Free-energy Lattice Boltzmann Model Research Articles

Implementation of a ternary lattice Boltzmann model in LAMMPS

The properties of multicomponent fluids are governed by the interplay of phase behavior, fluid dynamics, and interfacial thermodynamics. A mixture formulation that leverages this interplay is an important aspect in many fabrication processes based on emulsion templating. The lattice Boltzmann method (LBM) has become a popular approach for simulating hydrodynamic effects in complex fluids and soft matter. Here we present an implementation of a ternary lattice Boltzmann model that allows to simulate a mixture of three immiscible fluids. We build on the LATBOLTZ extension of the open-source package LAMMPS and implement a ternary free energy model recently introduced by Semprebon et al. [Phys. Rev. E 93, 033305 (2016)]. We validate the static and dynamic properties by simulating liquid lenses, double emulsions, and ternary mixtures. From the simulations, we obtain the complete morphology diagram of the ternary mixture in composition space. We further discuss an application of the method to phase segregation of ternary films. The implementation of the ternary LBM in LAMMPS opens vast opportunities for mesoscale simulations of interfacial phenomena and non-equilibrium transport processes in multicomponent fluid mixtures. Program summaryProgram Title: fix lb/multicomponentCPC Library link to program files:https://doi.org/10.17632/gn8znmyxrh.1Developer's repository link:https://github.com/uschille/lammps/tree/ternary-lbmLicensing provisions: GPLv2Programming language: C++Journal reference of previous version: C. Denniston, N. Afrasiabian, M.G. Cole-André, F.E. Mackay, S.T.T Ollila, T. Whitehead. LAMMPS lb/fluid fix version 2: Improved hydrodynamics forces implemented into LAMMPS through a lattice-Boltzmann fluid. Comput. Phys. Commun.275, 108318 (2022).Does the new version supersede the previous version?: NoReasons for the new version: The lb/multicomponent fix provides new capabilities for simulating ternary fluid mixtures.Summary of revisions: An implementation of the ternary free-energy lattice Boltzmann model [1] has been added to the LATBOLTZ extension of LAMMPS. The new functionality is available through the script command fix lb/multicomponent.Nature of problem: Multicomponent fluids involve non-local interactions that give rise to fluid-fluid surface tension and a non-ideal pressure tensor. This requires a modification of the standard lattice Boltzmann method.Solution method: The implementation extends the existing lattice Boltzmann code in LAMMPS [2] and introduces data structures and methods for the ternary algorithm introduced in [1]. The code carries out the streaming and collision steps for three fluid components using the appropriate equilibrium distributions. The equilibrium distribution is calculated based on thermodynamic quantities derived from the free energy.Additional comments including restrictions and unusual features: The current version of the lb/multicomponent fix is restricted to the D3Q19 lattice.

Fugacity-based lattice Boltzmann method for multicomponent multiphase systems.

The free-energy model can extend the lattice Boltzmann method to multiphase systems. However, there is a lack of models capable of simulating multicomponent multiphase fluids with partial miscibility. In addition, existing models cannot be generalized to honor thermodynamic information provided by any multicomponent equationof state of choice. In this paper, we introduce a free-energy lattice Boltzmann model where the forcing term is determined by the fugacity of the species, the thermodynamic property that connects species partial pressure to chemical potential calculations. By doing so, we are able to carry out multicomponent multiphase simulations of partially miscible fluids and generalize the methodology for use with any multicomponent equationof state of interest. We test this fugacity-based lattice Boltzmann method for the cases of vapor-liquid equilibrium for two- and three-component mixtures in various temperature and pressure conditions. We demonstrate that the model is able to reliably reproduce phase densities and compositions as predicted by multicomponent thermodynamics and can reproduce different characteristic pressure-composition and temperature-composition envelopes with a high degree of accuracy. We also demonstrate that the model can offer accurate predictions under dynamic conditions.

Improved well-balanced free-energy lattice Boltzmann model for two-phase flow with high Reynolds number and large viscosity ratio

Spurious velocities and inaccurate density properties arising from the imbalance of discretized forces at discrete level are frequently observed in numerical simulation of multiphase flows based on lattice Boltzmann equation (LBE) models. In this paper, an improved well-balanced free-energy LBE model is proposed for two phase flows with high Reynolds numbers and large viscosity differences based on the well-balanced LBE [Guo et al., Phys. Fluids 33, 031709 (2021)]. To this end, a free parameter associated with the shear rate is introduced into the equilibrium distribution function. This results in a fluid viscosity that is dependent not only on the relaxation time but also on the introduced parameter. The relaxation time can be chosen to improve the numerical stability and accuracy, while the viscosity is mainly determined by the new parameter. To test the capability of the present model in capturing discrete equilibrium states, both one-dimensional flat interface and two-dimensional stationary droplet are simulated. Numerical results show that the present model is capable of eliminating spurious velocities and maintaining a constant chemical potential when the system reaches an equilibrium state. To further validate the performance of the present LBE for dynamic problems, both binary droplet collision and single bubble rising are performed, which demonstrates that the present model has the capability to deal with two phase flows with high Reynolds number and large viscosity ratio.

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.

Achieving thermodynamic consistency in a class of free-energy multiphase lattice Boltzmann models.

The free-energy lattice Boltzmann (LB) model is one of the major multiphase models in the LB community. The present study is focused on a class of free-energy LB models in which the divergence of thermodynamic pressure tensor or its equivalent form expressed by the chemical potential is incorporated into the LB equation via a forcing term. Although this class of free-energy LB models may be thermodynamically consistent at the continuum level, it suffers from thermodynamic inconsistency at the discrete lattice level owing to numerical errors [Guo et al., Phys. Rev. E 83, 036707 (2010)10.1103/PhysRevE.83.036707]. The numerical error term mainly includes two parts: one comes from the discrete gradient operator and the other can be identified in a high-order Chapman-Enskog analysis. In this paper, we propose an improved scheme to eliminate the thermodynamic inconsistency of the aforementioned class of free-energy LB models. The improved scheme is constructed by modifying the equation of state of the standard LB equation, through which the discretization of ∇(ρc_{s}^{2}) is no longer involved in the force calculation and then the numerical errors can be significantly reduced. Numerical simulations are subsequently performed to validate the proposed scheme. The numerical results show that the improved scheme is capable of eliminating the thermodynamic inconsistency and can significantly reduce the spurious currents in comparison with the standard forcing-based free-energy LB model.

Multiphase lattice Boltzmann method and its applications in phase-change heat transfer

<p indent=0mm>In the past three decades, the lattice Boltzmann (LB) method has been developed into an efficient numerical method for simulating fluid flow and heat transfer. It is a mesoscopic numerical approach, sitting in the middle between the molecular dynamic method and conventional numerical methods based on the continuum assumption. Unlike conventional numerical methods, the LB method is built on the mesoscopic kinetic equation. It tracks the evolution of a particle distribution function and then accumulates the particle distribution function to obtain the macroscopic properties. Owing to its kinetic nature, the LB method has exhibited many advantages over conventional numerical methods. For example, in the LB equation the convective operator is linear, whereas the convective terms of the Navier-Stokes equations are nonlinear. Moreover, in the LB simulations the complex boundary conditions can be formulated with the elementary mechanical rules such as the bounce-back rule according to the interaction of the “LB particles” with the solid walls. Furthermore, the LB method is ideal for parallel computing because of its explicit scheme and local interactions. Particularly, the intermolecular interactions of fluids can be easily incorporated into the LB method. As a result, the interface between different phases can arise, deform, and migrate naturally in the LB modeling without using any techniques to track or capture the interface, which is often required in the VOF or Level Set method. The existing multiphase LB models can be generally classified into four categories, i.e., the color-gradient model, the pseudopotential model, the free-energy model, and the phase-field model. In the color-gradient LB model, two distribution functions are introduced to represent two different fluids and a color-gradient-based perturbation operator is employed to generate surface tension. Moreover, a recoloring step is used in the color-gradient model so as to separate different phases. The pseudopotential LB model is the simplest and the most popular multiphase LB model. In this model, the fluid interactions are mimicked by an interparticle potential, through which the separation of different phases or components can be achieved naturally. Accordingly, the interface between different phases can arise, deform and migrate naturally. The free-energy LB model was developed based on thermodynamics considerations, which modifies the second-order moment of the particle equilibrium distribution function so as to include a non-ideal thermodynamic pressure tensor. The phase-field LB model is based on the phase-field theory, in which the interface dynamics is described by an order parameter that obeys the Cahn-Hilliard equation or a Cahn-Hilliard-like equation. In this paper we briefly review some recent advances in these multiphase LB models. Particularly, recent progress in the pseudopotential LB model is highlighted. It is shown that the thermodynamic consistency of the pseudopotential model can be achieved by adjusting the mechanical stability condition via an improved forcing scheme. Furthermore, an alternative approach is presented to tune the surface tension of the pseudopotential model and an improved contact angle scheme is introduced, which retains the advantages of the original virtual-density scheme but does not suffer from an unphysical thick mass-transfer layer near the solid boundary. Moreover, the applications of the pseudopotential LB model to boiling and condensation heat transfer are outlined.

Study on the Bubble Growth and Departure with A Lattice Boltzmann Method

For the growth and departure of bubbles from an orifice, a free energy lattice Boltzmann model is adopted to deal with this complex multiphase flow phenomenon. A virtual layer is set at the boundary of the flow domain to deal with the no-slip boundary condition. Effects of the viscosity, surface tension, gas inertial force and buoyancy on the characteristics of bubbles when they grow and departure from an orifice in quiescent liquid are studied. The releasing period and departure diameter of the bubble are influenced by the residual gas at the orifice, and the interaction between bubbles is taken into consideration. The relations between the releasing period or departure diameter and the gravity acceleration show fair agreements with previous numerical and theoretical results. And the influence of the gas outflow velocity on bubble formation is discussed as well. For the bubbles growing in cross-flow field, effects of the cross-flow speed and the gas outflow velocity on the bubble formation are discussed, which is related to the application in ship resistance reduction. And optimal choice of the ship speed and gas outflow velocity is studied. Cases in this paper also prove that this high density ratio LBM model has its flexibility and effectiveness on multiphase flow simulations.

Lattice Boltzmann simulation of droplet condensation on a surface with wettability gradient

In this paper, a free energy lattice Boltzmann model of vapor condensation on a surface with a wettability gradient is developed and numerically analyzed to understand the microscopic behaviors of self-propelled droplet condensation. The effect of wettability gradient on droplet self-motion and coalescence as well as vapor condensation is examined and investigated. The condensation rate is presented during the whole droplet condensation process to analyze the role of wettability gradient on droplet condensation. The results indicate that the surface with wettability gradient is preferred for vapor condensation owing to the appearance of self-propelled droplet condensation. Condensate film is initially spread on the high surface energy region, and liquid nucleation sites form, grow and, subsequently, coalesce with other droplets on low surface energy regions. The condensation rate is higher on a surface with a larger wettability gradient due to the more effective removal of condensate. In addition, the condensation rate fluctuates with time at the quasi-steady-state stage. During the condensation process, the droplet coalescence triggers a sudden peak of condensation rate, and the generation of new nucleation results in a rapid increase in the condensation rate.

Thermodynamic consistency of a pseudopotential lattice Boltzmann fluid with interface curvature.

Thermodynamic consistency of pseudopotential lattice Boltzmann models is a major topic that needs comprehensive evaluations. When interface is flat, pseudopotential models can give density-pressure isotherms in excellent agreement with those from equation of state. When interface is curved, thermodynamic equilibriums are affected by interface curvature, and consistency of pseudopotential models has not been systematically evaluated. In this study, we show that the effect of Laplace pressure on phase equilibrium is quantitatively consistent with Kelvin equation at high reduced temperatures (≥0.7). At low temperatures, inconsistency that can be attributed to the effect of orientation of the interface was noted, and it can be improved by tuning of the pseudopotential. By relating interfacial tension of a simulated fluid to that of a real fluid, the lattice spacing of pseudopotential model is found to be on the order of several molecular diameters, the typical range of intermolecular interactions. Interfacial thickness at different temperatures in pseudopotential model compared well with experiments and molecular dynamics simulations, which confirms that the calculated length scale is reasonable. Evaluation of a free energy lattice Boltzmann model indicate that it is consistent with Kelvin equation at high temperatures. The free energy model, however, is not as accurate as the tested pseudopotential model, and discrepancies may come from the relative inaccuracies in the predictions of vapor densities and the thinner interfaces.

Pore-scale study of the effects of surface roughness on relative permeability of rock fractures using lattice Boltzmann method

Two-phase flow in fractured geo-material plays a significant role in various subsurface processes. Despite some success in studying two-phase flow in ideal smooth-walled fractures, the more realistic issues regarding two-phase flow in rough-walled fractures are still not well understood. In this study, an improved approach based on free energy lattice Boltzmann model is proposed to analyze the two-phase immiscible flow in the rough-walled fracture and to further estimate the relative permeability curves. The proposed model is validated by comparison with analytical results of individual fracture, and good agreement is achieved in general. Then it is used to investigate the effects of wall roughness and the initial phase distribution patterns on the co-current flow of two immiscible fluids. The simulation results demonstrate that roughness impacts phase distribution, flow patterns and relative permeability curves in a single rough-walled fracture. Rough surface increases two-phase interference and flow resistance, and an unsteady flow state appears for the non-wetting phase (NWP) when the wetting phase (WP) saturation is small. In addition, the NWP flow can either be lubricated or be resisted by the WP depending on the flow patterns and the WP saturation.

Wetting boundaries for a ternary high-density-ratio lattice Boltzmann method

We extend a recently proposed ternary free-energy lattice Boltzmann model with high density contrast [Phys. Rev. Lett. 120, 234501 (2018)PRLTAO0031-900710.1103/PhysRevLett.120.234501] by incorporating wetting boundaries at solid walls. The approaches are based on forcing and geometric schemes, with implementations optimized for ternary (and, more generally, higher-order multicomponent) models. Advantages and disadvantages of each method are addressed by performing both static and dynamic tests, including the capillary filling dynamics of a liquid displacing the gas phase and the self-propelled motion of a train of drops. Furthermore, we measure dynamic angles and show that the slip length critically depends on the equilibrium value of the contact angles and whether it belongs to liquid-liquid or liquid-gas interfaces. These results validate the model capabilities of simulating complex ternary fluid dynamic problems near solid boundaries, for example, drop impact solid substrates covered by a lubricant layer.

Entropic multi-relaxation free-energy lattice Boltzmann model for two-phase flows

The entropic multi-relaxation lattice Boltzmann method is extended to two-phase systems following the free-energy approach. Gain in stability is achieved by incorporating the force term due to Korteweg's stress into the redefined entropic stabilizer, which allows simulation of higher Weber and Reynolds numbers with an efficient and explicit algorithm. Results for head-on droplet collisions and droplet impact on super-hydrophobic substrates are matching experimental data accurately. Furthermore, it is demonstrated that the entropic stabilization leads to smaller spurious currents without affecting the interface thickness. The present findings demonstrate the universality of the simple and explicit entropic lattice Boltzmann models and provide a viable and robust alternative to existing methods.

Investigating the effects of shear rate on the collapse time in a gas–liquid system by lattice Boltzmann

In order to evaluate the effects of shear rate on the time of collapse process in a two-phase; one component system, a free energy lattice Boltzmann model is developed. Considering a system consists of two droplets in the surrounding vapor it is observed that, for various physical and geometrical conditions, imposing shear flow has similar effects on the collapse time. Actually, for each case there is an optimum shear rate of collapse that is corresponded to highest collapse rate and smallest time. In all test cases, for shear rates smaller than the optimum value, increasing the shear rate raise collapse rate and reduce its time. But for higher shear rates, increasing the shear rate has reverse effect and decreases the collapse rate. The results also show that, higher droplet radius ratio, viscosity ratio, surface tension and interface thickness lead to a higher optimum shear rate of collapse while higher density ratio of liquid and vapor decreases that.

Ternary free-energy lattice Boltzmann model with tunable surface tensions and contact angles.

We present a ternary free-energy lattice Boltzmann model. The distinguishing feature of our model is that we are able to analytically derive and independently vary all fluid-fluid surface tensions and the solid surface contact angles. We carry out a number of benchmark tests: (i) double emulsions and liquid lenses to validate the surface tensions, (ii) ternary fluids in contact with a square well to compare the contact angles against analytical predictions, and (iii) ternary phase separation to verify that the multicomponent fluid dynamics is accurately captured. Additionally we also describe how the model presented here can be extended to include an arbitrary number of fluid components.

Control of droplet collapse during coarsening process by imposing shear flow: a lattice Boltzmann simulation

The effects of shear flow on droplet collapse during the coarsening process in a vapor–liquid system are investigated by a free energy lattice Boltzmann model. To simulate different viscosity ratios, a local relaxation time parameter is integrated with LBM algorithm. The results show that for zero and small shear rates, droplet coarsening happens in its regular pattern where small droplet is collapsed while greater one grows (sub-critical regime). But if the shear rate be greater than a critical value, collapse is abated, droplet coarsening is inverted and smaller droplet grows (super-critical regime). Therefore during coarsening process, collapse mechanism can be controlled by imposing suitable shear flow. Also it is shown that, higher droplet radius ratio, viscosity ratio and surface tension lead to higher critical shear rate of collapse while higher density ratio of liquid and vapor decreases that.

A mean-field free energy lattice Boltzmann model for multicomponent fluids

We propose a mean-field free energy approach to simulate multi- component fluids. The model has been validated in terms of the Laplace equation of capillarity and dispersion relation of interfacial waves. Simulations of a ternary system shows that the total free energy decreases and reaches a minimum after phase separation has occurred. Different drop shapes can be obtained by adjusting the interaction strengths between individual components. These results demonstrate that both macroscopic free energy and microscopic fluid-fluid interactions have been well described in our multicomponent model.

Contact Line and Contact Angle Dynamics in Superhydrophobic Channels

The dynamics of the wetting and movement of a three-phase contact line confined between two superhydrophobic surfaces were studied using a mean-field free-energy lattice Boltzmann model. Principle features of superhydrophobic surfaces, such as trapped vapor/air between rough microstructures, high contact angles, reduced contact angle hysteresis, and low resistance to fluid flow, were all observed. Movement of the three-phase contact line over a well-patterned superhydrophobic surface displays a periodic stick-jump-slip behavior, while the dynamic contact angle changes accordingly from maximum to minimum. Two regimes were found for the flow velocity as a function of surface roughness and can be related directly to the balance between driving force and flow resistance. This work provides a better understanding of dynamic wetting and fluid flow behaviors over superhydrophobic surfaces and hence could be useful in related applications.