Cahn-Hilliard Model Research Articles

The Stabilized Finite Element Method for the Cahn-Hilliard Phase-Field Model of Diblock Copolymers on Evolving Surfaces

The Stabilized Finite Element Method for the Cahn-Hilliard Phase-Field Model of Diblock Copolymers on Evolving Surfaces

The Cahn–Hilliard model of coherent lamellar microstructure: application to alkali feldspar

The temporal evolution of coherent lamellar microstructures is significantly influenced by their elastic properties, particularly when the exsolved phases are coherent. This research utilizes the Cahn–Hilliard model to examine the morphological and energetic evolution of binary alkali feldspar, integrating anisotropic elastic energy into the Gibbs energy equation. The Cahn–Hilliard model successfully simulated the orientation of lamellae observed in natural samples and the elastic strain was consistent with previous research. We also computed the coherent solvus from the annealing simulation of various precursor compositions and temperatures. The temperature difference (ΔT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta T$$\\end{document}) between the strain-free solvus and the coherent solvus was ΔT=85∘C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta T = 85\\,{}^\\circ \ ext {C}$$\\end{document}, which is slightly lower than previously reported values obtained from similar parameters. This discrepancy is likely due to the presence of non-planar lamellae at the onset of phase separation, which are more stable than planar ones. We also simulated the binodal curves of the coherent solvi for different precursor phase compositions. The computed solvi were not unique but varied depending on the precursor composition. Our model is flexible because it does not assume any specific shapes for the lamellar interfaces and is applicable to various coherent binary systems.

المركبات غيرالحرجة لاشرطة السوائل الرقيقة للمزيج الثنائي

This study concentrates on the non-linear solutions of bounded films of binary mixtures with free-modulated surfaces, situated on an energetically neutral solid substrate and subject to periodic boundary conditions on the lateral walls. The Cahn-Hilliard model in its static limit is employed to investigate the problem, and the free surface is considered sharp with its own surface tension. The internal interfaces between the two polymers are allowed to diffuse. The problem is explored through its bifurcations varying several parameters, specifically the surface tension γ, film lateral domain size, and the mean film thickness of homogeneous and laterally (and horizontally) stratified compositions of two layers. The mixture is considered in the off-critical composition with 𝑐 = 0.4. The results reveal a rich family of solutions, providing valuable insights into the structural properties of thin films of binary mixtures،

Universal hyperuniformity in active field theories

We show that dry scalar-order active field theories (AFTs) are universally hyperuniform, i.e., density fluctuations are anomalously suppressed in the long-time limit regardless of the integrability or functional form of the active contributions up to third order in gradient terms. These AFTs include active model B, active model B+, and effective Cahn-Hilliard models. Moreover, density variances and spectral densities are virtually indistinguishable from that of passive phase-separated hyperuniform fields. Higher moments of the density fluctuations, however, reveal activity-dependent higher-order correlations that are not captured by conventional two-point measures that characterize hyperuniformity. Published by the American Physical Society 2024

Nonlocal-to-local convergence rates for strong solutions to a Navier-Stokes-Cahn-Hilliard system with singular potential

The main goal of this paper is to establish the nonlocal-to-local convergence of strong solutions to a Navier–Stokes–Cahn–Hilliard model with singular potential describing immiscible, viscous two-phase flows with matched densities, which is referred to as the Model H. This means that we show that the strong solutions to the nonlocal Model H converge to the strong solution to the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. Compared to previous results in the literature, our main novelty is to further establish corresponding convergence rates. Before investigating the nonlocal-to-local convergence, we first need to ensure the strong well-posedness of the nonlocal Model H. In two dimensions, this result can already be found in the literature, whereas in three dimensions, it will be shown in the present paper. Moreover, in both two and three dimensions, we establish suitable uniform bounds on the strong solutions of the nonlocal Model H, which are essential to prove the nonlocal-to-local convergence results.

Defect Solutions of the Nonreciprocal Cahn-Hilliard Model: Spirals and Targets.

We study the defect solutions of the nonreciprocal Cahn-Hilliard model. We find two kinds of defects, spirals with unit magnitude topological charge, and topologically neutral targets. These defects generate radially outward traveling waves and thus break the parity and time-reversal symmetry. For a given strength of nonreciprocity, spirals and targets with unique asymptotic wave number and amplitude are selected. We use large-scale simulations to show that at low nonreciprocity α, disordered states evolve into quasistationary spiral networks. With increasing α, we observe networks composed primarily of targets. Beyond a critical threshold α_{c}, a disorder-order transition from defect networks to traveling waves emerges. The transition is marked by a sharp rise in the global polar order.

Simplified conservative discretization of the Cahn-Hilliard-Navier-Stokes equations

In this paper we construct a novel discretization of the Cahn-Hilliard equation coupled with the Navier-Stokes equations. The Cahn-Hilliard equation models the separation of a binary mixture. We construct a very simple time integration scheme for simulating the Cahn-Hilliard equation, which is based on splitting the fourth-order equation into two second-order Helmholtz equations. We combine the Cahn-Hilliard equation with the Navier-Stokes equations to simulate phase separation in a two-phase fluid flow in two dimensions. The scheme conserves mass and momentum and exhibits consistency between mass and momentum, allowing it to be used with large density ratios. We introduce a novel discretization of the surface tension force from the phase-field variable that has finite support around the transition region. The model has a parameter that allows it to transition from a smoothed continuum surface force to a fully sharp interface formulation. We show that our method achieves second-order accuracy, and we compare our method to previous work in a variety of experiments.

Nonreciprocal collective dynamics in a mixture of phoretic Janus colloids

A multicomponent mixture of Janus colloids with distinct catalytic coats and phoretic mobilities is a promising theoretical system to explore the collective behavior arising from nonreciprocal interactions. An active colloid produces (or consumes) chemicals, self-propels, drifts along chemical gradients, and rotates its intrinsic polarity to align with a gradient. As a result the connection from microscopics to continuum theories through coarse-graining couples densities and polarization fields in unique ways. Focusing on a binary mixture, we show that these couplings render the unpatterned reference state unstable to small perturbations through a variety of instabilities including oscillatory ones which arise on crossing an exceptional point or through a Hopf bifurcation. For fast relaxation of the polar fields, they can be eliminated in favor of the density fields to obtain a microscopic realization of the Nonreciprocal Cahn–Hilliard model for two conserved species with two distinct sources of non-reciprocity, one in the interaction coefficient and the other in the interfacial tension. Our work establishes Janus colloids as a versatile model for a bottom-up approach to both scalar and polar active mixtures.

Asymptotically compatible energy of variable-step fractional BDF2 scheme for the time-fractional Cahn–Hilliard model

Abstract A novel discrete gradient structure of the variable-step fractional BDF2 formula approximating the Caputo fractional derivative of order $\alpha \in (0,1)$ is constructed by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula (BDF2) of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo derivative. Then a local discrete energy dissipation law of the variable-step fractional BDF2 implicit scheme is established for the time-fractional Cahn–Hilliard model under a weak step-ratio constraint $0.3960\le \tau _{k}/\tau _{k-1}<r^{*}(\alpha )$, where $\tau _{k}$ is the $k$th time-step size and $r^{*}(\alpha )\ge 4.660$ for $\alpha \in (0,1)$. The present result provides a practical answer to the open problem in [SINUM, 57: 218-237, Remark 6] and significantly relaxes the severe step-ratio restriction [Math. Comp., 90: 19–40, Theorem 3.2]. More interestingly, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn–Hilliard equation, respectively. To the best of our knowledge, such type energy dissipation law is established at the first time for the variable-step L2 type formula of Caputo’s derivative. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.

Sharp-Interface Limits of Cahn–Hilliard Models and Mechanics with Moving Contact Lines

Sharp-Interface Limits of Cahn–Hilliard Models and Mechanics with Moving Contact Lines

Mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces

Establishing multi-component phase field models on dynamic surfaces, and exploring the similarities and differences in phase field evolution between evolving and static surfaces, are interesting studies in multi-phase flow problems. This paper focuses on mathematical modeling and numerical simulation of the N-component Cahn-Hilliard model on evolving surfaces. In the modeling process, the evolution of the energy functional, componential mass conservation, and point-wise mass conservation (hyperplane link condition) are considered. On an evolving surface, the energy dissipation law does not hold because the surface velocity can be viewed as an external force in the system. Meanwhile, due to the velocity, the surface area may change, the componential mass conservation property and the hyperplane link condition can not be satisfied simultaneously. From the point of view of preserving these two physical properties of mass conservation, three types of N-component Cahn-Hilliard model are established on the evolving surface: componential mass conservation, point-wise mass conservation, and both componential and point-wise mass conservation. For the numerical simulation, the evolving surface finite element method is considered for the space-time discretization of the proposed model. In order to obtain a linear, decoupled, high-accurate, and stable scheme for long time numerical simulations, the stabilized semi-implicit method is integrated into the framework of the evolving surface finite element method. Through several numerical examples, the rationality of the model and the efficiency of the numerical method are shown. Additionally, three- and four-component phase separation phenomena are shown to investigate the N-component Cahn-Hilliard dynamic on various evolving surfaces.

On a generalization of the Cahn-Hilliard type equation with logarithmic nonlinearities for formation of islands

Abstract In this paper, main objective is to demonstrate the existence of solutions for an equation resembling the Cahn-Hilliard model, featuring a proliferation term and a logarithmic nonlinear term. This equation has been conceptualized within the context of interactions in liquid-gas systems, particularly in the context of island formation. The primary challenge lies in the departure from the original Cahn-Hilliard equation, as we no longer maintain conservation of the spatial average mean of the order parameter. This departure introduces the complexity in establishing uniform estimates for the solutions of the approximated problems concerning the regularization parameter, as it may potentially result in finite-time blow-up.

On the long-time behavior of the continuous and discrete solutions of a nonlocal Cahn–Hilliard type inpainting model

In this paper, we study the analytical and numerical long-time stability of a nonlocal Cahn–Hilliard model with a fidelity term for image inpainting introduced in our previous work (Jiang et al., 2024). First, we establish the uniform boundedness of the continuous problem in both L2 and H1 spaces, which is obtained by using the Gagliardo–Nirenberg inequality and the uniform Grönwall lemma. Then, for the temporal semi-discrete scheme, the uniform estimates in L2 and H1 spaces are derived with the aid of the discrete uniform Grönwall lemma under a suitable assumption on the nonlinear potential. This demonstrates the long-time stability of the proposed scheme in L2 and H1 spaces. Finally, we validate the long-time stability and the applicability of our method in signal reconstruction and image inpainting. These numerical experiments demonstrate the high effectiveness of our proposed model.

The divergence-free velocity formulation of the consistent Navier-Stokes Cahn-Hilliard model with non-matching densities, divergence-conforming discretization, and benchmarks

The prototypical diffuse-interface model that describes multi-component flows is the Navier-Stokes Cahn-Hilliard model (NSCH). Over the last decades many NSCH models have appeared that claim to describe the same physical phenomena, yet are distinct from one another. In a recent article [M.F.P. ten Eikelder, K.G. van der Zee, I. Akkerman, and D. Schillinger, Math. Mod. Meth. Appl. S. 33, pp 175-221, 2023.] we have established a unified framework of virtually all NSCH models. The framework reveals that there is only a single consistent NSCH model that naturally emanates from the underlying mixture theory. In the current article we present, verify and validate this novel consistent NSCH model by means of numerical simulation. To this purpose we discretize a divergence-free velocity formulation of the consistent NSCH model using divergence-conforming isogeometric spaces. We compare computations of our consistent model to results of existing models from literature. The predictive capability of the numerical methodology is demonstrated via three-dimensional computations of a rising bubble and the contraction of a liquid filament that compare well with experimental data.

Energy dissipation and evolutions of the nonlocal Cahn-Hilliard model and space fractional variants using efficient variable-step BDF2 method

In this work, an energy stable BDF2 scheme with general nonuniform time steps is developed for the nonlocal Cahn-Hilliard model to capture the multi-scale behavior of evolution efficiently and accurately. The fully discrete numerical scheme is demonstrated to inherit main physical properties of the continuous model, namely, mass conservation and energy dissipation. Recently, various variants of the Cahn-Hilliard model with nonlocal operators and fractional Laplacian have attracted great attention in terms of theoretical analysis and numerical approximation, while there is less emphasis on the comparisons among the different variant models. By comparing the impact of various parameters and coarsening behaviors, numerical results suggest that the nonlocal Cahn-Hilliard model bears a strong resemblance to the space fractional Cahn-Hilliard model derived from H−1 gradient flow of the nonlocal energy. Nevertheless, the order of the fractional Laplacian appears to affect only the rate of energy decay, without altering the profile of the solution for the fractional model relating to the H−β (0<β<1) gradient flow of the classical Ginzburg-Landau energy functional in one dimensional case.

High-order energy stable variable-step schemes for the time-fractional Cahn–Hilliard model

Based on the Alikhanov formula of the Caputo derivative and the exponential scalar auxiliary variable approach, two different second-order stable numerical schemes are constructed for the time-fractional Cahn–Hilliard model, including the nonlinear L2-1σ scheme and the linear L2-1σ-ESAV scheme. Applying the discrete gradient structure, we construct the asymptotically compatible energies and the associated energy dissipation laws of these two schemes for the time-fractional Cahn–Hilliard model. Several experiments are carried out to demonstrate the accuracy of our schemes and compare their computational efficiency, as well as to investigate the coarsening dynamics of the time-fractional Cahn–Hilliard model with different adaptive time-stepping strategies. In the long-time simulations, the linear L2-1σ-ESAV scheme may be more costly than the nonlinear L2-1σ scheme although the linear scheme may take less time at each time level. It is observed that the energy of the L2-1σ-ESAV scheme prones to produce small fluctuations when large time steps are used.

A multidimensional examination of phase separation in single-component fluids

A thermodynamic instability in a homogeneous fluid can lead to spontaneous formation of distinct domains within the fluid. This process involves not only the spatial redistribution of fluid density but also transient exchanges of pressure, temperature, and energy. However, classical theoretical frameworks, such as the Ginzburg–Landau and Cahn–Hilliard models, lack incorporation of these essential thermodynamic aspects. To investigate the dynamics of multiple physical fields during phase separation, we numerically solve a two-dimensional van der Waals fluid model. Thermodynamic consistency is demonstrated by verifying the coexistence curve. While the equilibrium pressure remains similar across the unstable region of the isotherm, we demonstrate that the energy in the system depends on the initial density. Although the majority of energy is stored as heat at typical values of the heat capacity, high-density domains contain less specific energy compared to their low-density counterparts due to interparticle attraction. Consequently, the transition of low-density domains into high-density through the process of coalescence releases excess energy, which redistributes in the form of longitudinal waves and heat. We also highlight the role of parameters, such as heat capacity and thermal conductivity, in less intuitive phenomena, including elevated temperature fluctuations and memory preservation.

Multifidelity methods for uncertainty quantification of a nonlocal model for phase changes in materials

This study is devoted to the construction of a multifidelity Monte Carlo (MFMC) method for the uncertainty quantification of a nonlocal, non-mass-conserving Cahn-Hilliard model for phase transitions with an obstacle potential. We are interested in estimating the expected value of an output of interest (OoI) that depends on the solution of the nonlocal Cahn-Hilliard model. As opposed to its local counterpart, the nonlocal model captures sharp interfaces without the need for significant mesh refinement. However, the computational cost of the nonlocal Cahn-Hilliard model is higher than that of its local counterpart with similar mesh refinement, inhibiting its use for outer-loop applications such as uncertainty quantification. The MFMC method augments the desired high-fidelity, high-cost OoI with a set of lower-fidelity, lower-cost OoIs to alleviate the computational burden associated with nonlocality. Most of the computational budget is allocated to sampling the cheap surrogate models to achieve speedup, whereas the high-fidelity model is sparsely sampled to maintain accuracy. For the non-mass-conserving nonlocal Cahn-Hilliard model, the use of the MFMC method results in, for a given computational budget, about an order of magnitude reduction in the mean-squared error of the expected value of the OoI relative to that of the Monte Carlo method.

Efficient surrogate models for materials science simulations: Machine learning-based prediction of microstructure properties

Determining, understanding, and predicting the so-called structure–property relation is an important task in many scientific disciplines, such as chemistry, biology, meteorology, physics, engineering, and materials science. Structure refers to the spatial distribution of, e.g., substances, material, or matter in general, while property is a resulting characteristic that usually depends in a non-trivial way on spatial details of the structure. Traditionally, forward simulations models have been used for such tasks. Recently, several machine learning algorithms have been applied in these scientific fields to enhance and accelerate simulation models or as surrogate models. In this work, we develop and investigate the applications of six machine learning techniques based on two different datasets from the domain of materials science: data from a two-dimensional Ising model for predicting the formation of magnetic domains and data representing the evolution of dual-phase microstructures from the Cahn–Hilliard model. We analyze the accuracy and robustness of all models and elucidate the reasons for the differences in their performances. The impact of including domain knowledge through tailored features is studied, and general recommendations based on the availability and quality of training data are derived from this.

Spinodal decomposition and domain coarsening in a multilayer Cahn-Hilliard model for lithium intercalation in graphite.

During the intercalation of lithium in layered host materials such as graphite, lithium atoms can move within the plane between two neighboring graphene sheets, but cannot cross the sheets. Repulsive interactions between atoms in different layers lead to the existence of ordered phases called "stages," with stage n consisting of one filled layer out of n, the others being empty. Such systems can be conveniently described by a multilayer Cahn-Hilliard model, which can be seen as a mean-field approximation of a lattice-gas model with intra- and interlayer interactions between lithium atoms. In this paper, the dynamics of stage formation after a rapid quench to lower temperature is analyzed, both by a linear stability analysis and by numerical simulation of the full equations. In particular, the competition between stages 2 and 3 is studied in detail. The linear stability analysis predicts that stage 2 always grows the fastest, even in the composition range where stage 3 is the stable equilibrium state. This is borne out by the numerical simulations, which show that stage 3 emerges only during the nonlinear coarsening of stage 2. Some consequences of this finding for the charge-discharge dynamics of electrodes in batteries are briefly discussed.