Scheme for contact angle and its hysteresis in a multiphase lattice Boltzmann method

Implementation of contact line motion based on the phase-field lattice Boltzmann method

This paper proposes a strategy to implement the free-energy-based wetting boundary condition within the phase-field lattice Boltzmann method. The greatest advantage of the proposed method is that the implementation of contact line motion can be significantly simplified while still maintaining good accuracy. For this purpose, the liquid-solid free energy is treated as a part of the chemical potential instead of the boundary condition, thus avoiding complicated interpolations with irregular geometries. Several numerical testing cases, including droplet spreading processes on the idea flat, inclined, and curved boundaries, are conducted, and the results demonstrate that the proposed method has good ability and satisfactory accuracy to simulate contact line motions.

Simplified wetting boundary scheme in phase-field lattice Boltzmann model for wetting phenomena on curved boundaries

In this work, a simplified wetting boundary scheme in the phase-field lattice Boltzmann model is developed for wetting phenomena on curved boundaries. The proposed scheme combines the advantages of the fluid-solid interaction scheme and geometric scheme-easy to implement (no need to interpolate the values of parameters exactly on solid boundaries and find proper characteristic vectors), the value of contact angle can be directly prescribed, and no unphysical spurious mass layer-and avoids mass leakage. Different from previous works, the values of the order parameter gradient on fluid boundary nodes are directly determined according to the geometric formulation rather than indirectly regulated through the order parameters on ghost solid nodes (i.e., ghost contact-line region). For this purpose, two numerical approaches to evaluate the order parameter gradient on fluid boundary nodes are utilized, one with the prevalent isotropic central scheme and the other with a local gradient scheme that utilizes the distribution functions. The simplified wetting boundary schemes with both numerical approaches are validated and compared through several numerical simulations. The results demonstrate that the proposed model has good ability and satisfactory accuracy to simulate wetting phenomena on curved boundaries under large density ratios.

An Analytical Model for Hysteretic Pressure-Sensitive Permeability of Nanoporous Media

Hysteretic pressure-sensitive permeability of nanohybrids composed of substantial nanopores is critical to characterizing fluid flow through nanoporous media. Due to the nanoscale effect (gas slippage), complex and heterogeneous pore structures of nanoporous media, the essential controls on permeability hysteresis of nanohybrids are not determined. In this study, a hysteretic pressure sensitive permeability model for nitrogen flow through dry nanoporous media is proposed. The derived model takes into account the nanoscale effect and pore deformation due to effective stress. The model is validated by comparing it with the experimental data. The results show that the calculated permeability and porosity are consistent with the measured results with the maximum relative error of 6.08% and 0.5%, respectively. Moreover, the hysteretic pressure-sensitive permeability of nanohybrids is related to effective stress, gas slippage, pore microstructure parameters, grain quadrilateral angle, and the loss rate of grain quadrilateral angle. The nanoscale effect is crucial to the permeability of nanoporous media. In addition, as impacted by the comprehensive impact of multiple relevant influential parameters, permeability during the pressure unloading process is not a monotonous function but presents complicated shapes. The proposed model can explain, quantify, and predict the permeability hysteresis effect of nanoporous media reasonably well.

Wicking fingering in electrospun membranes

Pronounced fingering of the waterfront is observed for in-plane wicking in thin, aligned electrospun fibrous membranes. We hypothesize that a perturbation in capillary pressure triggers the onset of fingering, which grows in a non-local manner based on the waterfront gradient. Vertical and horizontal wicking in thin electrospun membranes of poly(ethylene-co-vinyl alcohol) (EVOH) fibers with varying fiber alignment and degree of orientation is studied with backlight photography. A non-local transport model considering the gradient of the waterfront is developed, where fiber orientation is modeled with a correlated random field. The model shows that a transition from straight to highly fingered waterfront occurs during water uptake as observed in the experiment. The size and shape of the fingers depend on fiber orientation. Based on good model agreement, we show that, during wicking in thin electrospun membranes, fingering is initially triggered by a perturbation in capillary pressure caused by the underlying anisotropic and heterogeneous membrane structure which grows in a non-local manner depending on the waterfront gradient.

Unified lattice Boltzmann method with improved schemes for multiphase flow simulation: Application to droplet dynamics under realistic conditions

As a powerful mesoscale approach, the lattice Boltzmann method (LBM) has been widely used for the numerical study of complex multiphase flows. Recently, Luo et al. [Philos. Trans. R. Soc. A: Math. Phys. Eng. Sci. 379, 20200397 (2021)10.1098/rsta.2020.0397] proposed a unified lattice Boltzmann method (ULBM) to integrate the widely used lattice Boltzmann collision operators into a unified framework. In this study, we incorporate additional features into this ULBM in order to simulate multiphase flow under realistic conditions. A nonorthogonal moment set [Fei et al., Phys. Rev. E 97, 053309 (2018)10.1103/PhysRevE.97.053309] and the entropic-multi-relaxation-time (KBC) lattice Boltzmann model are used to construct the collision operator. An extended combined pseudopotential model is proposed to realize multiphase flow simulation at high-density ratio with tunable surface tension over a wide range. The numerical results indicate that the improved ULBM can significantly decrease the spurious velocities and adjust the surface tension without appreciably changing the density ratio. The ULBM is validated through reproducing various droplet dynamics experiments, such as binary droplet collision and droplet impingement on superhydrophobic surfaces. Finally, the extended ULBM is applied to complex droplet dynamics, including droplet pancake bouncing and droplet splashing. The maximum Weber number and Reynolds number in the simulation reach 800 and 7200, respectively, at a density ratio of 1000. The study demonstrates the generality and versatility of ULBM for incorporating schemes to tackle challenging multiphase problems.

Improved three-dimensional thermal multiphase lattice Boltzmann model for liquid-vapor phase change

Modeling liquid-vapor phase change using the lattice Boltzmann (LB) method has attracted significant attention in recent years. In this paper, we propose an improved three-dimensional thermal multiphase LB model for simulating liquid-vapor phase change. The proposed model has the following features. First, it is still within the framework of the thermal LB method using a temperature distribution function and therefore retains the fundamental advantages of the thermal LB method. Second, in the existing thermal LB models for liquid-vapor phase change, the finite-difference computations of the gradient terms ∇·u and ∇T usually require special treatment at boundary nodes, while in the proposed thermal LB model these two terms are calculated locally. Moreover, in some of the existing thermal LB models, the error term ∂_{t_{0}}(Tu) is eliminated by adding local correction terms to the collision process in the moment space, which causes these thermal LB models to be limited to the D2Q9 lattice in two dimensions and the D3Q15 or D3Q19 lattice in three dimensions. Conversely, the proposed model does not suffer from such an error term and therefore the thermal LB equation can be constructed on the D3Q7 lattice, which simplifies the model and improves the computational efficiency. Numerical simulations are carried out to validate the accuracy and efficiency of the proposed thermal multiphase LB model for simulating liquid-vapor phase change.

Simulations of surface acoustic wave interactions on a sessile droplet using a three-dimensional multiphase lattice Boltzmann model

This study reports the development of a three-dimensional numerical model for acoustic interactions with a microscale sessile droplet under surface acoustic wave (SAW) excitation using the lattice Boltzmann method (LBM). We first validate the model before SAW interactions are added. The results demonstrate good agreement with the analytical results for thermodynamic consistency, Laplace law, static contact angle on a flat surface, and droplet oscillation. We then investigate SAW interactions on the droplet, with resonant frequencies ranging 61.7-250.1 MHz. According to our findings, an increase in wave amplitude elicits an increase in streaming velocity inside the droplet, causing internal mixing, and further increase in wave amplitude leads to pumping and jetting. The boundaries of wave amplitude at various resonant frequencies are predicted for mixing, pumping, and jetting modes. The modeling predictions on the roles of forces (SAW, interfacial tension, inertia, and viscosity) on the dynamics of mixing, pumping, and jetting of a droplet are in good agreement with observations and experimental data. The model is further applied to investigate the effects of SAW substrate surface wettability, viscosity ratio, and interfacial tension on SAW actuation onto the droplet. This work demonstrates the capability of the LBM in the investigation of acoustic wave interactions between SAW and a liquid medium.

Lattice Boltzmann Modeling of Drying of Porous Media Considering Contact Angle Hysteresis

Drying of porous media is governed by a combination of evaporation and movement of the liquid phase within the porous structure. Contact angle hysteresis induced by surface roughness is shown to influence multi-phase flows, such as contact line motion of droplet, phase distribution during drainage and coffee ring formed after droplet drying in constant contact radius mode. However, the influence of contact angle hysteresis on liquid drying in porous media is still an unanswered question. Lattice Boltzmann model (LBM) is an advanced numerical approach increasingly used to study phase change problems including drying. In this paper, based on a geometric formulation scheme to prescribe contact angle, we implement a contact angle hysteresis model within the framework of a two-phase pseudopotential LBM. The capability and accuracy of prescribing and automatically measuring contact angles over a large range are tested and validated by simulating droplets sitting on flat and curved surfaces. Afterward, the proposed contact angle hysteresis model is validated by modeling droplet drying on flat and curved surfaces. Then, drying of two connected capillary tubes is studied, considering the influence of different contact angle hysteresis ranges on drying dynamics. Finally, the model is applied to study drying of a dual-porosity porous medium, where phase distribution and drying rate are compared with and without contact angle hysteresis. The proposed model is shown to be capable of dealing with different contact angle hysteresis ranges accurately and of capturing the physical mechanisms during drying in different porous media including flat and curved geometries.

SUPPLEMENTARY INFORMATION

The online version contains supplementary material available at 10.1007/s11242-021-01644-9.

Shape Optimization of a Microhole Surface for Control of Droplet Wettability via the Lattice Boltzmann Method and Response Surface Methodology

The chief aim is to explore the wetting state on a microhole surface and to optimize the shape parameters of a microhole surface. A two-dimensional pseudopotential model was established, and the effects of shapes on the wetting behavior were explored. The shape parameters were optimized via the response surface methodology. The results reveal that the microhole surface can achieve a superhydrophobic state. When the diameter varies from 25 to 200 μm, the droplet is gradually lifted. However, when the diameter of the microhole is too large, the contact angle decreases rapidly. When the microhole diameter increases, relative radii of the x- and y-directions exhibit increasing trends. With the increase of the spacing, the gaps between the microholes are gradually filled with the droplet. When spacing increases, relative radii of x- and y-directions exhibit decreasing trends. The largest contact angle of 171.246° at the diameter of 76 μm and the spacing of 48 μm is observed.

Three-dimensional pseudopotential lattice Boltzmann model for multiphase flows at high density ratio

In this study, we extend the pseudopotential lattice Boltzmann model proposed by Huang and Wu [J. Comput. Phys. 327, 121 (2016)10.1016/j.jcp.2016.09.030] to a three-dimensional model for practical simulations of multiphase flows with high density ratio. In this model, an additional source term is introduced into the evolution function, and the performed high-order Chapman-Enskog analysis demonstrates that the Navier-Stokes equations with accurate pressure tensor are recovered. Also, an alternative geometric formulation is developed to obtain various contact angles and an iteration scheme is involved in the initialization to improve the stability of the model. Theoretical and numerical investigations both validate that the thermodynamic consistency and tuning surface tension independently of density ratio is achieved through varying the two free parameters in the source term. Numerical simulations of droplet wetting indicate that a large degree range of contact angles can be precisely realized with the implementation of the wetting boundary scheme. Further dynamic examinations of droplet impingement on a thin film and a dry surface also verify the stability and capability of the proposed pseudopotential lattice Boltzmann model.

Effects of hysteresis window on contact angle hysteresis behaviour at large Bond number

Contact angle hysteresis, defined as the difference between advancing and receding contact angles, is an important phenomenon in multiphase flow on a wetting surface. In this study, a modified pseudo-potential lattice Boltzmann (LB) multiphase model with tunable surface tension is proposed, which is further coupled with the geometrical formulation contact angle scheme to investigate the motion of droplets invoking the contact angle hysteresis. We focus on the dynamic behaviour of droplets driven by a body force at the Bond number ranging from 1 to 6, which is defined as the ratio of the body force to the capillary force. The droplet morphology change is examined by varying (i) the Bond number and (ii) the hysteresis window. Results show the droplet morphology evolution can be classified into different stages, including stretch, relaxation, and equilibrium. The droplet oscillation phenomenon at large Bond numbers at the equilibrium stage is observed for the first time. In addition, it is found that such oscillation can lead to the breakup and/or coalescence of droplets when the surface waves spread on the top of the droplet. Furthermore, there is slight oscillation of the normalized length, width and height at the equilibrium stage for the neutral hysteresis window while more dramatic oscillation will appear for the hydrophobic hysteresis window.

Lattice Boltzmann method for contact-line motion of binary fluids with high density ratio

Within the phase-field framework, we present an accurate and robust lattice Boltzmann (LB) method for simulating contact-line motion of immiscible binary fluids on the solid substrate. The most striking advantage of this method lies in that it enables us to handle two-phase flows with mass conservation and a high density contrast of 1000, which is often unavailable in the existing multiphase LB models. To simulate binary fluid flows, the present method utilizes two LB evolution equations, which are respectively used to solve the conservative Allen-Cahn equation for interface capturing, and the incompressible Navier-Stokes equations for hydrodynamic properties. Besides, to account for the substrate wettability, two popular contact angle models including the cubic surface-energy model and the geometrical one are incorporated into the present method, and their performances are numerically evaluated over a wide range of contact angles. The contact-angle hysteresis effect, which is inherent to a rough or chemically inhomogeneous substrate, is also introduced in the present LB approach through the strategy proposed by Ding and Spelt [J. Fluid Mech. 599, 341 (2008)10.1017/S0022112008000190]. The present method is first validated by simulating droplet spreading and capillary intrusion on the ideal or smooth pipes. It is found that the cubic surface-energy and geometrical wetting schemes both offer considerable accuracy for predicting a static contact angle within its middle region, while the former is more stable at extremely small contact angles. Besides, it is shown that the geometrical wetting scheme enables us to obtain better accuracy for predicting dynamic contact points in capillary pipe. Then we use the present LB method to simulate the droplet shearing processes on a nonideal substrate with contact angle hysteresis. The geometrical wetting model is found to be capable of reproducing four typical motion modes of contact line, while the surface-energy wetting scheme fails to predict the hysteresis behaviors in some cases. At last, a complex contact-line dynamic problem of three-dimensional microscale droplet impact on a wettable solid is simulated, and it is found that the numerical results for droplet shapes agree well with the experimental data.

Finite-element lattice Boltzmann simulations of contact line dynamics

The lattice Boltzmann method has become one of the standard techniques for simulating a wide range of fluid flows. However, the intrinsic coupling of momentum and space discretization restricts the traditional lattice Boltzmann method to regular lattices. Alternative off-lattice Boltzmann schemes exist for both single- and multiphase flows that decouple the velocity discretization from the underlying spatial grid. The current study extends the applicability of these off-lattice methods by introducing a finite element formulation that enables simulating contact line dynamics for partially wetting fluids. This work exemplifies the implementation of the scheme and furthermore presents benchmark experiments that show the scheme reduces spurious currents at the liquid-vapor interface by at least two orders of magnitude compared to a nodal implementation and allows for predicting the equilibrium states accurately in the range of moderate contact angles.

Improved thermal lattice Boltzmann model for simulation of liquid-vapor phase change

In this paper, an improved thermal lattice Boltzmann (LB) model is proposed for simulating liquid-vapor phase change, which is aimed at improving an existing thermal LB model for liquid-vapor phase change [S. Gong and P. Cheng, Int. J. Heat Mass Transfer 55, 4923 (2012)10.1016/j.ijheatmasstransfer.2012.04.037]. First, we emphasize that the replacement of ∇·(λ∇T)/∇·(λ∇T)ρc_{V}ρc_{V} with ∇·(χ∇T) is an inappropriate treatment for diffuse interface modeling of liquid-vapor phase change. Furthermore, the error terms ∂_{t_{0}}(Tv)+∇·(Tvv), which exist in the macroscopic temperature equation recovered from the previous model, are eliminated in the present model through a way that is consistent with the philosophy of the LB method. Moreover, the discrete effect of the source term is also eliminated in the present model. Numerical simulations are performed for droplet evaporation and bubble nucleation to validate the capability of the model for simulating liquid-vapor phase change. It is shown that the numerical results of the improved model agree well with those of a finite-difference scheme. Meanwhile, it is found that the replacement of ∇·(λ∇T)/∇·(λ∇T)ρc_{V}ρc_{V} with ∇·(χ∇T) leads to significant numerical errors and the error terms in the recovered macroscopic temperature equation also result in considerable errors.

Rebound suppression of a droplet impacting on an oscillating horizontal surface

The behavior of a droplet impinging onto a solid substrate can be influenced significantly by the horizontal motion of the substrate. The coupled interactions between the moving wall and the impacting droplet may result in various outcomes, which may be different from the usual normal droplet impact on a stationary wall. In this paper, we present a method to suppress drop rebound on hydrophobic surfaces via transverse wall oscillations, normal to the impact direction. The numerical investigation shows that the suppression of droplet rebound has a direct relationship with the oscillation phase, amplitude, and frequency. For a particular range of oscillation frequencies and amplitudes, a lateral shifting of the droplet position is observed along the oscillating direction. While large oscillation amplitude favors the process of droplet deposition, a high frequency promotes droplet rebound from the oscillating wall. A linear trend in the transition region between deposition and rebound is observed from a scaled phase diagram of the oscillation amplitude versus frequency. We provide a systematic investigation of drop deposition and elucidate the mechanism of rebound suppression through the temporal evolution of the nonaxial kinetic energy and the velocity flow field.

Contact angle adjustment in equation-of-state-based pseudopotential model

The single component pseudopotential lattice Boltzmann model has been widely applied in multiphase simulation due to its simplicity and stability. In many studies, it has been claimed that this model can be stable for density ratios larger than 1000. However, the application of the model is still limited to small density ratios when the contact angle is considered. The reason is that the original contact angle adjustment method influences the stability of the model. Moreover, simulation results in the present work show that, by applying the original contact angle adjustment method, the density distribution near the wall is artificially changed, and the contact angle is dependent on the surface tension. Hence, it is very inconvenient to apply this method with a fixed contact angle, and the accuracy of the model cannot be guaranteed. To solve these problems, a contact angle adjustment method based on the geometry analysis is proposed and numerically compared with the original method. Simulation results show that, with our contact angle adjustment method, the stability of the model is highly improved when the density ratio is relatively large, and it is independent of the surface tension.

Lattice Boltzmann simulation of coalescence of multiple droplets on nonideal surfaces

The interaction dynamics of droplets on a solid surface is a fundamental problem that is important to a wide variety of industrial applications, such as inkjet printing. Most previous research has focused on a single droplet and little research has been reported on the dynamics of multiple-droplet interactions on surfaces. Recently, Zhou et al. [W. Zhou, D. Loney, A. G. Fedorov, F. L. Degertekin, and D. W. Rosen, Lattice Boltzmann simulations of multiple-droplet interaction dynamics, Phys. Rev. E 89, 033311 (2014)] reported an efficient numerical solver based on the lattice Boltzmann method (LBM) that enabled the simulation of the multiple-droplet interaction dynamics on an ideal surface (i.e., smooth and homogeneous). In order to predict the interaction dynamics in the real world, it is necessary to take into consideration the contact angle hysteresis phenomenon on a nonideal surface, which is possibly caused by the surface roughness and chemical inhomogeneity of the surface. In this paper a dynamic contact angle boundary condition is developed to take into account the contact angle hysteresis effect based on the previously reported LBM. The improved LBM is validated with experimental data from the literature. The influence of the droplet impact conditions (e.g., fluid properties and impingement velocity), droplet spacing, and surface conditions on the two-droplet interaction dynamics is investigated with the validated LBM. Interesting phenomena are observed and discussed. The interaction of a line of six droplets on a nonideal surface is simulated to demonstrate the powerful capability of the developed numerical solver in simulating the multiple-droplet interaction dynamics in the real world.

Lattice Boltzmann modeling of contact angle and its hysteresis in two-phase flow with large viscosity difference

Contact angle hysteresis is an important physical phenomenon omnipresent in nature and various industrial processes, but its effects are not considered in many existing multiphase flow simulations due to modeling complexity. In this work, a multiphase lattice Boltzmann method (LBM) is developed to simulate the contact-line dynamics with consideration of the contact angle hysteresis for a broad range of kinematic viscosity ratios. In this method, the immiscible two-phase flow is described by a color-fluid model, in which the multiple-relaxation-time collision operator is adopted to increase numerical stability and suppress unphysical spurious currents at the contact line. The contact angle hysteresis is introduced using the strategy proposed by Ding and Spelt [Ding and Spelt, J. Fluid Mech. 599, 341 (2008)JFLSA70022-112010.1017/S0022112008000190], and the geometrical wetting boundary condition is enforced to obtain the desired contact angle. This method is first validated by simulations of static contact angle and dynamic capillary intrusion process on ideal (smooth) surfaces. It is then used to simulate the dynamic behavior of a droplet on a nonideal (inhomogeneous) surface subject to a simple shear flow. When the droplet remains pinned on the surface due to hysteresis, the steady interface shapes of the droplet quantitatively agree well with the previous numerical results. Four typical motion modes of contact points, as observed in a recent study, are qualitatively reproduced with varying advancing and receding contact angles. The viscosity ratio is found to have a notable impact on the droplet deformation, breakup, and hysteresis behavior. Finally, this method is extended to simulate the droplet breakup in a microfluidic T junction, with one half of the wall surface ideal and the other half nonideal. Due to the contact angle hysteresis, the droplet asymmetrically breaks up into two daughter droplets with the smaller one in the nonideal branch channel, and the behavior of daughter droplets is significantly different in both branch channels. Also, it is found that the contact angle hysteresis is strengthened with decreasing the viscosity ratio, leading to an earlier droplet breakup and a decrease in the maximum length that the droplet can reach before the breakup. These simulation results manifest that the present multiphase LBM can be a useful substitute to Ba et al. [Phys. Rev. E 88, 043306 (2013)PLEEE81539-375510.1103/PhysRevE.88.043306] for modeling the contact angle hysteresis, and it can be easily implemented with higher computational efficiency.

Lattice Boltzmann modeling of directional wetting: comparing simulations to experiments

Lattice Boltzmann Modeling (LBM) simulations were performed on the dynamic behavior of liquid droplets on chemically striped patterned surfaces, ultimately with the aim to develop a predictive tool enabling reliable design of future experiments. The simulations accurately mimic experimental results, which have shown that water droplets on such surfaces adopt an elongated shape due to anisotropic preferential spreading. Details of the contact line motion such as advancing of the contact line in the direction perpendicular to the stripes exhibit pronounced similarities in experiments and simulations. The opposite of spreading, i.e., evaporation of water droplets, leads to a characteristic receding motion first in the direction parallel to the stripes, while the contact line remains pinned perpendicular to the stripes. Only when the aspect ratio is close to unity, the contact line also starts to recede in the perpendicular direction. Very similar behavior was observed in the LBM simulations. Finally, droplet movement can be induced by a gradient in surface wettability. LBM simulations show good semiquantitative agreement with experimental results of decanol droplets on a well-defined striped gradient, which move from high- to low-contact angle surfaces. Similarities and differences for all systems are described and discussed in terms of the predictive capabilities of LBM simulations to model direction wetting.