Wetting Boundary Condition Research Articles

Normal capillary forces on a spherical particle protruding from a thin liquid film and its application to swarming bacteria

When a particle is placed in a liquid film thinner than its diameter, the liquid surface curves to satisfy its wetting boundary condition. We solve for the meniscus profile and the capillary force on the spherical object numerically as a function of film thickness, contact angle, particle diameter, liquid-vapor surface tension, and gravitational force. In contrast to previous work, our approach does not require the meniscus inclination angle at its origin as an input parameter. This angle is typically unknown for microscopic particles, while the film thickness can be often either experimentally controlled or measured. We use our method for calculating the capillary force on surface-associated bacteria living in a thin liquid film. We show that if the film is thinner than their diameter, the capillary force causes strong confinement and therefore plays a crucial role in the bacteria’s ability to move. This approach is useful in a wide range of disciplines, especially in which liquid film thickness is a natural parameter, such as in the self-assembly of macroscopic and microscopic particles.

Lattice Boltzmann simulation of drop splitting in a fractal tree-like microchannel

Fractal tree-like microchannel is advantageous to faster emulsification and droplet production in microchannels. Although computer simulation is becoming a powerful tool for design and optimization, precise treatment of a large number of walls in fractal tree-like microchannel is troublesome. The numerical roughness caused by traditional bounce-back boundary condition may accumulate and generate large errors in flow pattern recognization. In this paper, an approach integrating Immersed Boundary Method (IBM), Phase-Field Model (PFM) and Lattice Boltzmann Method (LBM) is developed, aiming to accurately simulate droplet splitting in fractal tree-like microchannel system involving surfactants and wetting boundary walls. Then the operating conditions including capillary number, flow rate ratio and wetting boundary conditions were optimized. We found that the capillary number in 0.02 ∼ 0.05 and flow rate ratio in 1/3.6 – 1/9 can accelerate emulsification. Hydrophobic-lipophilic walls generate a slug-like water drop of bullet shape and thin oil film between the drop and walls, facilitating drop movement in microchannel. The effects of surfactant, Marangoni stress and interfacial tension force on droplet splitting were investigated. The contact walls at the corner and forks resist the surfactant migration, and therefore enrich surfactant at the upstream end of drop interface, leading to uneven distribution of interfacial tension. The breaking interfaces and the interfaces contacting with wall or fork are subjected to opposite reaction of Marangoni stress and tend to be thinner and finally deform or break.

Numerical study of microfluidic emulsion dynamics under the influence of heterogeneous surface wettability

Surface wettability plays a vital role in various natural phenomena and technical applications, such as coating, printing, membrane technology, microfluidic devices, and porous media flow. We numerically investigate the dynamics and stability of liquid–liquid dispersion flowing in a microfluidic channel with heterogeneous surface wettability, using a conservative phase-field lattice Boltzmann method. First, we validate the implementation of the wetting boundary conditions. We then perform three-dimensional simulations of a single drop motion immersed in another immiscible liquid in a square microchannel with alternating surface wettability: hydrophilic, hydrophobic, and again hydrophilic sections. The viscosity of the dispersed liquid (oil) is ten times higher than that of the continuous one (water), while the two liquids have equal densities. Depending on the drop length and Capillary number, calculated based on the drop velocity and the dynamic viscosity of the continuous phase, four different flow patterns are observed when the drop passes through the hydrophobic section. The distinct flow patterns include passing without any changes in drop shape or dynamics, adhesion of the dispersed liquid to the walls, phase inversion (i.e., water becomes the dispersed phase), and drop breakage. These outcomes are in close agreement with the experimental data. A detailed explanation of the choice of the numerical parameters is presented. The critical aspect of capturing the observed drop behavior is introducing an intervening thin-film of the continuous phase of at least three diffuse interface thicknesses (between the drop and channel walls).

Development of a Phase-Field Method for Phase Change Simulations Using a Conservative Allen–Cahn Equation

Abstract Two-phase flows including a phase change such as liquid–vapor flows play an important role in many industrial applications. A deeper understanding of the phase change phenomena is required to improve the performance and safety of nuclear power plants. For this purpose, we developed a phase change simulation method based on the phase-field method (PFM). The low computational efficiency of the conventional PFM based on the Cahn–Hilliard equation is an obstacle in practical simulations. To resolve this problem, we presented a new PFM based on the conservative Allen–Cahn equation including a phase change model. The wettability also needs to be considered in the phase change simulation. When we apply the conventional wetting boundary condition to the conservative Allen–Cahn equation, there is a problem that the mass of each phase is not conserved on the boundary. To resolve this issue, we developed the mass correction method which enables mass conservation in the wetting boundary. The proposed PFM was validated in benchmark problems. The results agreed well with the theoretical solution and other simulation results, and we confirmed that this PFM is applicable to the two-phase flow simulation including the phase change. We also investigated the computational efficiency of the PFM. In a comparison with the conventional PFM, we found that our proposed PFM was more than 100 times faster. Since computational efficiency is an important factor in practical simulations, the proposed PFM will be preferable in many industrial simulations.

Phase-field-based lattice Boltzmann model for ternary fluid flows considering the wettability effect

A phase-field-based lattice Boltzmann model for ternary fluid flows is proposed, where the conservative Allen-Cahn equation for ternary fluids is used to track the interface of fluids. A time derivative term is introduced into the lattice Boltzmann equation for the conservative Allen-Cahn equation and a forcing distribution function is used in the lattice Boltzmann equation for the Navier-Stokes equations. The computational accuracy and capability of the proposed model are verified by benchmark cases related to stationary droplets, spreading of a liquid lens, and phase separation of a ternary fluid mixture. Moreover, a wetting boundary scheme based on the framework of the lattice Boltzmann method applicable to ternary fluid flows is developed. This model performs well in eliminating the undesired numerical artifact after consideration of the wettability effect. Typical wettability problems are simulated to verify the applicability of the wetting boundary scheme, including the spreading of binary droplets and compound droplets on a surface. The numerical results for static contact angles and spreading length show a good fit of the data to the theoretical results within the scope of permissible error. Furthermore, the reliability of this wetting boundary condition for the dynamic case is verified by our modeling the motion of compound droplets subjected to shear flow on a substrate. This boundary scheme performs well for the motion of ternary fluids with a small density ratio and a large density ratio. In conclusion, the phase-field-based lattice Boltzmann model for ternary fluid flows considering the wetting effect has good applicability for modeling ternary fluid flows with a density ratio of 1000.

Lattice Boltzmann simulation of three-phase flows with moving contact lines on curved surfaces.

A numerical method for simulating three-phase flows with moving contact lines on arbitrarily complex surfaces is developed in the framework of the lattice Boltzmann method. In this method, the immiscible three-phase flow is modeled through a multiple-relaxation-time color-gradient model, which not only allows for a full range of interfacial tensions but also produces stable outcomes for a wide range of viscosity ratios. A characteristic line model is introduced to implement the wetting boundary condition, which is not only easy to implement but is also able to handle arbitrarily complex boundaries with prescribed contact angles. The developed method is first validated by the simulation of a Janus droplet resting on a flat surface, a perfect Janus droplet deposited on a cylinder, and the capillary intrusion of ternary fluids for various viscosity ratios. It is then used to study a compound droplet subject to a uniform incoming flow passing through a multipillar structure, where three different values of surface wettability are considered. The simulated results show that the surface wettability has significant impact on the droplet dynamic behavior and final fluid distribution.

Alternative wetting boundary condition for the chemical-potential-based free-energy lattice Boltzmann model.

The free-energy lattice Boltzmann (LB) method is a multiphase LB approach based on the thermodynamic theory. Compared with traditional free-energy LB models, which employ a nonideal thermodynamic pressure tensor, the chemical-potential-based free-energy LB model has attracted much attention in recent years as it avoids computing the thermodynamic pressure tensor and its divergence. In this paper, we propose an improved wetting boundary condition for the chemical-potential-based free-energy LB model. Different from the original wetting boundary condition in the literature, the improved wetting boundary condition utilizes a surface chemical potential that is compatible with the chemical potential of the fluid domain. Accordingly, the thermodynamic consistency of the chemical-potential-based free-energy LB model can be retained by the improved wetting boundary condition. Numerical simulations are performed for droplets resting on flat and cylindrical surfaces with different contact angles. The numerical results show that the improved wetting boundary condition yields more reasonable results and the maximum spurious velocities are found to be smaller by 2 ∼ 3 orders of magnitude than those produced by the original wetting boundary condition.

Hybrid lattice-Boltzmann finite-difference simulation of ternary fluids near immersed solid objects of general shapes

In this paper, a hybrid lattice-Boltzmann finite-difference method is developed for the simulation of ternary fluids near immersed solid objects of general shapes. The flow equations are solved by the lattice-Boltzmann method and the coupled Cahn–Hilliard equations for interface evolutions are solved by the finite-difference method. A special implementation of the wetting boundary condition on a surface of general shapes immersed inside the domain was extended for ternary fluids within the phase-field framework with no need to use complicated interpolations. Several two and three dimensional problems with three immiscible fluids were studied by using the proposed method and the results agree well with analytical predictions and/or previous numerical and experimental studies. In particular, the inclusion of properly chosen free energy to handle total spreading enabled us to numerically reproduce the encapsulation of a small droplet by another bigger one of different component on a round fiber. The proposed method is expected to be useful to investigate a variety of multiphase problems involving ternary fluids and surfaces with different configurations, including the challenging total spreading regime.

Numerical study of thermocapillary migration behaviors of droplets on a grooved surface with a three-dimensional color-gradient lattice Boltzmann model

Thermocapillary actuation is used extensively in droplet-based microfluidic devices to manipulate the dynamic behaviors of droplets. In this study, a three-dimensional color-gradient lattice Boltzmann model is used to investigate the migration behaviors of droplets in the Wenzel state on a grooved surface that is subject to a uniform temperature gradient. On the solid surface, the wetting boundary condition is used to improve the accuracy of the simulations and to suppress spurious velocities at the contact line. The model is used to simulate the thermocapillary migration of a three-dimensional deformable droplet and the thermocapillary migration of a two-dimensional droplet on a solid substrate, and its accuracy is verified against theoretical predictions. The migration behavior of droplets on a smooth surface is investigated, and the flow field and corresponding temperature field around the droplets are analyzed. The experimental findings numerically confirm that a surface with micro-grooves parallel to the temperature gradient can accelerate thermocapillary migration to a greater extent than a smooth surface, indicating the influence of the grooves. The influence of the viscosity ratio is investigated, and it is found that the use of high-viscosity fluids is an effective means of obstructing migration. To determine the influence of surface roughness, a systematic and parametric study of groove depth and width is conducted. Finally, the influence of the orientation of the surface topography is investigated, and it is demonstrated that a surface with micro-grooves perpendicular to the temperature gradient can obstruct migration.

A thermodynamically consistent gradient theory for diffusion–reaction–deformation in solids: Application to conversion-type electrodes

We develop a thermodynamically consistent phase-field finite strain theory for problems in solids mechanics that couple transport of species into a host material, sharp interface reactions of the species with the host, mechanical deformation and stress. The theory distinguishes between diffusion-limited and reaction-limited kinetics, resolving the manner in which a sharp reaction front can be developed in either case. The phase field formulation has the added benefit of enabling the application of wetting (surface energy) boundary conditions which are critical in reproducing experimentally relevant reaction front morphologies. The theory is fully coupled with diffusion and reaction phenomena impacting mechanical deformation and subsequent stress generation, and conversely these phenomena are coupled to mechanical stress. We derive thermodynamically consistent driving forces for diffusion and reaction through a continuum treatment of these phenomena. Importantly, the resulting formulation makes precise the nature of the material properties driving these thermodynamic forces and in turn makes it amenable to being specialized and calibrated for application.While the framework is quite general, we apply it to modeling conversion electrodes for energy storage using a three-dimensional finite element implementation. We demonstrate the manner in which the theory can be specialized and calibrated in straightforward fashion. Simulations are performed of chemical reactions of FeS2 crystals with lithium and sodium ions, both of which proceed through the formation and propagation of a sharp interface, and are compared to experimental observations of the same system. Our simulations show good qualitative agreement with experimental observations, and elucidate the critical role mechanics plays in determining the morphology of the sharp reaction interface and subsequent stress generation which can lead to mechanical deterioration of these materials. Beyond this application, the theoretical framework should serve useful in a number of engineering problems of relevance in which diffusion and sharp interface reactions occur.

Numerical study of drop behavior in a pore space

Deformation and breakup of a liquid drop immersed in another immiscible liquid and flowing through a single pore has been studied numerically using a conservative phase-field lattice Boltzmann method. Several benchmarks were conducted to validate the code, including the recovery of Laplace pressure, the layered flow of two immiscible liquids, and the implementation of wetting boundary conditions on a curved surface. Gravity-driven motion of a drop through the pore space was qualitatively compared to the available experimental results. Quantitative assessment of the pressure field across the interface of the moving and deforming drop was performed. Our results show that high Weber number due to low surface tension and low Reynolds number due to low velocity of the continuous liquid promote drop breakage. More viscous drops break easier than less viscous drops. We present the phase charts (Weber vs capillary number) and the critical conditions (Weber as a function of Reynolds number) of drop breakage.

A unified lattice Boltzmann model for immiscible and miscible ternary fluids

Based on the phase-field theory, we develop a unified lattice Boltzmann model for ternary flow, in which the miscibility between the three fluid components can be adjusted independently. Based on a modified free energy function, we derive the conservative phase-field equations using the gradient flow method. A generalized continuous surface tension force formulation is deduced by using the virtual work method. The wetting boundary condition is derived based on mass conservation law. The proposed model for ternary fluids is consistent with the binary-fluid models in the absence of one fluid. A lattice Boltzmann (LB) model is developed to solve the phase-field equations and hydrodynamic equations, and this model can deal with problems involving high density and viscosity contrasts. The proposed method is examined through several test cases. A layered Poiseuille flow and droplet coalescence problems are carried out to validate the present LB model. Several dynamic problems in ternary fluid problems involving a solid are simulated, including the wetting of two droplets on a circular cylinder and impacting of a multiphase droplet on a fixed particle. Finally, we apply the model to a three-dimensional multi-bubbles rising problem to access its numerical accuracy and stability.

Lattice Boltzmann model for ternary fluids with solid particles

On the basis of phase-field theory, we develop a lattice Boltzmann model for ternary fluids containing solid. We develop a modified bounce-back method to describe the interactions between the solid and N-phase (N≥2) fluids. We derive a wetting boundary condition for three-phase flows from the point of mass conservation and propose a scheme for implementing the wetting condition on curved boundaries. We develop a diffuse interface method to compute the capillary force acting on the moving solid objects at the ternary fluids-sold contact lines. In addition, this model can deal with problems involving high density and viscosity contrasts. The proposed method is examined through several test cases. We test the modified bounce-back scheme, wetting boundary condition, and capillary force model in three different cases, and the numerical results agree well with the analytical solutions. Finally, we apply the model to two three-dimensional problems to assess its numerical accuracy and stability.

Improvement of Three-Dimensional Curved Wetting Boundary Condition on Two-Phase LBM

Improvement of Three-Dimensional Curved Wetting Boundary Condition on Two-Phase LBM

Multiple-relaxation time color-gradient lattice Boltzmann model for simulating contact angle in two-phase flows with high density ratio

The contact line dynamics is an important phenomenon in natural and industrial processes and is still a challenging problem. The complexity of this problem in computational simulations is much more than theoretical and experimental works. In this paper, a color-gradient (CG) lattice Boltzmann method (LBM) for simulating wetting phenomena and the dynamics of contact line in two-phase flows with very high density-ratio (∼ 1000 was used. This method applies a multi-relaxation time (MRT) collision operator to enhance the stability of numerical scheme. Both static and dynamic contact angles were enforced at the wall through geometrical wetting boundary condition. Note, the main nobility of this paper is the supplementation of geometrical wetting boundary condition to the color gradient LBM and applying the present computational methodology for two-phase flow physics with a very high density ratio. This method is first validated by simulating a stationary drop and static contact angles. Also, the dynamical behavior of a drop on an ideal surface in shear flow was computationally validated. Finally, simulation of a drop motion subjected to gravitational force was the fourth test case studied. According to the results of these test cases, small values of spurious velocities and equilibrium contact angle errors were found in the simulations of steady cases studied. While, in our unsteady test cases, the behavior of the interface shapes and contact angles were in good agreements with previous reliable studies.

A phase-field-based lattice Boltzmann method for moving contact line problems on curved stationary boundaries in two dimensions

In this work, we propose a phase-field-based lattice Boltzmann method to simulate moving contact line (MCL) problems on curved boundaries. The key point of this method is to implement the boundary conditions on curved solid boundaries. Specifically, we use our recently proposed single-node scheme for the no-slip boundary condition and a new scheme is constructed to deal with the wetting boundary conditions (WBCs). In particular, three kinds of WBCs are implemented: two wetting conditions derived from the wall free energy and a characteristic MCL model based on geometry considerations. The method is validated with several MCL problems and numerical results show that the proposed method has utility for all the three WBCs on both straight and curved boundaries.

Lattice Boltzmann modeling of wall-bounded ternary fluid flows

• A novel wetting boundary scheme in the lattice Boltzmann method is proposed for ternary fluids. • The proposed scheme can preserve the reduction consistency property of the diphasic case. • The wetting scheme is demonstrated to be accurate for a wide range of contact angles. • Fascinating interfacial phenomena are found in simulations of compound drop impact dynamics. Starting from the phase-field perspective, we first formulate a novel wetting boundary condition to describe the interactions among ternary fluids and a solid and then we propose a boundary scheme for its implementation in the framework of the lattice Boltzmann (LB) method. This scheme for three-phase fluids can preserve the reduction consistency property of the diphasic case such that it can give physically relevant results. Combining this wetting boundary scheme and the LB ternary fluid model based on multicomponent phase-field theory, we simulated several ternary fluid flow problems involving a solid substrate, including the spreading of binary drops on a substrate, the spreading of a compound drop on a substrate, the capillary intrusion of ternary fluids, and the shear of a compound liquid drop on a substrate. The numerical results are found to be good agreement with the analytical solutions and some available results. Finally, as an application, we use the LB model coupled with the present wetting boundary scheme to numerically investigate the impact of a compound drop on a solid circular cylinder. It is found that the dynamics of a compound drop can be remarkably influenced by the wettability of the solid surface and the dimensionless Weber number .

A linear, second-order, energy stable, fully adaptive finite element method for phase-field modelling of wetting phenomena

We propose a new numerical method to solve the Cahn-Hilliard equation coupled with non-linear wetting boundary conditions. We show that the method is mass-conservative and that the discrete solution satisfies a discrete energy law similar to the one satisfied by the exact solution. We perform several tests inspired by realistic situations to verify the accuracy and performance of the method: wetting of a chemically heterogeneous substrate in three dimensions, wetting-driven nucleation in a complex two-dimensional domain and three-dimensional diffusion through a porous medium.

A linear, second-order, energy stable, fully adaptive finite-element method for phase-field modelling of wetting phenomena

We propose a new numerical method to solve the Cahn–Hilliard equation coupled with non-linear wetting boundary conditions. We show that the method is mass-conservative and that the discrete solution satisfies a discrete energy law similar to the one satisfied by the exact solution. We perform several tests inspired by realistic situations to verify the accuracy and performance of the method: wetting of a chemically heterogeneous substrate in three dimensions, wetting-driven nucleation in a complex two-dimensional domain and three-dimensional diffusion through a porous medium.

Modeling Oil Recovery in Mixed-Wet Rocks: Pore-Scale Comparison Between Experiment and Simulation

To examine the need to incorporate in situ wettability measurements in direct numerical simulations, we compare waterflooding experiments in a mixed-wet carbonate from a producing reservoir and results of direct multiphase numerical simulations using the color-gradient lattice Boltzmann method. We study the experiments of Alhammadi et al. (Sci Rep 7(1):10753, 2017. https://doi.org/10.1038/s41598-017-10992-w) where the pore-scale distribution of remaining oil was imaged using micro-CT scanning. In the experiment, in situ contact angles were measured using an automated algorithm (AlRatrout et al. in Adv Water Resour 109:158–169, 2017. https://doi.org/10.1016/j.advwatres.2017.07.018), which indicated a mixed-wet state with spatially non-uniform angles. In our simulations, the pore structure was obtained from segmented images of the sample used in the experiment. Furthermore, in situ measured angles were also incorporated into our simulations using our previously developed wetting boundary condition (Akai et al. in Adv Water Resour 116(March):56–66, 2018. https://doi.org/10.1016/j.advwatres.2018.03.014). We designed six simulations with different contact angle assignments based on experimentally measured values. Both a constant contact angle based on the average value of the measured values and non-uniform contact angles informed by the measured values gave a good agreement for fluid pore occupancy between the simulation and the experiment. However, the constant contact angle assignment predicted 54% higher water effective permeability after waterflooding than that estimated for the experimental result, whereas the non-uniform contact angle assignment gave less than 1% relative error. This means that to correctly predict fluid conductivity in mixed-wet rocks, a spatially heterogeneous wettability state needs to be taken into account. The novelty of this work is to provide a direct pore-scale comparison between experiments and simulations employing experimentally measured contact angles, and to demonstrate how to use measured contact angle data to improve the predictability of direct numerical simulation, highlighting the difference between the contact angle required for the simulation of dynamic displacement process and the contact angle measured at equilibrium after waterflooding.