized by contact angle, contact radius and drop height. Although, wetting has been studied using contact angles of drops on solids for more than 200 years, the question remains unanswered: Is wetting of a rough and chemically heterogeneous surface controlled by the interactions within the solid/liquid contact area beneath the droplet or only at the three-phase contact line? After the publications of Pease in 1945, Extrand in 1997, 2003 and Gao and McCarthy in 2007 and 2009, it was proposed that advancing, receding contact angles, and contact angle hysteresis of rough and chemically heterogeneous surfaces are determined by interactions of the liquid and the solid at the three-phase contact line alone and the interfacial area within the contact perimeter is irrelevant. As a consequence of this statement, the well-known Wenzel (1934) and Cassie (1945) equations which were derived using the contact area approach are proposed to be invalid and should be abandoned. A hot debate started in the field of surface science after 2007, between the three-phase contact line and interfacial contact area approach defenders. This paper presents a review of the published articles on contact angles and summarizes the views of the both sides. After presenting a brief history of the contact angles and their measurement methods, we discussed the basic contact angle theory and applications of contact angles on the characterization of flat, rough and micropatterned superhydrophobic surfaces. The weak and strong sides of both three-phase contact line and contact area approaches were discussed in detail and some practical conclusions were drawn.     

Effect of contact line curvature on solid-fluid surface tensions without line tension

For sessile droplets partially wetting a solid surface, it has been observed experimentally that the value of the contact angle depends on the contact line curvature and this dependence has been attributed to tension in the contact line. But previous analyses of these observations have neglected adsorption at the solid-liquid interface and its effect on the surface tension of this interface. We show that if this adsorption is taken into account the relation between the contact angle and contact line curvature is completely accounted for without introducing line tension. Further, from the observed relation between the contact angle and contact line curvature, the adsorption at the solid-liquid interface can be determined, as can the surface tensions of the solid-liquid and solid-vapor interfaces.     

平衡接触角对受热液滴在水平壁面上铺展特性的影响

壁面温度是影响壁面润湿性的重要外部条件. 为解决液滴铺展中三相接触线处应力集中问题, 已有研究多采用预置液膜假设, 但无法探究壁面温度对润湿性的影响. 本文针对受热液滴在固体壁面上的铺展过程, 基于润滑理论建立了演化模型, 通过数值模拟, 从平衡接触角角度分析了温度影响壁面润湿性及铺展过程的内部机理. 研究表明: 随温度梯度增大, 液滴所受Marangoni效应增强, 致使液滴向低温区的铺展速率加快; 铺展过程中, 位于高温区的接触线与液滴主体部分间形成一层薄液膜, 重力与热毛细力先后主导该区域的铺展; 当液-固或气-液界面张力对温度的敏感度高于另两个界面时, 低温区方向的平衡接触角不断增大, 使壁面润湿性恶化, 导致液滴铺展减慢; 而当气-固界面张力对温度的敏感度高于其他两个界面时, 低温区方向上的平衡接触角将减小, 由此改善壁面润湿性, 加快液滴铺展; 在温度影响壁面润湿性和液滴铺展过程中, 平衡接触角起关键作用.     

Structure of the surface microrelief of a droplet evaporating from a rough substrate as a possible cause of contact angle hysteresis

Based on the refraction images of a droplet evaporating on a rough substrate, we simultaneously observed the dynamics of its surface microrelief, contact angle, and contact line deformations along the entire perimeter of the contact line. This has led us conclude that the microrelief structure is directly related to the phenomenon of contact angle hysteresis and the jump-like pattern of contact line deformation. We suggest a possible mechanism for the occurrence of contact angle hysteresis during droplet evaporation and derive the relations that specify the range of possible contact angles at known microrelief parameters.     

液滴在不同润湿性表面上蒸发时的动力学特性

基于润滑理论,采用滑移边界条件建立了二维液滴厚度的演化模型和移动接触线动力学模型,利用数值计算方法模拟了均匀加热基底上固着液滴蒸发时的动力学特性,分析了液-气、固-气和液-固界面张力温度敏感性对壁面润湿性和液滴动态特性的影响.结果表明,液滴的运动过程受毛细力、重力、热毛细力和蒸发的影响,重力对液滴铺展起促进作用,而毛细力、热毛细力则起抑制作用;通过改变界面张力温度敏感性系数,可使液滴蒸发过程中的接触线呈现处于钉扎或部分钉扎模式,且接触线钉扎模式下的液滴存续时间低于部分钉扎模式;提高液-气与液-固界面张力温度敏感系数均可改善壁面润湿性能,加快液滴铺展速率;而增大固-气界面张力温度敏感系数则导致壁面润湿性能恶化、延缓液滴铺展过程;通过改变固-气界面张力温度敏感系数更有利于调控处于蒸发状态下的液滴运动.     

Thoughts on some outstanding issues in the physics of equilibrium wetting and conceptual understanding of contact lines

Equilibrium wetting is a fundamental phenomenon, relevant to many scientific areas. Since the pioneering work on equilibrium wetting of Thomas Young (1805) [1], researchers strived to advance our understanding of this fundamental problem. Despite its apparent simplicity, equilibrium wetting phenomenon still holds many unanswered questions and represents a challenge to modern physicists and engineers. The relationship between quantities amenable to measurements, like macroscopic wetting contact angle, and other surface ener- gies and physical properties remains to be fully elucidated. Wetting is a physical problem which spans over two length scales, inner region (“microscopic”) length scale and outer region (“macroscopic”). The three-phase contact line, where the macroscopic region meets the micro- scopic one, and underlying surface forces, represents a challenge to fully understand and model. In this paper, a brief review of the basics of wetting and existing concepts is first presented. Then two important questions are discussed in the light of the latest experimental findings: first the relevance of the continuum concept when describing interfaces near the three-phase contact line, and second the effect of adsorption on interfacial energies and its use to explain some interesting observations like the dependence of equilibrium contact angle on pressure and size of droplets. These recent observations raise some fundamental questions about how the three-phase contact line is conceptualised.     

Interface profiles near three-phase contact lines in electric fields

Long-range electrostatic fields deform the surface profile of a conductive liquid in the vicinity of the contact line. We have investigated the equilibrium profiles by balancing electrostatic and capillary forces locally at the liquid vapor interface. Numerical results show that the contact angle at the contact line approaches Young's angle. Simultaneously, the local curvature displays a weak algebraic divergence. Furthermore, we present an asymptotic analytical model, which confirms these results and elucidates the scaling behavior of the profile close to the contact line.     

Vibrated sessile drops: Transition between pinned and mobile contact line oscillations

We study the effects of vertical vibrations on non-wetting large water sessile drops flattened by gravity. The solid substrate is characterized by a finite contact angle hysteresis (10-15 degrees). By varying the frequency and the amplitude of the vertical displacement, we observe two types of oscillations. At low amplitude, the contact line remains pinned and the drop presents eigen modes at different resonance frequencies. At higher amplitude, the contact line moves: it remains circular but its radius oscillates at the excitation frequency. The transition between these two regimes arises when the variations of contact angle exceed the contact angle hysteresis. We interpret different features of these oscillations, such as the decrease of the resonance frequencies at larger vibration amplitudes. The hysteresis acts as solid friction on the contour oscillations, and gives rise to a stick-slip regime at intermediate amplitude.Received: 4 April 2004, Published online: 10 August 2004PACS: 47.55.Dz Drops and bubbles - 68.08.Bc Wetting - 47.35. + i Hydrodynamic waves     

Numerical simulation of an evaporative meniscus on a moving substrate

A hydrodynamic model based on lubrication theory has been developed to describe an evaporative meniscus in a complete wetting configuration, when evaporation takes place in ambient air. We focus on combined effects of evaporation and the substrate motion on the effective contact angle. Numerical simulations show two distinct regimes when varying the substrate velocity on several orders of magnitude. At a small velocity, the effective contact angle is governed by evaporation and is independent of the substrate velocity, while the substrate motion is dominant at a high velocity. In the latter case, a Landau-Levich regime is obtained for the receding contact line, and a Cox-Voinov regime for the advancing contact line. Finally, we use our numerical results to test the simplified model developed by Pham et al. [5,6].     

Scaling of dynamic contact angles in a lattice-Boltzmann model

We study the origins of the dynamic contact angle in a two-dimensional lattice-Boltzmann model of immiscible fluids. We show that the dynamic contact angle changes as a function of capillary number as observed in laboratory experiments and explain how this dependence arises in the lattice-Boltzmann model. We also explain how the fluid-fluid interface can move while retaining its shape. The interface has an apparent slip length. The apparent slip follows the classical Navier slipping rule where the velocity of the fluid at the wall is proportional to the viscous stress at the wall. This apparent slip length is proportional to the viscous length scale associated with the spurious flow induced by uncompensated stress at the three-phase contact point.     

Deformation of an elastic substrate by a three-phase contact line

Young's classic analysis of the equilibrium of a three-phase contact line ignores the out-of-plane component of the liquid-vapor surface tension. While it is expected that this unresolved force is balanced by the elastic response of the solid, a definitive analysis has remained elusive because of an apparent divergence of stress at the contact line. While a number of theories have been presented to cut off the divergence, none of them have provided reasonable agreement with experimental data. We measure surface and bulk deformation of a thin elastic film near a three-phase contact line using fluorescence confocal microscopy. The out-of-plane deformation is well fit by a linear elastic theory incorporating an out-of-plane restoring force due to the surface tension of the solid substrate. This theory predicts that the deformation profile near the contact line is scale-free and independent of the substrate elastic modulus.     

A conservative level set method for contact line dynamics

A new model for simulating contact line dynamics is proposed. We apply the idea of driving contact-line movement by enforcing the equilibrium contact angle at the boundary, to the conservative level set method for incompressible two-phase flow [E. Olsson, G. Kreiss, A conservative level set method for two phase flow, J. Comput. Phys. 210 (2005) 225–246]. A modified reinitialization procedure provides a diffusive mechanism for contact-line movement, and results in a smooth transition of the interface near the contact line without explicit reconstruction of the interface. We are able to capture contact-line movement without loosing the conservation. Numerical simulations of capillary dominated flows in two space dimensions demonstrate that the model is able to capture contact line dynamics qualitatively correct.     

The vicinity of an equilibrium three-phase contact line using density-functional theory: density profiles normal to the fluid interface

The paper by Nold et al. [Phys. Fluids 26 (7), 072001 (2014)] examined density profiles and the micro-scale structure of an equilibrium three-phase (liquid–vapour–solid) contact line in the immediate vicinity of the wall using elements from the statistical mechanics of classical fluids, namely density-functional theory. The present research note, building on the above work, further contributes to our understanding of the nanoscale structure of a contact line by quantifying the strong dependence of the liquid–vapour density profile on the normal distance to the interface, when compared to the dependence on the vertical distance to the substrate. A recent study by Benet et al. [J. Phys. Chem. C 118 (38), 22079 (2014)] has shown that this could explain the emergence of a film-height-dependent surface tension close to the wall, with implications for the Frumkin–Derjaguin theory.     

Film transitions of receding contact lines

When a solid plate is withdrawn from a liquid bath, a receding contact line is formed where solid, liquid, and gas meet. Above a critical speed U_cr, a stationary contact line can no longer exist and the solid will eventually be covered completely by a liquid film. Here we show that the bifurcation diagram of this coating transition changes qualitatively, from discontinuous to continuous, when decreasing the inclination angle θ_p of the plate. We show that this effect is governed by the presence of capillary waves, illustrating that the large scale flow strongly effects the maximum speed of dewetting.     

溶液液滴蒸发变干的环状沉积

液体蒸发驱动的颗粒自组装现象在许多的工业技术中有重要应用. 本文利用显微镜观测含有颗粒物质的液滴变干后留在固体表面的颗粒形成的环状沉积图案. 采用微米粒径的SiO₂小球水溶液液滴蒸发变干模拟咖啡环的形成过程, 结果发现液滴蒸发过程中接触线的钉扎是环状沉积的必要条件. 在液滴蒸发过程中颗粒随着补偿流不断的向液滴边缘移动, 聚集在接触线处形成环. 液滴蒸发变干后残留在液滴内部的颗粒数随颗粒质量分数的增加而增加, 可以达到单层的颗粒排列. 而玻璃衬底上的颗粒环在颗粒质量分数很小时, 形成单层排列, 且一排一排地生长. 蒸发过程中颗粒环由于液滴边缘的尺寸限制向液滴中心缓慢移动. 这会导致液滴中不同大小颗粒的分离. 关键词： 液滴 接触线 蒸发 颗粒     

Quasi-static motion of microparticles at the depinning contact line of an evaporating droplet on PDMS surface

In this paper, evaporation of sessile water droplets containing fluorescent polystyrene(PS) microparticles on polydimethylsiloxane(PDMS) surfaces with different curing ratios was studied experimentally using laser confocal microscopy. At the beginning, there were some microparticles located at the contact line and some microparticles moved towards the line. Due to contact angle hysteresis, at first both the contact line and the microparticles were pinned. With the depinning contact line, the microparticles moved together spontaneously. Using the software ImageJ, the location of contact lines at different time were acquired and the circle centers and radii of the contact lines were obtained via the least square method. Then the average distance of two neighbor contact lines at a certain time interval was obtained to characterize the motion of the contact line. Fitting the distance-time curve at the depinning contact line stage with polynomials and differentiating the polynomials with time, we obtained the velocity and acceleration of both the contact line and the microparticles located at the line. The velocity and the maximum acceleration were,respectively, of the orders of 1 μm/s and 20-200 nm/s~2, indicating that the motion of the microparticles located at the depinning contact line was quasi-static. Finally, we presented a theoretical model to describe the quasi-static process, which may help in understanding both self-pinning and depinning of microparticles.     

An elastic-plastic contact model for line contact structures

Although numerical simulation tools are now very powerful,the development of analytical models is very important for the prediction of the mechanical behaviour of line contact structures for deeply understanding contact problems and engineering applications.For the line contact structures widely used in the engineering field,few analytical models are available for predicting the mechanical behaviour when the structures deform plastically,as the classic Hertz’s theory would be invalid.Thus,the present study proposed an elastic-plastic model for line contact structures based on the understanding of the yield mechanism.A mathematical expression describing the global relationship between load history and contact width evolution of line contact structures was obtained.The proposed model was verified through an actual line contact test and a corresponding numerical simulation.The results confirmed that this model can be used to accurately predict the elastic-plastic mechanical behaviour of a line contact structure.     

A mesh-dependent model for applying dynamic contact angles to VOF simulations

Typical VOF algorithms rely on an implicit slip that scales with mesh refinement, to allow contact lines to move along no-slip boundaries. As a result, solutions of contact line phenomena vary continuously with mesh spacing; this paper presents examples of that variation. A mesh-dependent dynamic contact angle model is then presented, that is based on fundamental hydrodynamics and serves as a more appropriate boundary condition at a moving contact line. This new boundary condition eliminates the stress singularity at the contact line; the resulting problem is thus well-posed and yields solutions that converge with mesh refinement. Numerical results are presented of a solid plate withdrawing from a fluid pool, and of spontaneous droplet spread at small capillary and Reynolds numbers.     

Geometry-dependent electrostatics near contact lines

Long-ranged electrostatic interactions in electrolytes modify contact angles on charged substrates in a scale and geometry-dependent manner. For angles measured at scales smaller than the typical Debye screening length, the wetting geometry near the contact line must be explicitly considered. Using variational and asymptotic methods, we derive new transcendental equations for the contact angle as functions of the electrostatic potential only at the three phase contact line. Analytic expressions are found in certain limits and compared with predictions for contact angles measured with lower resolution. An estimate for electrostatic contributions to line tension is also given.