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![Spinodal decomposition for ϕ¯0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\phi }_0=0$$\end{document} using parameters (4.1). Snapshots of phase variables 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} and ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} are taken at t=1,10,20,50,100,200,400,1000,1500,2000\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1, 10,20, 50,100, 200, 400, 1000,1500,2000$$\end{document}. For each subfigure, the left is the profile of 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}, and the right is the profile of ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}](publication/312946804/figure/fig3/AS:961816637222921@1606326364250/Spinodal-decomposition-for-ph00documentclass12ptminimal-usepackageamsmath.gif)
Spinodal decomposition for ϕ¯0=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\phi }_0=0$$\end{document} using parameters (4.1). Snapshots of phase variables 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} and ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document} are taken at t=1,10,20,50,100,200,400,1000,1500,2000\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1, 10,20, 50,100, 200, 400, 1000,1500,2000$$\end{document}. For each subfigure, the left is the profile of 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}, and the right is the profile of ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{document}
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In this paper, we consider the numerical approximations for the commonly used binary fluid-surfactant phase field model that consists two nonlinearly coupled Cahn-Hilliard equations. The main challenge in solving the system numerically is how to develop easy-to-implement time stepping schemes while preserving the unconditional energy stability. We...
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Citations
... This process is usually performed numerically. On the planar, surface and three-dimensional domains, there are quite a few literature on binary fluid-surfactant phase field models and their corresponding numerical methods, see [11,37,[42][43][44]46] for an incomplete list. However, practical simulations of binary fluid-surfactant phase field model on an evolving surface for interfacial sciences, which corresponds to disinfection or cleaning of the skin or sheet for example, are rarely studied. ...
... The method is energy stable but produces a nonlinear format in most cases and has a highly computational cost. Invariant energy quantization method [42,43] and scalar auxiliary variable method [37,44,46] write the energy generalization as a modified quadratic form by introducing appropriate auxiliary variables. Then, a new equivalent system of equations with modified energy can be derived. ...
... Next, we discuss the evolution property of energy (2.4). On the planar domain, the binary fluid-surfactant phase field model has the energy dissipation property [43]. On the stationary surfaces, i.e., v(x, t) = 0, the model also has the energy dissipation property [37]. ...
In this paper, the binary fluid-surfactant phase field model on evolving surfaces is presented and numerically studied for interfacial sciences. Based on the evolving surface finite element method, the first-order and second-order standard semi-implicit schemes are shown and analyzed to be conditionally stable. The energy dissipation law of the binary fluid-surfactant phase field model does not hold for evolving surfaces with arbitrary surface velocities which can be viewed as an action of external force. Therefore, this paper is dedicated to addressing the difficulties of conditional stability and numerical stiffness caused by the small interfacial parameters instead of energy stability. The first-order and second-order stabilizing approaches are employed to improve the stabilities of semi-implicit schemes. For the efficiency of long time simulation, an adaptive time-stepping technique considering energy evolution and surface velocity is developed. The effectiveness of the proposed stabilizing and adaptive time-stepping approaches is verified by numerical experiments. Various numerical simulations are performed to demonstrate the effectiveness and efficiency of the proposed method, and numerically investigate the behavior of the binary fluid-surfactant on evolving surfaces.
... The challenge of the time advancement for this particular model involves how to deal with the nonlinear and coupled terms, so that not only the energy dissipation laws of the PDE system can be extended to the discrete level, but also the high-order time accuracy and the computational efficiency can be as convenient and practical as possible. In a rather extensive numerical study of different versions of the surfactant phase-field model, we notice that there have been many successful attempts at the time discretization, for instance, the nonlinear convex-splitting approach applied in [40], the nonlinear implicit approach given in [41], the IEQ (Invariant Energy Quadratization) approach (or its various version the so-called Scalar Auxiliary Variable approach) developed in [42][43][44][45][46], etc. However, these known methods are either fully discrete methods based on regular regions or only semi-discrete methods in time. ...
... We consider in this subsection the Cahn-Hilliard phase field model of the binary fluid-surfactant system [41] ...
... Yang [41] developed a second-order linear scheme by introducing two auxiliary functions ...
... where f (φ) is a local polynomial in φ (typically of the 4th or 6th order). Choosing again model A/B dynamics leads to a 4th/6th order Allen-Cahn/Cahn-Hiliard evolution equation for the order parameter φ(r, t), a problem which has been extensively covered in the mathematical literature [56][57][58][59]. When not specifically interested in studying phase coexistence, one can consider this model within the Gaussian approximation by truncating f (φ) τ φ 2 to the quadratic order, again leading to ...
We present a simple and systematic procedure to determine the effective dynamics of a Brownian particle coupled to a rapidly fluctuating correlated medium, modeled as a scalar Gaussian field, under spatial confinement. The method allows us, in particular, to address the case in which the fluctuations of the medium are suppressed in the vicinity of the particle, as described by a quadratic coupling in the underlying Hamiltonian. As a consequence of the confinement of the correlated medium, the resulting effective Fokker–Planck equation features spatially dependent drift and diffusion coefficients. We apply our method to simplified fluid models of binary mixtures and microemulsions near criticality containing a colloidal particle, and we analyze the corrections to the stationary distribution of the particle position and the diffusion coefficient.
... But the lack of geometric fluxing in diffuse-interface method poses a challenge towards modeling scalars and to prevent the artificial numerical diffusion (numerical leakage) of the scalar across the interface. Here, it is worth mentioning the related previous studies on surfactant modeling (Liu and Zhang, 2010, Teigen et al., 2011, Yang, 2018, two-phase flows with non-Newtonian constitutive laws (Yue et al., 2005), and multicomponent systems in the context of diffuse-interface methods (Huang et al., 2021). ...
... where f (φ) is a local polynomial in φ (typically of the 4th or 6th order). Choosing again model A/B dynamics leads to a 4th/6th order Allen-Cahn/Cahn-Hiliard evolution equation for the order parameter φ(r, t), a problem which has been extensively covered in the mathematical literature [56][57][58][59]. When not specifically interested in studying phase coexistence, one can consider this model within the Gaussian approximation by truncating f (φ) τ φ 2 to the quadratic order, again leading to ...
We present a simple and systematic procedure to determine the effective dynamics of a Brownian particle coupled to a rapidly fluctuating correlated medium, modeled as a scalar Gaussian field, under spatial confinement. The method allows us, in particular, to address the case in which the fluctuations of the medium are suppressed in the vicinity of the particle, as described by a quadratic coupling in the underlying Hamiltonian. As a consequence of the confinement of the correlated medium, the resulting effective Fokker-Planck equation features spatially dependent drift and diffusion coefficients. We apply our method to simplified fluid models of binary mixtures and microemulsions near criticality containing a colloidal particle, and we analyze the corrections to the stationary distribution of the particle position and the diffusion coefficient.
... One is that it is very easy to construct linear, second-order and unconditionally energy stable schemes. Until now, it has been applied successfully to simulate many classical gradient flows such as Allen-Cahn models [51, 37], Cahn-Hilliard models [48,28], phase field crystal models [31,26], molecular beam epitaxial growth model [52,10], Cahn-Hilliard-Navier-Stokes models [24] and so on. It is worth mentioning that the dissipative system without gradient flow structure, such as Navier-Stokes models [27,30,25] can also be simulated effectively by the SAV approach. ...
... The gradient flow models are very important and popular dissipative systems which cover a lot of fields such as alloy casting, new material preparation, image processing, finance and so on [3,17,31,33,34,35,40,45]. Many classical gradient flow models such as Allen-Cahn model [1,11,16,39,51,54], Cahn-Hilliard model [6,12,18,39,44,48,55] and phase field crystal model [21,22,29,33,49] have been widely used to solve a series of physical problems. Gradient flow models are generally derived from the functional variation of free energy. ...
For the past few years, scalar auxiliary variable (SAV) and SAV-type approaches became very hot and efficient methods to simulate various gradient flows. Inspired by the new SAV approach in \cite{huang2020highly}, we propose a novel technique to construct a new exponential scalar auxiliary variable (E-SAV) approach to construct high-order numerical energy stable schemes for gradient flows. To improve its accuracy and consistency noticeably, we propose an E-SAV approach with relaxation, which we named the relaxed E-SAV (RE-SAV) method for gradient flows. The RE-SAV approach preserves all the advantages of the traditional SAV approach. In addition, we do not need any the bounded-from-below assumptions for the free energy potential or nonlinear term. Besides, the first-order, second-order and higher-order unconditionally energy stable time-stepping schemes are easy to construct. Several numerical examples are provided to demonstrate the improved efficiency and accuracy of the proposed method.
... Taking the hydrodynamic effects into account could be another challenging issue (see, for instance, [24] and its references) as well as replacing the standard Cahn-Hilliard equation with its nonlocal counterpart (see [13] and references therein). We also recall that some models of surfactants are represented by coupled Cahn-Hilliard equations (see [20], cf. also [30] and its references for the numerical approximations) possibly with hydrodynamic effects (see, e.g., [10] and references therein). The present approach could be extended to these models. ...
We consider a system which consists of a Cahn-Hilliard equation coupled with a Cahn-Hilliard-Oono equation in a bounded domain of $\mathbb{R}^d$, $d = 2,3$. This system accounts for macrophase and microphase separation in a polymer mixture through two order parameters $u$ and $v$. The free energy of this system is a bivariate interaction potential which contains the mixing entropy of the two order parameters and suitable coupling terms. The equations are endowed with initial conditions and homogeneous Neumann boundary conditions both for $u$, $v$ and for the corresponding chemical potentials. We first prove that the resulting problem is well posed in a weak sense. Then, in the conserved case, we establish that the weak solution regularizes instantaneously. Furthermore, in two spatial dimensions, we show the strict separation property for $u$ and $v$, namely, they both stay uniformly away from the pure phases ±1 in finite time. Finally, we investigate the long-time behavior of a finite energy solution showing, in particular, that it converges to a single stationary state.
... Adaptive mesh refinement (AMR) based schemes have also been incorporated for the numerical solutions of the Cahn-Hilliard and Cahn-Hilliard Navier-Stokes (CHNS) equations in previous studies [8,10]. More recently, Invariant Energy Quadratization (IEQ) scheme has been adopted in some phase field based studies [42,65,66]. Lee et al. [32] performed numerical simulations of the binary Cahn-Hilliard equation using finite difference, finite element and spectral methods. ...
In this paper, we have developed an efficient volume penalization based diffuse-filtering scheme to solve variable mobility based Cahn-Hilliard equation in complex geometries. The main novelty of the work is that a dealiased pseudo-spectral scheme with immersed interface method (IIM) is proposed for solving any generalized concentration-dependent mobility function-based Cahn-Hilliard (CH) equation in complicated computational domains. An indicator function based mobility parameter is introduced to perform simulation of binary spinodal decomposition problem at a lower computational expense in complex geometries by solving a single phase field equation. Due to the smooth removal of high frequency Fourier components, the solution of the present RK4 based diffuse-filtering scheme does not display spurious currents when suitable low-pass filtering strategy and adequately resolved mobility indicator are incorporated. The traditional and memory optimized zero padding schemes are also implemented to show the comparative performance of different dealiasing schemes for the variable mobility based Cahn-Hilliard equation. It is found that the diffuse-filtering scheme displays reasonable accuracy similar to the zero padding based schemes but its average CPU time is significantly lower, which indicates better computational performance of the scheme for the variable mobility Cahn-Hilliard equation. Time variation of the characteristic length scale during spinodal decomposition of a binary mixture agrees well with the analytical prediction. The optimal three stage SSPRK3 temporal scheme is employed and it is found that time step size can be increased approximately 1.4 times than the classical RK4 scheme reducing total CPU time. Oscillation free numerical solution and conservation of order parameter are obtained for the complex geometry based spinodal decomposition problem. A radially-averaged structure factor is introduced to quantify resolution issues of the dealiasing schemes for the spinodal decomposition problem in different complex geometries.
... All the phase-field models mentioned above have gained particular interest as far as numerical simulations are concerned. For example, the models with regular potentials have been numerically investigated in [41,42,43,44]. However, it was also noted in [41] that even this modification does not simplify a rigorous proof that the resulting energy functional is bounded from below. ...
We investigate a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant. The resulting system of partial differential equations consists of a sixth-order Cahn-Hilliard equation for the difference of local concentrations of the binary fluid mixture coupled with a fourth-order Cahn-Hilliard equation for the local concentration of the surfactant. The former has a smooth potential, while the latter has a singular potential. Both equations are coupled with a Navier-Stokes system for the (volume averaged) fluid velocity. The evolution system is endowed with suitable initial conditions, a no-slip boundary condition for the velocity field and homogeneous Neumann boundary conditions for the phase functions as well as for the chemical potentials. We first prove the existence of a global weak solution, which turns out to be unique in two dimensions. Stronger regularity assumptions on the initial data allow us to prove the existence of a unique global (resp. local) strong solution in two (resp. three) dimensions. In the two dimensional case, we can derive a continuous dependence estimate with respect to the norms controlled by the total energy. Then we establish instantaneous regularization properties of global weak solutions for $t>0$. In particular, we show that the surfactant concentration stays uniformly away from the pure states $0$ and $1$ after some positive time.
