scholarly journals Deformation of an elastic substrate due to a resting sessile droplet

2017 ◽  
Vol 29 (2) ◽  
pp. 281-300 ◽  
Author(s):  
AARON BARDALL ◽  
KAREN E. DANIELS ◽  
MICHAEL SHEARER
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Substrate Surface ◽  
Fluid Pressure ◽  
Force Balance ◽  
Energy Conditions ◽  
Equilibrium Contact Angle ◽  
Solution Technique ◽  
Vertical Force ◽  
Tangential Contact

On a sufficiently soft substrate, a resting fluid droplet will cause significant deformation of the substrate. This deformation is driven by a combination of capillary forces at the contact line and the fluid pressure at the solid surface. These forces are balanced at the surface by the solid traction stress induced by the substrate deformation. Young's Law, which predicts the equilibrium contact angle of the droplet, also indicates an a priori radial force balance for rigid substrates, but not necessarily for soft substrates that deform under loading. It remains an open question whether the contact line transmits a non-zero force tangent to the substrate surface in addition to the conventional normal (vertical) force. We present an analytic Fourier transform solution technique that includes general interfacial energy conditions, which govern the contact angle of a 2D droplet. This includes evaluating the effect of gravity on the droplet shape in order to determine the correct fluid pressure at the substrate surface for larger droplets. Importantly, we find that in order to avoid a strain singularity at the contact line under a non-zero tangential contact line force, it is necessary to include a previously neglected horizontal traction boundary condition. To quantify the effects of the contact line and identify key quantities that will be experimentally accessible for testing the model, we evaluate solutions for the substrate surface displacement field as a function of Poisson's ratio and zero/non-zero tangential contact line forces.

scholarly journals Dynamics of droplets under electrowetting effect with voltages exceeding the contact angle saturation threshold

2021 ◽  
Vol 925 ◽  
Author(s):  
Quoc Vo ◽  
Tuan Tran
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Applied Voltage ◽  
Droplet Surface ◽  
Capillary Waves ◽  
Force Balance ◽  
Equilibrium Contact Angle ◽  
Initial Contact ◽  
Viscous Effects ◽  
Electrowetting On Dielectric

Electrowetting on dielectric (EWOD) is a powerful tool in many droplet-manipulation applications with a notorious weakness caused by contact-angle saturation (CAS), a phenomenon limiting the equilibrium contact angle of an EWOD-actuated droplet at high applied voltage. In this paper, we study the spreading behaviours of droplets on EWOD substrates with the range of applied voltage exceeding the saturation limit. We experimentally find that at the initial stage of spreading, the driving force at the contact line still follows the Young–Lippmann law even if the applied voltage is higher than the CAS voltage. We then theoretically establish the relation between the initial contact-line velocity and the applied voltage using the force balance at the contact line. We also find that the amplitude of capillary waves on the droplet surface generated by the contact line's initial motion increases with the applied voltage. We provide a working framework utilising EWOD with voltages beyond CAS by characterising the capillary waves formed on the droplet surface and their self-similar behaviours. We finally propose a theoretical model of the wave profiles taking into account the viscous effects and verify this model experimentally. Our results provide avenues to utilise the EWOD effect with voltages beyond the CAS threshold, and have strong bearing on emerging applications such as digital microfluidic and ink-jet printing.

scholarly journals Prediction of Contact Angle of Nanofluids by Single-Phase Approaches

Energies ◽  
2019 ◽  
Vol 12 (23) ◽  
pp. 4558 ◽  
Author(s):  
Nur Çobanoğlu ◽  
Ziya Haktan Karadeniz ◽  
Patrice Estellé ◽  
Raul Martínez-Cuenca ◽  
Matthias H. Buschmann
Keyword(s):  
Contact Angle ◽  
Single Phase ◽  
Substrate Surface ◽  
Droplet Surface ◽  
Force Balance ◽  
Radius Of Curvature ◽  
Geometrical Parameters ◽  
Ambient Conditions ◽  
Sessile Droplet ◽  
Droplet Shape

Wettability is the ability of the liquid to contact with the solid surface at the surrounding fluid and its degree is defined by contact angle (CA), which is calculated with balance between adhesive and cohesive forces on droplet surface. Thermophysical properties of the droplet, the forces acting on the droplet, atmosphere surrounding the droplet and the substrate surface are the main parameters affecting on CA. With nanofluids (NF), nanoparticle concentration and size and shape can modify the contact angle and thus wettability. This study investigates the validity of single-phase CA correlations for several nanofluids with different types of nanoparticles dispersed in water. Geometrical parameters of sessile droplet (height of the droplet, wetting radius and radius of curvature at the apex) are used in the tested correlations, which are based on force balance acting on the droplet surface, energy balance, spherical dome approach and empirical expression, respectively. It is shown that single-phase models can be expressed in terms of Bond number, the non-dimensional droplet volume and two geometrical similarity simplexes. It is demonstrated that they can be used successfully to predict CA of dilute nanofluids’ at ambient conditions. Besides evaluation of CA, droplet shape is also well predicted for all nanofluid samples with ±5% error.

An Electromechanical Model for Electrowetting With Finite Droplet Size

2020 ◽  
Vol 142 (7) ◽  
Author(s):  
Deng Huang ◽  
Fang Qian ◽  
Wenyao Zhang ◽  
Wenbo Li ◽  
Rui Chuan ◽  
...  
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Finite Size ◽  
Surface Tension Force ◽  
Equilibrium Contact Angle ◽  
Electric Force ◽  
Electromechanical Model ◽  
Flat Substrate ◽  
Three Phase ◽  
Maxwell Stresses

Abstract We present an electromechanical model for the analysis of electrowetting by considering the balance between an electric force and a surface tension force acting on the contact line of three phases, namely the droplet (D) phase, the substrate (S) phase, and the ambiance (A) phase. We show that the Maxwell stresses at the ambiance–substrate (A–S) interface, the droplet–substrate (D–S) interface, and the droplet–ambiance (D–A) interface induce an electric force on the three-phase contact line which is responsible for the modification of the apparent contact angle in electrowetting. For a classical electrowetting configuration with a flat substrate, we show that the electric force on the contact line (or the electrowetting number) is mainly due to the Maxwell stresses at the D–A interface. The model is validated by its excellent agreement with the classical Young-Lippmann (Y-L) model for sufficiently large droplets and comparable electric permittivities between A and S phases. Interestingly, our new model reveals that the finite size of droplet produces profound effects on the electrowetting that the electrowetting number becomes dependent on the permittivity of A phase and the equilibrium contact angle, which is in stark contrast to the Y-L model. The reasons for these remarkable effects are elaborated and clarified. The findings in the current study are complementary to the classical Y-L model and provide new insights into the electrowetting phenomenon.

Visualization of Dynamic Contact Angle

2013 ◽  
Author(s):  
Brandon S. Field
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
High Speed ◽  
Solid Phase ◽  
Capillary Rise ◽  
Dynamic Contact Angle ◽  
Fluid Phase ◽  
Dynamic Contact ◽  
Force Balance ◽  
Receding Contact

Capillary rise of air-water-solid systems have been recorded with high-speed video. Glass and metal have been used as the solid phase, and the dynamic shape of the meniscus and contact angle have been characterized. The advancing and receding contact angle is of interest in computational simulations of boiling flow, and the present visualizations attempt to quantify the dynamic aspects of contact line motion. The centroid of the capillary meniscus has been tracked in order to determine the force at the contact line based on a force balance of the elevated fluid phase. The solid phase is raised and lowered in the fluid at different rates to observe advancing and receding contact lines.

Dynamics of Droplet Motion Under Electrowetting Actuation

2011 ◽  
Author(s):  
S. Ravi Annapragada ◽  
Susmita Dash ◽  
Suresh V. Garimella ◽  
Jayathi Y. Murthy
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Fluid Motion ◽  
Dynamic Contact Angle ◽  
Transient Behavior ◽  
Force Balance ◽  
Contact Radius ◽  
Droplet Motion ◽  
Droplet Shape ◽  
Frictional Forces

The static shape of droplets under electrowetting actuation is well-understood. The steady-state shape of the droplet is obtained based on the balance of surface tension and electrowetting forces, and the change in apparent contact angle is well-characterized by the Young-Lippmann equation. However, the transient droplet shape behavior when a voltage is suddenly applied across a droplet has received less attention. Additional dynamic frictional forces are at play during this transient process. We present a model to predict this transient behavior of the droplet shape under electrowetting actuation. The droplet shape is modeled using the volume of fluid method. The electrowetting and dynamic frictional forces are included as an effective dynamic contact angle through a force balance at the contact line. The model is used to predict the transient behavior of water droplets on smooth hydrophobic surfaces under electrowetting actuation. The predictions of transient behavior of droplet shape and contact radius are in excellent agreement our experimental measurements. The internal fluid motion is explained and the droplet motion is shown to initiate from the contact line. An approximate mathematical model is also developed to understand the physics of the droplet motion and to describe the overall droplet motion and the contact line velocities.

Contact-line dynamics of a diffuse fluid interface

2000 ◽  
Vol 402 ◽  
pp. 57-88 ◽  
Author(s):  
DAVID JACQMIN
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Chemical Potential ◽  
Equilibrium Contact Angle ◽  
Moving Contact Line ◽  
Moving Contact ◽  
Potential Gradients ◽  
Inner Region ◽  
Chemical Potential Gradients ◽  
Contact Line Dynamics

An investigation is made into the moving contact line dynamics of a Cahn–Hilliard–van der Waals (CHW) diffuse mean-field interface. The interface separates two incompressible viscous fluids and can evolve either through convection or through diffusion driven by chemical potential gradients. The purpose of this paper is to show how the CHW moving contact line compares to the classical sharp interface contact line. It therefore discusses the asymptotics of the CHW contact line velocity and chemical potential fields as the interface thickness ε and the mobility κ both go to zero. The CHW and classical velocity fields have the same outer behaviour but can have very different inner behaviours and physics. In the CHW model, wall–liquid bonds are broken by chemical potential gradients instead of by shear and change of material at the wall is accomplished by diffusion rather than convection. The result is, mathematically at least, that the CHW moving contact line can exist even with no-slip conditions for the velocity. The relevance and realism or lack thereof of this is considered through the course of the paper.The two contacting fluids are assumed to be Newtonian and, to a first approximation, to obey the no-slip condition. The analysis is linear. For simplicity most of the analysis and results are for a 90° contact angle and for the fluids having equal dynamic viscosity μ and mobility κ. There are two regions of flow. To leading order the outer-region velocity field is the same as for sharp interfaces (flow field independent of r) while the chemical potential behaves like r−ξ, ξ = π/2/max{θeq, π − θeq}, θeq being the equilibrium contact angle. An exception to this occurs for θeq = 90°, when the chemical potential behaves like ln r/r. The diffusive and viscous contact line singularities implied by these outer solutions are resolved in the inner region through chemical diffusion. The length scale of the inner region is about 10√μκ – typically about 0.5–5 nm. Diffusive fluxes in this region are O(1). These counterbalance the effects of the velocity, which, because of the assumed no-slip boundary condition, fluxes material through the interface in a narrow boundary layer next to the wall.The asymptotic analysis is supplemented by both linearized and nonlinear finite difference calculations. These are made at two scales, experimental and nanoscale. The first set is done to show CHW interface behaviour and to test the qualitative applicability of the CHW model and its asymptotic theory to practical computations of experimental scale, nonlinear, low capillary number flows. The nanoscale calculations are carried out with realistic interface thicknesses and diffusivities and with various assumed levels of shear-induced slip. These are discussed in an attempt to evaluate the physical relevance of the CHW diffusive model. The various asymptotic and numerical results together indicate a potential usefullness for the CHW model for calculating and modelling wetting and dewetting flows.

Evolution of Liquid Meniscus Shape in a Capillary Tube

2007 ◽  
Vol 129 (8) ◽  
pp. 957-965 ◽  
Author(s):  
Shong-Leih Lee ◽  
Hong-Draw Lee
Keyword(s):  
Contact Angle ◽  
Contact Line ◽  
Capillary Rise ◽  
Stress Singularity ◽  
Dynamic Contact Angle ◽  
Dynamic Contact ◽  
Surface Flow ◽  
Force Balance ◽  
Liquid Meniscus ◽  
Meniscus Shape

There are still many unanswered questions related to the problem of a capillary surface rising in a tube. One of the major questions is the evolution of the liquid meniscus shape. In this paper, a simple geometry method is proposed to solve the force balance equation on the liquid meniscus. Based on a proper model for the macroscopic dynamic contact angle, the evolution of the liquid meniscus, including the moving speed and the shape, is obtained. The wall condition of zero dynamic contact angle is allowed. The resulting slipping velocity at the contact line resolves the stress singularity successfully. Performance of the present method is examined through six well-documented capillary-rise examples. Good agreements between the predictions and the measurements are observable if a reliable model for the dynamic contact angle is available. Although only the capillary-rise problem is demonstrated in this paper, the concept of this method is equally applicable to free surface flow in the vicinity of a contact line where the capillary force dominates the flow.

Kinetics of Low-Salinity-Flooding Effect

SPE Journal ◽  
2014 ◽  
Vol 20 (01) ◽  
pp. 8-20 ◽  
Author(s):  
H.. Mahani ◽  
S.. Berg ◽  
D.. Ilic ◽  
W.-B.. -B. Bartels ◽  
V.. Joekar-Niasar
Keyword(s):  
Contact Angle ◽  
Time Constant ◽  
Oil Recovery ◽  
Contact Line ◽  
Contact Area ◽  
Force Balance ◽  
Oil Droplets ◽  
Low Salinity ◽  
Fundamental Understanding ◽  
Kinetics Of

Summary Low-salinity waterflooding (LSF) is one of the least-understood enhanced-oil-recovery (EOR)/improved-oil-recovery (IOR) methods, and proper understanding of the mechanism(s) leading to oil recovery in this process is needed. However, the intrinsic complexity of the process makes fundamental understanding of the underlying mechanism(s) and the interpretation of laboratory experiments difficult. Therefore, we use a model system for sandstone rock of reduced complexity that consists of clay minerals (Na-montmorillonite) deposited on a glass substrate and covered with crude-oil droplets and in which different effects can be separated to increase our fundamental understanding. We focus particularly on the kinetics of oil detachment when exposed to low-salinity (LS) brine. The system is equilibrated first under high-salinity (HS) brine and then exposed to brines of varying (lower) salinity while the shape of the oil droplets is continuously monitored at high resolution, allowing for a detailed analysis of the contact angle and the contact area as a function of time. It is observed that the contact angle and contact area of oil with the substrate reach a stable equilibrium at HS brine and show a clear response to the LS brine toward less-oil-wetting conditions and ultimately detachment from the clay substrate. This behavior is characterized by the motion of the three-phase (oil/water/solid) contact line that is initially pinned by clay particles at HS conditions, and pinning decreases upon exposure to LS brine. This leads to a decrease in contact area and contact angle that indicates wettability alteration toward a more-water-wet state. When the contact angle reaches a critical value at approximately 40 to 50°, oil starts to detach from the clay. During detachment, most of the oil is released, but in some cases a small amount of oil residue is left behind on the clay substrate. Our results for different salinity levels indicate that the kinetics of this wettability change correlates with a simple buoyancy- over adhesion-force balance and has a time constant of hours to days (i.e., it takes longer than commonly assumed). The unexpectedly long time constant, longer than expected by diffusion alone, is compatible with an electrokinetic ion-transport model (Nernst-Planck equation) in the thin water film between oil and clay. Alternatively, one could explain the observations only by more-specific [non- Derjaguin–Landau–Verwey–Overbeek (DLVO) type] interactions between oil and clay such as cation-bridging, direct chemical bonds, or acid/base effects that tend to pin the contact line. The findings provide new insights into the (sub) pore-scale mechanism of LSF, and one can use them as the basis for upscaling to, for example, pore-network scale and higher scales (e.g., core scale) to assess the impact of the slow kinetics on the time scale of an LSF response on macroscopic scales.

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