Wetting Boundary Condition Research Articles

Numerical modelling of moisture transfers with hysteresis within cementitious materials: Verification and investigation of the effects of repeated wetting–drying boundary conditions

In natural environment, the cover layer of reinforced concrete structures is affected by periodic variations of external relative humidity (RH). However, most moisture transport models in the literature only focus on drying of materials. In this study, a method coupling a moisture transport model with any kind of hysteresis modelling is presented. Two hysteresis models (conceptual and empirical) have been implemented and compared. The scope of the study is limited to cyclic variations of RH with no direct contact with liquid water during the wetting steps. Experimental data verifications show that the conceptual approach yields better results than the empirical one. Comparisons of non-hysteresis and hysteresis modellings have been carried out for different cycle durations, RH amplitudes and initial moisture states. All comparisons and investigations enhance the necessity of considering hysteresis to quantify moisture transport under repeated drying–wetting boundary conditions.

Modelling thermocapillary migration of a microfluidic droplet on a solid surface

A multiphase lattice Boltzmann model is developed to simulate immiscible thermocapillary flows with the presence of fluid–surface interactions. In this model, interfacial tension force and Marangoni stress are included by introducing a body force term based on the concept of continuum surface force, and phase segregation is achieved using the recolouring algorithm proposed by Latva-Kokko and Rothman. At a solid surface, fluid–surface interactions are modelled by a partial wetting boundary condition that uses a geometric formulation to specify the contact angle, and a colour-conserving boundary closure scheme to improve the numerical accuracy and suppress spurious velocities at the contact line. An additional convection–diffusion equation is solved by the passive scalar approach to obtain the temperature field, which is coupled to the hydrodynamic equations through an equation of state. This model is first validated by simulations of static contact angle and dynamic capillary intrusion process when a constant interfacial tension is considered. It is then used to simulate the thermocapillary migration of a microfluidic droplet on a horizontal solid surface subject to a uniform temperature gradient. We for the first time demonstrate numerically that the droplet motion undergoes two different states depending on the surface wettability: the droplet migrates towards the cooler regions on hydrophilic surfaces but reverses on hydrophobic surfaces. Decreasing the viscosity ratio can enhance the intensity of thermocapillary vortices, leading to an increase in migration velocity. The contact angle hysteresis, i.e., the difference between the advancing and receding contact angles, is always positive regardless of the contact angle and viscosity ratio. The contact angle hysteresis and the migration velocity both first decrease and then increase with the contact angle, and their minimum values occur at the contact angle of 90 degrees.

Numerical study of drop motion on a surface with stepwise wettability gradient and contact angle hysteresis

In this work, the motion of a two-dimensional drop on a surface with stepwise wettability gradient (WG) is studied numerically by a hybrid lattice-Boltzmann finite-difference method. We incorporate the geometric wetting boundary condition that allows accurate implementation of a contact angle hysteresis (CAH) model. The method is first validated through a series of tests that check different constituents of the numerical model. Then, simulations of a drop on a wall with given stepwise WG are performed under different conditions. The effects of the Reynolds number, the viscosity ratio, the WG, as well as the CAH on the drop motion are investigated in detail. It was discovered that the shape of the drop in steady motion may be fitted by two arcs that give two apparent contact angles, which are related to the respective contact line velocities and the relevant contact angles (that specify the WG and CAH) through the relation derived by Cox [“The dynamics of the spreading of liquids on a solid surface. Part 1. viscous flow,” J. Fluid Mech. 168, 169–194 (1986)] if the slip length in simulation is defined according to Yue et al. [“Sharp-interface limit of the Cahn-Hilliard model for moving contact lines,” J. Fluid Mech. 645, 279–294 (2010)]. It was also found that the steady capillary number of the drop is significantly affected by the viscosity ratio, the magnitudes of the WG, and the CAH, whereas it almost shows no dependence on the Reynolds number.

Tear film dynamics with evaporation, wetting, and time-dependent flux boundary condition on an eye-shaped domain.

We study tear film dynamics with evaporation on a wettable eye-shaped ocular surface using a lubrication model. The mathematical model has a time-dependent flux boundary condition that models the cycles of tear fluid supply and drainage; it mimics blinks on a stationary eye-shaped domain. We generate computational grids and solve the nonlinear governing equations using the OVERTURE computational framework. In vivo experimental results using fluorescent imaging are used to visualize the influx and redistribution of tears for an open eye. Results from the numerical simulations are compared with the experiment. The model captures the flow around the meniscus and other dynamic features of human tear film observed in vivo.

Lattice Boltzmann simulations of multiple-droplet interaction dynamics.

A lattice Boltzmann (LB) formulation, which is consistent with the phase-field model for two-phase incompressible fluid, is proposed to model the interface dynamics of droplet impingement. The interparticle force is derived by comparing the macroscopic transport equations recovered from LB equations with the governing equations of the continuous phase-field model. The inconsistency between the existing LB implementations and the phase-field model in calculating the relaxation time at the phase interface is identified and an approximation is proposed to ensure the consistency with the phase-field model. It is also shown that the commonly used equilibrium velocity boundary for the binary fluid LB scheme does not conserve momentum at the wall boundary and a modified scheme is developed to ensure the momentum conservation at the boundary. In addition, a geometric formulation of the wetting boundary condition is proposed to replace the popular surface energy formulation and results show that the geometric approach enforces the prescribed contact angle better than the surface energy formulation in both static and dynamic wetting. The proposed LB formulation is applied to simulating droplet impingement dynamics in three dimensions and results are compared to those obtained with the continuous phase-field model, the LB simulations reported in the literature, and experimental data from the literature. The results show that the proposed LB simulation approach yields not only a significant speed improvement over the phase-field model in simulating droplet impingement dynamics on a submillimeter length scale, but also better accuracy than both the phase-field model and the previously reported LB techniques when compared to experimental data. Upon validation, the proposed LB modeling methodology is applied to the study of multiple-droplet impingement and interactions in three dimensions, which demonstrates its powerful capability of simulating extremely complex interface phenomena.

Simulation of Droplet Resting on Micro-structured Surface by Lattice Boltzmann Method

Three-dimensional liquid droplet resting on micro-structured pillar surface is simulated with lattice Boltzmann method incorporat- ing the Cahn-Hilliard equation. This is based on the two-phase model of Zheng et al. [14] combined with partial wetting boundary condition of Briant et al. [12]. Two Simulations are conducted with the same area fractions but different pillar heights, i.e. surface roughness, which is shown to have minor influence on the apparent contact angles at the Cassie state investigated. The predicted apparent contact angles are contrasted with measurements. Predictions show lower apparent contact angles and the deviations are around 10%. This might be due to the inadequate treatment of wetting boundary condition of the pillar surface, and this is deserved to be further studied.

Color-gradient lattice Boltzmann model for simulating droplet motion with contact-angle hysteresis

Lattice Boltzmann method (LBM) is an effective tool for simulating the contact-line motion due to the nature of its microscopic dynamics. In contact-line motion, contact-angle hysteresis is an inherent phenomenon, but it is neglected in most existing color-gradient based LBMs. In this paper, a color-gradient based multiphase LBM is developed to simulate the contact-line motion, particularly with the hysteresis of contact angle involved. In this model, the perturbation operator based on the continuum surface force concept is introduced to model the interfacial tension, and the recoloring operator proposed by Latva-Kokko and Rothman is used to produce phase segregation and resolve the lattice pinning problem. At the solid surface, the color-conserving wetting boundary condition [Hollis et al., IMA J. Appl. Math. 76, 726 (2011)] is applied to improve the accuracy of simulations and suppress spurious currents at the contact line. In particular, we present a numerical algorithm to allow for the effect of the contact-angle hysteresis, in which an iterative procedure is used to determine the dynamic contact angle. Numerical simulations are conducted to verify the developed model, including the droplet partial wetting process and droplet dynamical behavior in a simple shear flow. The obtained results are compared with theoretical solutions and experimental data, indicating that the model is able to predict the equilibrium droplet shape as well as the dynamic process of partial wetting and thus permits accurate prediction of contact-line motion with the consideration of contact-angle hysteresis.

Lattice Boltzmann phase-field modeling of thermocapillary flows in a confined microchannel

To understand how thermocapillary forces manipulate the droplet motion in a confined microchannel, a lattice Boltzmann phase-field model is developed to simulate immiscible thermocapillary flows with consideration of fluid–surface interactions. Based on our recent work of Liu et al., 2013 [54], an interfacial force of potential form is proposed to model the interfacial tension force and the Marangoni stress. As only the first-order derivatives are involved, the proposed interfacial force is easily combined with the wetting boundary condition to account for fluid–surface interactions. The hydrodynamic equations are solved using a multiple-relaxation-time algorithm with the interfacial force treated as a forcing term, while an additional convection–diffusion equation is solved by a passive-scalar approach to obtain the temperature field, which is coupled to the interfacial tension by an equation of state. The model is first validated against analytical solutions for the thermocapillary-driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then demonstrated to produce the correct equilibrium contact angle for a binary fluid with different viscosities when a constant interfacial tension is taken into account. Finally, we numerically simulate the thermocapillary flows for a microfluidic droplet adhering on a solid wall and subject to a simple shear flow when a laser is applied to locally heat the fluids, and investigate the influence of contact angle and fluid viscosity ratio on the droplet dynamical behavior. The droplet motion can be completely blocked provided that the contact angle exceeds a threshold value, below which the droplet motion successively undergoes four stages: constant velocity, deceleration, acceleration, and approximately constant velocity. When the droplet motion is completely blocked, three steady vortices are clearly visible, and the droplet is fully filled by two counter-rotating vortices with the smaller one close to the external vortex. The thermocapillary convection is strengthened with decreasing viscosity ratio of the droplet to the carrier fluid. For low viscosity ratios, the droplet motion is completely blocked and exhibits the similar behavior, but the structure of the internal vortices is more complicated at the lowest viscosity ratio. For high viscosity ratios, the droplet motion is partially blocked and undergoes a series of complex transitions, which can be explained as a result of the dynamically varying Marangoni forces.

Investigation of Lattice Boltzmann wetting boundary conditions for capillaries with irregular polygonal cross-section

We have investigated the performance of an alternative wetting boundary condition for complex geometries in a phase field Lattice Boltzmann scheme, which is an alternative to the commonly used formulation by Yeomans and coworkers. Though our boundary condition is much simpler in its implementation, all investigated schemes show proper droplet spreading behaviour following the Cox–Voinov law. Still, numerical artefacts like spurious velocities or chequer board effects in the pressure field can be significantly reduced by the use of a two-relaxation-time (TRT) scheme, likewise recent studies by the Yeomans group. The outstanding property of our implementation is the presence of an (artificial) thin wetting layer, which influences the relation between the saturation (Sw) and capillary pressure pcap in channels with irregular polygonal cross section. The pcap(Sw) relation from our simulation follows the shifted-Young–Laplace (sYL) law, showing that the physics of this wetting layer is similar to precursor films due to Van der Waals forces. With the knowledge of the thickness of the wetting layer, simulation results can be translated back to realistic pore configurations with thinner wetting layers.

A HYBRID PHASE-FIELD BASED LATTICE BOLTZMANN METHOD FOR CONTACT LINE DYNAMICS

A hybrid phase-field based lattice Boltzmann method (LBM) is proposed in this paper to simulate the contact line dynamics. The flow field is obtained through the lattice Boltzmann equation (LBE). Concurrently, the interface capturing is accomplished by directly solving Cahn-Hilliard equation, which is the governing equation of interface evolution. A symmetric spatial discretization scheme is adopted to enhance the stability. Compared with the conventional algorithms which solve two sets of LBEs, the present method has several advantages such as reduction of the number of variables in the solution process, decoupling the mobility with relaxation time and enabling a more direct manner to implement wetting boundary conditions. The proposed algorithm is first validated through recovering the analytical profile of a surface layer. It is then applied to simulate droplet spreading on surfaces with different wettability.

A Lattice Boltzmann Simulation of Contact Line Motion and Internal Fluid Flows in an Evaporating Droplet on Homogenous Substrates

A two-dimensional lattice Boltzmann model is constructed to simulate the evaporation of a single droplet on a homogenous wetting substrate. An evaporation model is used in the simulation to describe the mass loss from the surface of the evaporating droplet. In addition, a partial wetting boundary condition is imposed on the lattice sites on the substrate surface to determine the wettability of the substrate. The evaporation of a single droplet in ambient vapor shows a linear decrease in the square of the droplet radius, which agrees well with the d2 law for an evaporating droplet. During the evaporation of a droplet on a hydrophobic substrate, the wetting diameter decreases while the contact angle remains constant. During the evaporation of a droplet on a hydrophilic substrate, the wetting diameter decreases while the contact angle remains almost constant when a partial wetting boundary condition is imposed. However, three stages of contact line motion are observed when the partial wetting boundary condition is modified to describe the dynamical change in the contact angle, and this observation agrees well with a previous study by other investigtors. In addition, internal fluid flows towards the contact line of the droplet in the pinning stage of the evaporation are discussed.

Dissipation in rapid dynamic wetting

In this article, we present a modelling approach for rapid dynamic wetting based on the phase field theory. We show that in order to model this accurately, it is important to allow for a non-equilibrium wetting boundary condition. Using a condition of this type, we obtain a direct match with experimental results reported in the literature for rapid spreading of liquid droplets on dry surfaces. By extracting the dissipation of energy and the rate of change of kinetic energy in the flow simulation, we identify a new wetting regime during the rapid phase of spreading. This is characterized by the main dissipation to be due to a re-organization of molecules at the contact line, in a diffusive or active process. This regime serves as an addition to the other wetting regimes that have previously been reported in the literature.

Boundary condition considerations in lattice Boltzmann formulations of wetting binary fluids

We propose a new lattice Boltzmann numerical scheme for binary-fluid surface interactions. The new scheme combines the existing binary free energy lattice Boltzmann method [Swift et al., Phys. Rev. E 54 (1996)] and a new wetting boundary condition for diffuse interface methods in order to eliminate spurious variations in the order parameter at solid surfaces. We use a cubic form for the surface free energy density and also take into account the contribution from free energy in the volume when discretizing the wetting boundary condition. This allows us to eliminate the spurious variation in the order parameter seen in previous implementations. With the new scheme a larger range of equilibrium contact angles are possible to reproduce and capillary intrusion can be simulated at higher accuracy at lower resolution.

Dynamic wetting boundary condition for continuum hydrodynamics with multi-component lattice Boltzmann equation simulation method

We present methodological innovations to the multicomponent lattice Boltzmann equation simulation method of Lishchuk, Care and Halliday (2003, Phys. Rev. E, 67, 036701) which allow for the simulation of dynamic contact lines (DCLs) in the continuum approximation. The scope of the improvements is demonstrated by examples validated by quantitative results. Our innovations allow the simulator access to an expanded range of simulation parameters like viscosity, viscosity contrast and interfacial tensions, and to obtain data with low levels of interfacial micro-current activity, in the region of the DCL.

濡れ性を考慮した多孔質構造内の気液二相流シミュレーション

Gas-liquid two-phase flows in porous structures are simulated by the lattice Boltzmann method for two-phase fluids with large density differences, in which the wetting boundary condition on solid walls is incorporated. The behavior of the liquid phase penetrating the hydrophobic and hydrophilic microchannels is investigated. The curvature radius and the average height of the liquid phase in equilibrium are calculated, and found to be in good agreement with the theoretical predictions. In addition, the behavior of the liquid phase in porous structures with various porosities is simulated for hydrophilic and hydrophobic cases. From these results, it is found that the liquid phase in the porous structure is transported from the hydrophobic to hydrophilic regions, and that the liquid tends to penetrate in locally narrow spaces owing to the effect of large pressure gradient.

Numerical simulation of dynamic behavior of droplet on solid surface by the two-phase lattice Boltzmann method

The dynamic behavior of a droplet on a solid surface is simulated by the lattice Boltzmann method (LBM) for two-phase fluids with large density differences; the wetting boundary condition on solid walls is incorporated in this simulation. By using the method, the dynamic behavior of a droplet impinging on a horizontal wall is investigated in terms of various Weber numbers. The dynamic contact angle, the contact line velocity, and the wet length are calculated, and found to be in good agreement with available experimental data. In addition, the method is applied to simulations of the collision of a falling droplet with a stationary droplet on a solid surface. The behavior of the droplets and the mixing process during their collision are simulated in terms of various impact velocities and several static contact angles on the solid surface. It is seen that mixing occurs around the rim of the coalescent droplet due to the circular flows. Also, the relationship between the mixing rate of the primary coalescent droplet and Weber number is investigated.

Modal analysis of liquid sloshing with different contact line boundary conditions using FEM

A finite element method (FEM) for liquid sloshing modal analysis is established. Surface tension and three kinds of contact line boundary conditions, namely, free-end, pin-end and wetting boundary conditions, are taken into account. Sloshing damping caused by energy dissipation at the wall, in the interior fluid and at the contact line is calculated. Numerical results are compared with the analytical values and measurements. For the pin-end and free-end boundary conditions the differences between numerical value and analytical value are small, and for the wetting boundary condition, because approximation is used, the differences are more significant.

Lattice Boltzmann simulation of liquid–gas flows through solid bodies in a square duct

The lattice Boltzmann method for two-phase immiscible fluids with large density differences proposed by Inamuro et al. [T. Inamuro, T. Ogata, S. Tajima, N. Konishi, A lattice Boltzmann method for incompressible two-phase flows with large density differences, J. Comput. Phys. 198 (2004) 628–644] is applied to the problem of liquid–gas flows through solid bodies in a square duct. A wetting boundary condition is introduced so that partial wetting on solid surfaces is realized to agree with Cahn theory. Using this method, we investigate the characteristics of wettability in terms of dynamic contact angles between two fluids and a solid wall. Also, we carry out simulations of liquid–gas rising flows through solid bodies in a square duct. It is found from these simulations that the present method can be useful for the problems of liquid–gas flows through complicated geometries.

Wetting of a particle in a thin film

When a particle is placed in a thin liquid film on a planar substrate, the liquid either climbs or descends the particle surface to satisfy its wetting boundary condition. Analytical solutions for the film shape, the degree of particle immersion, and the downward force exerted by the wetting meniscus on the particle are presented in the limit of small Bond number. When line tension is significant, multiple solutions for the equilibrium meniscus position emerge. When the substrate is unyielding, a dewetting transition is predicted; that is, it is energetically favorable for the particle to rest on top of the film rather than remain immersed in it. If the substrate can bend, the energy to drive this bending is found in the limits of slow or rapid solid deflection. These results are significant in a wide array of disciplines, including controlled delivery of drugs to pulmonary airways, the probing of liquid film/particle interface properties using particles affixed to AFM tips and the positioning of small particles in thin films to create patterned media.

Theoretical investigation of crystal growth shaping process with the wetting boundary condition

A theoretical investigation of crystal growth shaping process (to elaborate crystals in the form of tubes or rods with different cross sections, and other complicated forms) is carried out on the basis of the dynamic stability concept. The capillary dynamic stability of shaped crystal growth from the melt is analyzed using a mathematical model based on the proposal of axisymmetry of crystal and setup geometry. The study is carried out for the different conditions of growth and various configurations of melt meniscus. We study shapers with complex geometry for which a wetting boundary condition has to be considered. A general method is proposed to design the shaper geometry that satisfies the capillary stability conditions of the melt meniscus during the whole crystal growth process. The static stability of the liquid-free surface is analyzed by means of the Jacobi equation.