Diffuse Interface Theory Research Articles

A free-energy based multiple-distribution-function lattice Boltzmann method for multi-component and multi-phase flows

This study presents the development of a multiple-distribution-function lattice Boltzmann model (MDF-LBM) for the accurate simulation of multi-component and multi-phase flow. The model is based on the diffuse interface theory and free energy model, which enable the derivation of hydrodynamic equations for the system. These equations comprise a Cahn-Hilliard (CH) type mass balance equation, which accounts for cross diffusion terms for each species, and a momentum balance equation. By establishing a relationship between the total chemical potential and the general pressure, the momentum balance equation is reformulated in a potential form. This potential form, together with the CH type mass balance equation, is then utilized to construct the MDF-LBM as a coupled convection–diffusion system. Numerical simulations demonstrate that the proposed MDF-LBM accurately captures phase behavior and ensures mass conservation. Additionally, the calculated interface tension exhibits good agreement with experimental data obtained from laboratory studies.

Dynamics of nucleation in binary alloys

ABSTRACT Nucleation in binary alloys is studied in the capillary approximation of classical theory. By allowing both the size of the cluster and its composition to vary, the phase transition is investigated in a two-dimensional space. This parametrisation allows to derive diffusivities of the Fokker-Planck equation for the distribution function of clusters from the Cahn–Hilliard equation. It is shown how kinetics, together with thermodynamics, determine the direction of the nucleation current as well as the magnitude of the nucleation rate. The properties of the critical clusters and the direction of the nucleation current at the saddle point are derived for a generic model, from the binodal line to the spinodal limit. The critical clusters are found to exhibit properties that are very similar to that of non-classical theories. Once moderate supersaturation is reached, the interplay between kinetics and thermodynamics is found to invalidate the classical picture by modifying the direction of the nucleation current. The consequences on the magnitude of the rate of nucleation are discussed. The model is then applied to decomposition of FeCr solid solutions and is shown to constitute a reasonable sharp-interface approximation of the diffuse-interface theory of nucleation for the determination of both the cluster properties and the nucleation rate.

Martingale solutions to stochastic nonlocal Cahn–Hilliard- Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

Abstract In the diffuse interface theory, the motion of two incompressible viscous fluids and the evolution of their interface are described by the well-known model H. The model consists of the Navier–Stokes equations, nonlinearly coupled with a convective Cahn–Hilliard type equation. Here we consider a stochastic version of the model driven by a noise of Lévy type, and where the standard Cahn–Hilliard equation is replaced by its nonlocal version with a singular (e.g., logarithmic) potential. The case of smooth potentials with arbitrary polynomial growth has been already analyzed in [G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 2019, 23–68]. Taking advantage of this previous result, we investigate this more challenging and physically relevant case. Global existence of a martingale solution is proved with no-slip and no-flux boundary conditions in both 2D and 3D bounded domains. In the two-dimensional case, we prove the uniqueness of weak solutions when the viscosity is constant.

Global regularity and asymptotic stabilization for the incompressible Navier–Stokes-Cahn–Hilliard model with unmatched densities

We study an initial-boundary value problem for the incompressible Navier–Stokes–Cahn–Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well-known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes towards an equilibrium state as t→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t \\rightarrow \\infty $$\\end{document}. More precisely, the concentration function ϕ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\phi $$\\end{document} is a strong solution of the Cahn–Hilliard equation for (arbitrary) positive times, whereas the velocity field u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{{u}}}$$\\end{document} becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn–Hilliard equation with divergence-free velocity belonging only to L2(0,∞;H0,σ1(Ω))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2(0,\\infty ; {\ extbf{H}}^1_{0,\\sigma }(\\Omega ))$$\\end{document}, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz–Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.

Molecular dynamics simulations of spontaneous and seeded nucleation and theoretical calculations for zinc selenide

Understanding and controlling the liquid to crystal transformation is a central topic for numerous natural phenomena and technological applications. However, the microscopic mechanism of crystal nucleation is still elusive, which leads to strong controversies regarding the ability of the most used model, the Classical Nucleation Theory (CNT), to describe nucleation rates in supercooled liquids. In this work, we were able to deeply supercool Zinc Selenide (ZnSe), and determine spontaneous homogeneous steady-state nucleation rates, JMD, by MD simulations using the mean lifetime method. At moderate supercoolings, where the nucleation rates are much smaller, we used the seeding method to compute the nucleation rates by the CNT formalism, JCNT, without any fitting parameter, using the physical properties obtained by MD simulations: the melting temperature, Tm, density, melting enthalpy, diffusion coefficient, D+, and the critical nucleus size, N∗, combined with two expressions for the thermodynamic driving force, Δμ. The values of interfacial free energy, γ, calculated by the CNT expression using the MD simulation data, via both the seeding method and the mean lifetime method at moderate and deep supercoolings show a weak temperature dependence, which is in line with the Diffuse Interface Theory. The extrapolated values of γ, from the spontaneous nucleation regime to the seeding nucleation region cover the range of values of γ calculated via the seeding method and the CNT formalism. Finally, the JCNT extrapolated from moderate supercoolings to deep supercoolings are in good agreement with the JMD. These results confirm the validity of the CNT.

Crystallization kinetics in a 5BaO·8SiO2 glass

The accuracy of a Differential Thermal Analysis (DTA) technique for predicting the temperature range of significant nucleation is examined in a 5BaO·8SiO2 glass based on iterative numerical calculations. The diffusion coefficient is calculated from the measured induction time and the crystal growth velocity. The Classical Theory of Nucleation and the Diffuse Interface Theory were used to model crystal nucleation and growth. However, in both theories, two different diffusion coefficients were required for the calculated crystallization peaks to match the measured DTA data. The development of small nuclei near the critical size during nucleation was calculated using the diffusion coefficient from the measured induction time for nucleation. The growth of nuclei much larger than the critical size was determined by the diffusion coefficient from the measured growth velocity. The difference could reflect a complicated nucleation process in which the primary nucleating phase is not the 5BaO·8SiO2 crystal.

Weak and strong solutions to the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system

We study the nonhomogeneous incompressible Navier-Stokes-Cahn-Hilliard system in a bounded smooth domain in Rd, d=2,3. This model arises from the Diffuse Interface theory of binary mixtures accounting for density variation, capillarity effects at the interface and partial mixing. We consider the case of initial density away from zero and concentration-depending viscosity with free energy potential equal to either the Landau potential or the Flory-Huggins logarithmic potential. In this setting, we prove the existence of global weak solutions in two and three dimensions, and the existence of strong solutions with bounded and strictly positive density. The strong solutions are local in time in three dimensions and global in time in two dimensions.

Quantification of nucleation delay in magmatic systems: experimental and theoretical approach

We developed a model to predict nucleation delay of olivine, plagioclase, and clinopyroxene in basaltic melts, and alkali feldspar and quartz in felsic melts. The model, based upon classical nucleation theory, was improved by applying modifications inspired by diffuse interface theory. The theoretical calculations were compared to results available in the literature (basalts) and from new experiments on a hydrous (~ 4 wt% water), metaluminous granitic composition conducted in piston cylinder apparatus at 600 MPa and temperatures from 500 to 800 °C. The experimental results and the modelled nucleation delays in silicate melts agree within a factor of ~ 5, which creates an opportunity for better understanding of the nucleation kinetics and crystallization timing of melts of various compositions. We demonstrate the implications of nucleation delay and discuss possible applications of this newly developed model to dry and hydrous felsic melts. We present the nucleation delay of feldspar and quartz in dry rhyolitic melt and its influence on the formation of obsidian, and we compare the nucleation delay of these minerals in a hydrous granitic melt against the typical internal structure of a granitic pegmatite dike.

Modeling nonisothermal crystallization in a BaO∙2SiO2 glass

AbstractThe accuracy of a differential thermal analysis (DTA) technique for predicting the temperature range of significant nucleation is examined in a BaO∙2SiO2 glass by iterative numerical calculations. The numerical model takes account of time‐dependent nucleation, finite particle size, size‐dependent crystal growth rates, and surface crystallization. The calculations were made using the classical and, for the first time, the diffuse interface theories of nucleation. The results of the calculations are in agreement with experimental measurements, demonstrating the validity of the DTA technique. They show that this is independent of the DTA scan rate used and that surface crystallization has a negligible effect for the glass particle sizes studied. A breakdown of the Stokes‐Einstein relation between viscosity and the diffusion coefficient is demonstrated for low temperatures, near the maximum nucleation rate. However, it is shown that accurate values for the diffusion coefficient can be obtained from the induction time for nucleation and the growth velocity in this temperature range.

A two-step nucleation model based on diffuse interface theory (DIT) to explain the non-classical view of calcium carbonate polymorph formation

A two-step nucleation model to explain the non-classical pathway of crystallization of calcium carbonate polymorphs.

Uniqueness and Regularity for the Navier--Stokes--Cahn--Hilliard System

The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier--Stokes equations, which ...

Partial and complete wetting in a micro-tube

This study investigates the flow of a thin annular film driven by an axial force in a micro-tube. Partial wetting is taken into account using the diffuse interface theory and the film dynamics is approximated by a long-wave mesoscopic model. One of the goals is to understand the transition between partial wetting and complete wetting when the driving force increases. Two scenarios, depending on the curvature, lead to complete wetting. For low curvature, complete wetting arises by ridge spreading, whereas for a sufficiently large curvature, it arises via coating of the sliding ridge. The latter regime has no equivalent for plane substrates. The transition between the two scenarios is explained as an exchange of solution branches involving a solution composed of merged ridges.

A Review of Classical and Nonclassical Nucleation Theories

Nucleation, the initial process in vapor condensation, crystal nucleation, melting, and boiling, is the localized emergence of a distinct thermodynamic phase at the nanoscale that macroscopically grows in size with the attachment of growth units. These phase changes are the result of atomistic events driven by thermal fluctuations. The occurrence of atomistic level events with the length scales on the order of 10–10 m and time scales of 10–13 S equivalent to the vibrational frequency of atoms makes the nucleation a very complicated phenomenon to study. Even though abundant literature is available about fundamental aspects of nucleation, the knowledge on these phenomena is far from complete. The classical pathway to nucleation which was once considered to have general applicability to all nucleating systems is gradually giving way to a nonclassical pathway which is now considered as a dominating mechanism in solution crystallization and other systems. In this review, an attempt is made to compare underlying physical principles involved in various nucleating systems and their theoretical treatment based on classical nucleation theory, and other important theories such as a density functional approach and diffuse interface theory. The limitations of classical theory, the gradual evolution of a nonclassical two-step pathway to nucleation, and the questions that have to be addressed in the future are discussed systematically.

Meso-Scale Phase-Field Modeling of Microstructural Evolution in Solid Oxide Fuel Cells

Microstructural evolution occurs in solid oxide fuel cells (SOFCs) during operation, which cause severe electrochemical performance degradation as well as structural failure such as crack formation. Nickel (Ni) particle coarsening is believed to cause microstructural evolution in the Ni-based anode of SOFCs. Furthermore, the accumulation of expanded pores during the microstructural evolution is responsible for crack formation. Based on the diffuse-interface theory and the phase-field method, integrating both Cahn-Hilliard equation and Ginzburg-Landau equation, a meso-scale phase-field model was established to investigate microstructural evolution and crack formation in the Ni / Yttria Stabilized Zirconia (YSZ) anode of SOFCs. Then, the model was applied to explore the coupling effects of microstructural evolution on SOFCs performance degradation. Simulation results show that particle size, porosity, and particle size ratio of Ni-YSZ are most influential parameters in microstructural evolution. It was found that the triple phase boundaries (TPBs) area was decreased by 24% and power density was decreased by 11.03% after 1000 hours of operation. In addition, the power density was decreased by 27%. This work is expected to provide us a comprehensive understanding of SOFCs’ microstructural evolution as well as a tool for SOFC's performance degradation analysis.

Lattice Boltzmann model for thermal behavior of a droplet on the solid surface

In this work, a thermal lattice Boltzmann model is applied to simulate thermal behavior of a droplet on the solid surface. For this reason Lee's model which uses the Cahn–Hilliard diffuse interface theory, is utilized to capture the droplet interface. Also contact angle between solid, liquid and gas phases has been considered in simulations. To take into account the thermal effects, passive scalar model is conjugated with the Lee's method. Since in this model the solution of temperature distribution is independent of flow field, application of the Boussinesq approximation couples the energy and momentum equations. Numerical results for two thermal boundary conditions; constant wall temperature and constant heat flux on the wall, are presented and results have been compared with previous numerical results. Results show that by increasing the Prandtl number ratio between droplet and its surrounding, thermal diffusion within the droplet will be delayed and this causes reduction in the droplet average temperature. Also it was shown that the wall heat flux is concentrated around the droplet, while that in the gas phase is negligible.

Conduction-only transport phenomena in compressible bivelocity fluids: diffuse interfaces and Korteweg stresses.

"Diffuse interface" theories for single-component fluids—dating back to van der Waals, Korteweg, Cahn-Hilliard, and many others—are currently based upon an ad hoc combination of thermodynamic principles (built largely upon Helmholtz's free-energy potential) and so-called “nonclassical” continuum-thermomechanical principles (built largely upon Newtonian mechanics), with the latter originating with the pioneering work of Dunn and Serrin [Arch. Ration. Mech. Anal. 88, 95 (1985)]. By introducing into the equation governing the transport of energy the notion of an interstitial work-flux contribution, above and beyond the usual Fourier heat-flux contribution, namely, jq = -k∇T, to the energy flux, Dunn and Serrin provided a rational continuum-thermomechanical basis for the presence of Korteweg stresses in the equation governing the transport of linear momentum in compressible fluids. Nevertheless, by their failing to recognize the existence and fundamental need for an independent volume transport equation [Brenner, Physica A 349, 11 (2005)]—especially for the roles played therein by the diffuse volume flux j v and the rate of production of volume πν at a point of the fluid continuum—we argue that diffuse interface theories for fluids stand today as being both ad hoc and incomplete owing to their failure to recognize the need for an independent volume transport equation for the case of compressible fluids. In contrast, we point out that bivelocity hydrodynamics, as it already exists [Brenner, Phys. Rev. E 86, 016307 (2012)], provides a rational, non-ad hoc, and comprehensive theory of diffuse interfaces, not only for single-component fluids, but also for certain classes of crystalline solids [Danielewski and Wierzba, J. Phase Equilib. Diffus. 26, 573 (2005)]. Furthermore, we provide not only what we believe to be the correct constitutive equation for the Korteweg stress in the class of fluids that are constitutively Newtonian in their rheological response to imposed stresses but, equally importantly, we establish the explicit functional forms of Korteweg's phenomenological thermocapillary coefficients appearing therein.

Pore-Scale Simulations of Gas Displacing Liquid in a Homogeneous Pore Network Using the Lattice Boltzmann Method

A lattice Boltzmann high-density-ratio model, which uses diffuse interface theory to describe the interfacial dynamics and was proposed originally by Lee and Liu (J Comput Phys 229:8045–8063, 2010), is extended to simulate immiscible multiphase flows in porous media. A wetting boundary treatment is proposed for concave and convex corners. The capability and accuracy of this model is first validated by simulations of equilibrium contact angle, injection of a non-wetting gas into two parallel capillary tubes, and dynamic capillary intrusion. The model is then used to simulate gas displacement of liquid in a homogenous two-dimensional pore network consisting of uniformly spaced square obstructions. The influence of capillary number (Ca), viscosity ratio (\(M\)), surface wettability, and Bond number (Bo) is studied systematically. In the drainage displacement, we have identified three different regimes, namely stable displacement, capillary fingering, and viscous fingering, all of which are strongly dependent upon the capillary number, viscosity ratio, and Bond number. Gas saturation generally increases with an increase in capillary number at breakthrough, whereas a slight decrease occurs when Ca is increased from \(8.66\times 10^{-4}\) to \(4.33\times 10^{-3}\), which is associated with the viscous instability at high Ca. Increasing the viscosity ratio can enhance stability during displacement, leading to an increase in gas saturation. In the two-dimensional phase diagram, our results show that the viscous fingering regime occupies a zone markedly different from those obtained in previous numerical and experimental studies. When the surface wettability is taken into account, the residual liquid blob decreases in size with the affinity of the displacing gas to the solid surface. Increasing Bo can increase the gas saturation, and stable displacement is observed for \(Bo>1\) because the applied gravity has a stabilizing influence on the drainage process.

A modeling approach to droplet contact-line motion dynamics in high-density-ratio two-phase flow

The present study is to investigate the contact line dynamics in high-density-ratio two-phase flow. A two-dimensional lattice Boltzmann model for immiscible fluids flow with gravity is developed, which combines the stabilized numerical discretization for the continuous Boltzmann equation and the diffuse-interface theory. The model is developed on the basis of previous work. However, some significant modifications are made to suit the purpose of the present study, including direct calculation of macroscopic variables without post-collision process and consideration of gravity. The model and numerical method are firstly validated by various flow problems with analytical solutions. It is then used to investigate the contact line motion of a droplet attached on a substrate in shear flow. The influences of a series of flow parameters are systematically investigated and some conclusions are obtained. Above the critical Capillary number, the droplet motion exhibits break-up characteristics. In steady-slip mode, the receding contact angle, θR, decreases and advancing contact angle, θA, increases linearly with the Capillary number, Ca, while the normalized droplet velocity nearly holds constant with different values of Ca. The contact line motion is not sensitive to Reynolds number, Re. When Bond number Bo>1, both the dynamic contact angles and the droplet velocity decrease significantly with increasing Bo. The obtained results are partially compared with those reported by other investigators, and a good agreement has been reached in several aspects, such as break-up characteristics and Bond number effect.

Bayesian calibration, validation, and uncertainty quantification of diffuse interface models of tumor growth

The idea that one can possibly develop computational models that predict the emergence, growth, or decline of tumors in living tissue is enormously intriguing as such predictions could revolutionize medicine and bring a new paradigm into the treatment and prevention of a class of the deadliest maladies affecting humankind. But at the heart of this subject is the notion of predictability itself, the ambiguity involved in selecting and implementing effective models, and the acquisition of relevant data, all factors that contribute to the difficulty of predicting such complex events as tumor growth with quantifiable uncertainty. In this work, we attempt to lay out a framework, based on Bayesian probability, for systematically addressing the questions of Validation, the process of investigating the accuracy with which a mathematical model is able to reproduce particular physical events, and Uncertainty quantification, developing measures of the degree of confidence with which a computer model predicts particular quantities of interest. For illustrative purposes, we exercise the process using virtual data for models of tumor growth based on diffuse-interface theories of mixtures utilizing virtual data.

Phase-field modeling of three-phase electrode microstructures in solid oxide fuel cells

A phase-field model for describing three-phase electrode microstructure (i.e., electrode-phase, electrolyte-phase, and pore-phase) in solid oxide fuel cells is proposed using the diffuse-interface theory. Conserved composition and non-conserved grain orientation order parameters are simultaneously used to describe the coupled phase coarsening and grain growth in the three-phase electrode. The microstructural evolution simulated by the phase-field approach demonstrates the significant dependence of morphological microstructure and output statistic material features on the prescribed kinetic parameters and three-phase volume fractions. The triple-phase boundary fraction is found to have a major degradation in the early evolution.