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     

On the sound radiation of a rolling tyre

The sound radiation from rolling tyres is still not very well understood. Although details such as horn effect or directivity during rolling have been investigated, it is not clear which vibrational modes of the tyre structure are responsible for the radiated sound power. In this work an advanced tyre model based on Wave Guide Finite Elements is used in connection with a contact model validated in previous work. With these tools the tyre vibrations during rolling on an ISO surface are simulated. Starting from the calculated contact forces in time the amplitudes of the modes excited during rolling are determined as function of frequency. A boundary element model also validated in previous work is applied to predict the sound pressure level on a reference surface around a tyre placed on rigid ground as function of the modal composition of the tyre vibrations. Taking into account different modes when calculating the vibrational field as input into the boundary element calculations, it is possible to identify individual modes or groups of modes of special relevance for the radiated sound power. The results show that mainly low-order modes with relative low amplitudes but high radiation efficiency in the frequency range around 1 kHz are responsible for the radiated sound power at these frequencies, while those modes which are most strongly excited in that frequency range during rolling are irrelevant for the radiated sound power. This fact is very essential when focusing on the design of quieter tyres.     

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.     

Physical properties of dynamic force microscopies in contact and noncontact operation

In the field of Scanning Force Microscopy several dynamical contact and noncontact modes have been introduced increasing the range of detectable surface and interface properties, and allowing to detect material properties such as elasticity and mass density on the nanometer scale. A detailed understanding of tip/surface interactions and the dynamic processes involved is required to understand the origin of a material contrast using these techniques. Here a general method to solve the equation of motion of a vibrating SFM cantilever/tip system in an external force field is presented. Contact modes as well as intermittent contact modes are discussed using a single set of equations describing the cantilever/tip motion, and by varying the size of amplitudes of the vibrating cantilever/tip system. To quantitatively describe the oscillation behavior of the SFM cantilever at large amplitudes the computer simulations are based on the MYD/BHW model providing a realistic contact model with respect to the contact area, the size of the contact forces as well as the transition from repulsive to attractive forces. The results are compared with the experiment and with different approaches based on analytical and numerical models.     

High frequency ultrasonic detection of C-crack defects in silicon nitride bearing balls

A non-destructive testing method for silicon nitride bearing balls based on ultrasonic resonance spectroscopy is proposed here. Through the theoretical study of their elastic vibrations, it is possible to characterize the balls using a vibration mode that is similar to surface wave propagation. The study of the influence of C-crack defects on the resonances of Rayleigh modes is presented here. These C-cracks are typically formed by impacts between balls during finishing or handling. They are frequently found on the surface of silicon nitride bearing balls and these C-cracks decrease the rolling contact fatigue life considerably. This kind of defect is difficult to detect because the C-shaped surface crack is very small and narrow (500 μm × 5 μm), and its depth does not exceed 50 μm. The proposed methodology can both excite spheroidal vibrations in the ceramic balls and detect such vibrations over a large frequency range. In particular, high frequency vibrations are considered because these are similar to the surface waves propagating in the cortical zone of the ceramic balls and consequently they can be used to detect C-crack defects.     

Seismic effects of incident P waves on an embedded foundation in poroelastic half-space

Dynamic vibrations of a circular rigid foundation, which is embedded in poroelastic soil and subjected to incident P waves, are studied by semi-analytical methods in this present work. The motion of the soil is governed by Biot's dynamic poroelastic theory. A set of potentials are introduced to represent the incident waves, and the scattering waves caused by the foundation are considered based on the decomposition of the total wave field in soil. The soil along the vertical side of the foundation is assumed to be composed of series of infinitesimally thin poroelastic layers, while the soil under the foundation base is regarded as the poroelastic half-space and to be independent of the overlying soil. The interaction problem is solved by Hankel transforms. Then, combining the boundary conditions along the contact surface between the soil and the foundation and the dynamic equilibrium equation of the foundation, expressions of the vertical and rocking vibration amplitudes of the embedded foundation excited by the incident P waves are acquired. Numerical results are presented to demonstrate the influences of embedded depth, foundation mass, pore water in the soil and incident angle on the vibrations of the foundation.     

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.     

Triplon modes of puddles

Free fluctuations of the contact line of large drops ("puddles") of wavelength lambda > kappa(-1), the capillary length, cannot be seen on a solid substrate because even a small but finite hysteresis is enough to block these slow modes. We show here that vertical vibrations of the substrate (at frequency omegaE, acceleration Lambda) above a threshold amplitude Lambda(c) release the line and excite contour oscillations (triplons). We observe harmonic modes and parametric excitations at omegaE/2. We construct the phase diagram (Lambda, omegaE) of these subharmonic modes and we study their growth dynamics: they slow down near the threshold of the contour instability.     

Adsorption sites of CO on Pt (111)

The surface vibrations of CO adsorbed on Pt(111) single crystal surfaces at 320 K have been studied by electron-energy-loss spectroscopy. At low coverages two vibration modes at 58 and ∼260 meV are observed. For exposures >0.2 Langmuir two additional modes at 45 and 232 meV develop. Considering also the observed LEED structures these vibrations are attributed to CO molecules being adsorbed upright in on-top and bridge sites, respectively.     

Reflection of elastic waves in a crystal of Heusler alloy Ni<Subscript>2</Subscript>MnGa in the phase transition range

The reflection of longitudinal and transverse acoustic waves from the free surface of the ferromagnetic shape memory alloy Ni₂MnGa that is located in the ranges of the premartensite and martensite phase transformations is considered. The propagation directions and amplitudes of the waves reflected in the (001) plane of the crystal are determined. They acquire the character of substantially quasi-longitudinal and quasi-transverse vibrations rather than being pure modes. The angles of wave reflection and conversion are shown to be effectively controlled by temperature and a magnetic field due to the colossal acoustic anisotropy of the crystal over the wide range of its phase transitions. Beginning from a certain critical angle of incidence of a quasi-transverse wave, the quasi-longitudinal wave having appeared upon reflection becomes an accompanying surface vibration, and it can be emitted into the bulk of the crystal when the phase transition point is approached. Two angles of full conversion of an incident quasi-longitudinal wave into a quasi-transverse wave are established, and their temperature dependences are found. Trivisonno’s experimental data for the ultrasound velocity and absorption in an Ni₂MnGa crystal are used to numerically estimate these acoustic effects.     

倾斜均质表面上非等径液滴的聚合特性

对倾斜均匀表面上非等径液滴的聚合过程及特性进行了可视化实验研究,获得了液滴半径和表面倾角等参数对液滴聚合过程中液滴液桥半径、接触角和接触线变化特性的影响规律,进一步说明了倾斜表面上液滴聚合可以加快液滴的运动.     

Contact angle hysteresis on nano-structured surfaces

We present results from an experimental study on the phenomenon of contact angle hysteresis on solid surfaces decorated by a random array of nanometric hollows. For weak values of the areal density of defects φ_d, the hysteresis H increases linearly with φ_d. This evolution is described by a pinning–depinning process of the contact line by individual defects. At higher values of φ_d, a collective pinning effect appears and H decreases with increasing φ_d. In the linear regime, our experimental results are compared to theoretical predictions for contact angle hysteresis induced by a single isolated defect on the solid surface. We suggest that the crossover from the individual to the collective pinning effects could be interpreted in terms of an overlapping of wetting cross sections. Finally, we analyse the influence of both the size and the morphology (hollows/hillocks) of defects on the anchorage of the contact line.     

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.     

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.     

容器内角处流体界面特性与Surface Evolver程序适用性的研究

微重力条件下内角处液体行为的研究对于认识表面张力主导的液体行为, 预测和控制空间微重力条件下的液体位置、瞬时状态, 以及对空间流体进行有效的管理等方面都非常重要. 通过分析接触角与流体界面在容器内角处的接触线方向之间的关系, 并与Concus-Finn理论进行比较, 提出了内角处接触线、接触角和几何形状之间相互关联的机理, 并探讨了Concus和Finn等 的相关理论结果的物理内涵. 在此基础上, 进一步将内角处的相关理论结果与Surface Evolver程序得出的数值结果进行了比较, 指出当容器中的内角小于180°时, Surface Evolver程序通过自动划分网格即可比较准确地预言内角处的接触线和液面行为, 但是当内角大于180°时, 自动划分网格得到的数值结果有较大的误差, 需要通过手动划分网格减少网格奇异才能减小误差. 因此, 对于具有复杂几何形状的容器, 需注意网格的奇异性, 并对内角处的液面进行定量的验证, 才能有效判断Surface Evolver程序结果的正确性. 本工作对于深入认识内角处的液面特性和机理, 理解Surface Evolver程序的适用条件, 以及分析微重力条件下容器内角处的液体行为等方面都具有明显意义.     

ASYMPTOTIC APPROACH FOR NON-LINEAR PERIODICAL VIBRATIONS OF CONTINUOUS STRUCTURES

An asymptotic approach for determining periodic solutions of non-linear vibration problems of continuous structures (such as rods, beams, plates, etc.) is proposed. Starting with the well-known perturbation technique, the independent displacement and frequency is expanded in a power series of a natural small parameter. It leads to infinite systems of interconnected non-linear algebraic equations governing the relationships between modes, amplitudes and frequencies. A non-trivial asymptotic technique, based on the introduction of an artificial small parameter is used to solve the equations. An advantage of the procedure is the possibility to take into account a number of vibration modes. As examples, free longitudinal vibrations of a rod and lateral vibrations of a beam under cubically non-linear restoring force are considered. Resonance interactions between different modes are investigated and asymptotic formulae for corresponding backbone curves are derived.     

Droplets on inclined rough surfaces

The behaviour of liquid droplets on inclined heterogeneous surfaces was simulated by the lattice-Boltzmann method using the Shan-Chen multiphase model. The effect of topography of the surface on the contact angle hysteresis was investigated. It is shown in particular, by using anisotropic rough surfaces, how surface topography and thereby the continuity of the three-phase contact line, affect this hysteresis. Our results clearly indicate that the superhydrophobicity of a surface cannot be judged by the contact angle alone.     

Generation of nonstationary Görtler vortices by localized surface nonuniformities. Receptivity coefficients

The mechanism of production of nonstationary Görtler vortices in a boundary layer on concave wall by surface nonuniformities (vibrations and roughness) has been experimentally examined. The nonuniformities were produced by a specially developed disturbance source. They were controlled, localized along the streamwise coordinate, and periodic over the span of the experimental model. Tests in a low-turbulence wind tunnel have proved that the disturbance source is an efficient means of experimental study of the receptivity and stability problem for boundary layers dominated by Görtler instability. The operation of the disturbance source leads to the production of small-amplitude nonstationary Görtler vortices (tenth or hundredth fractions of a per cent of the free-stream velocity) with predefined characteristics (frequency and spanwise wavelength). In our experiments, we quantitatively examined the problem of linear receptivity of boundary layer to surface nonuniformities in a broad range of frequencies for the most dangerous spanwise scales of Görtler vortices. The values of the amplitudes and phases of the receptivity coefficients were determined. The amplitudes proved to be much smaller in magnitude in comparison with the excitation of modes of hydrodynamic instabilities of other types (Tollmien-Schlichting waves and cross-flow-instability modes). It was found that, with increasing the frequency, the amplitudes of the receptivity coefficients showed a distinct growth while for high frequencies those amplitudes also exhibited a growth with the spanwise scale of perturbations, although for stationary surface roughness no effect due to this scale was observed. It was found that the dependences on frequency of the efficiency of the mechanisms of stability and receptivity showed opposing behaviors, were in competition, and could partially compensate each other, promoting, thus, the production of boundary-layer Görtler vortices in a broad range of frequencies.     

Effect of contact angle hysteresis on water droplet evaporation from super-hydrophobic surfaces

Small water drops demonstrate different evaporation modes on super-hydrophobic polymer surfaces with different hysteresis of contact angle. While on the high-hysteresis surface evaporation follows the constant-contact-diameter mode, the constant-contact-angle mode dominates on the low-hysteresis surface. These modes were previously reported for smooth hydrophilic and hydrophobic surfaces, respectively. The experimental data are compared to the previous models describing spherical cap drops that evaporate in different modes, and good fitting is obtained.     

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.