Cahn-Hilliard Energy Research Articles

Minimizers for the de Gennes–Cahn–Hilliard energy under strong anchoring conditions

AbstractIn this article, we use the Nehari manifold and the eigenvalue problem for the negative Laplacian with Dirichlet boundary condition to analytically study the minimizers for the de Gennes–Cahn–Hilliard energy with quartic double‐well potential and Dirichlet boundary condition on the bounded domain. Our analysis reveals a bifurcation phenomenon determined by the boundary value and a bifurcation parameter that describes the thickness of the transition layer that segregates the binary mixture's two phases. Specifically, when the boundary value aligns precisely with the average of the pure phases, and the bifurcation parameter surpasses or equals a critical threshold, the minimizer assumes a unique form, representing the homogeneous state. Conversely, when the bifurcation parameter falls below this critical value, two symmetric minimizers emerge. Should the boundary value be larger or smaller from the average of the pure phases, symmetry breaks, resulting in a unique minimizer. Furthermore, we derive bounds of these minimizers, incorporating boundary conditions and features of the de Gennes–Cahn–Hilliard energy.

Gamma convergence for the de Gennes–Cahn–Hilliard energy

Gamma convergence for the de Gennes–Cahn–Hilliard energy

Approximation of surface diffusion flow: A second-order variational Cahn–Hilliard model with degenerate mobilities

This paper addresses the approximation of surface diffusion flow using Cahn–Hilliard-type models. We introduce and analyze a new variational phase field model that associates the classical Cahn–Hilliard energy with two degenerate mobilities and guarantees a second-order accuracy in the approximation of the sharp limit. We also introduce simple and efficient numerical schemes to approximate the solutions to three Cahn–Hilliard-type models for surface diffusion flow, including the one we propose, and we provide 2D and 3D numerical experiments that illustrate the advantages of our approach.

Cahn–Hilliard equations on an evolving surface

We describe a functional framework suitable to the analysis of the Cahn–Hilliard equation on an evolving surface whose evolution is assumed to be given a priori. The model is derived from balance laws for an order parameter with an associated Cahn–Hilliard energy functional and we establish well-posedness for general regular potentials, satisfying some prescribed growth conditions, and for two singular non-linearities – the thermodynamically relevant logarithmic potential and a double-obstacle potential. We identify, for the singular potentials, necessary conditions on the initial data and the evolution of the surfaces for global-in-time existence of solutions, which arise from the fact that integrals of solutions are preserved over time, and prove well-posedness for initial data on a suitable set of admissible initial conditions. We then briefly describe an alternative derivation leading to a model that instead preserves a weighted integral of the solution and explain how our arguments can be adapted in order to obtain global-in-time existence without restrictions on the initial conditions. Some illustrative examples and further research directions are given in the final sections.

Symmetry of constrained minimizers of the Cahn–Hilliard energy on the torus

We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to so-called volume-constrained minimizers of the Cahn–Hilliard energy. The resulting connectedness of superlevel sets is used in two dimensions together with the Bonnesen inequality to quantitatively estimate the sphericity of minimizers. We also show how two-point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the Cahn–Hilliard model.

A High-Order Convex Splitting Method for a Non-Additive Cahn–Hilliard Energy Functional

Various Cahn–Hilliard (CH) energy functionals have been introduced to model phase separation in multi-component system. Mathematically consistent models have highly nonlinear terms linked together, thus it is not well-known how to split this type of energy. In this paper, we propose a new convex splitting and a constrained Convex Splitting (cCS) scheme based on the splitting. We show analytically that the cCS scheme is mass conserving and satisfies the partition of unity constraint at the next time level. It is uniquely solvable and energy stable. Furthermore, we combine the convex splitting with the specially designed implicit–explicit Runge–Kutta method to develop a high-order (up to third-order) cCS scheme for the multi-component CH system. We also show analytically that the high-order cCS scheme is unconditionally energy stable. Numerical experiments with ternary and quaternary systems are presented, demonstrating the accuracy, energy stability, and capability of the proposed high-order cCS scheme.

A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

We develop a novel finite element method for a phase field model of nematic liquid crystal droplets. The continuous model considers a free energy comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energy representing the interfacial energy of the droplet, and a weak anchoring energy representing the interaction of the liquid crystal molecules with the surface tension on the interface (i.e. anisotropic surface tension). Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics. We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy, as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part. Next, we present a discrete gradient flow method and show that it is monotone energy decreasing. Furthermore, we show that global discrete energy minimizers $\Gamma$-converge to global minimizers of the continuous energy. We conclude with numerical experiments illustrating different gradient flow dynamics, including droplet coalescence and break-up.

Existence and properties of certain critical points of the Cahn-Hilliard energy

The Cahn-Hilliard energy landscape on the torus is explored in the critical regime of large system size and mean value close to $-1$. Existence and properties of a droplet-shaped local energy minimizer are established. A standard mountain pass argument leads to the existence of a saddle point whose energy is equal to the energy barrier, for which a quantitative bound is deduced. In addition, finer properties of the local minimizer and appropriately defined constrained minimizers are deduced. The proofs employ the $\Gamma$-limit (identified in a previous work), quantitative isoperimetric inequalities, variational arguments, and Steiner symmetrization.

An efficient, unconditionally energy stable local discontinuous Galerkin scheme for the Cahn–Hilliard–Brinkman system

In this paper, we present an efficient and unconditionally energy stable fully-discrete local discontinuous Galerkin (LDG) method for approximating the Cahn–Hilliard–Brinkman (CHB) system, which is comprised of a Cahn–Hilliard type equation and a generalized Brinkman equation modeling fluid flow. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit scheme which consists of the implicit Euler method combined with a convex splitting of the discrete Cahn–Hilliard energy strategy for the temporal discretization. The unconditional energy stability of this fully-discrete convex splitting scheme is also proved. Obviously, the fully-discrete equations at the implicit time level are nonlinear, and to enhance the efficiency of the proposed approach, the nonlinear Full Approximation Scheme (FAS) multigrid method has been employed to solve this system of algebraic equations. We also show the nearly optimal complexity numerically. Numerical experiments based on the overall solution method of combining the proposed LDG method, convex splitting scheme and the nonlinear multigrid solver are given to validate the theoretical results and to show the effectiveness of the proposed approach for the CHB system.

An efficient fully-discrete local discontinuous Galerkin method for the Cahn–Hilliard–Hele–Shaw system

In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction (Δt=O(Δx4)) of explicit time discretization methods for stability, we introduce a semi-implicit time integration scheme which is based on a convex splitting of the discrete Cahn–Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.

Modeling Tumor Growth in an Evolving Tissue Structure: A Diffuse Domain Approach

While cancer has been one of the leading causes of human death, predicting and thwarting cancerous tumor development are challenging because of interacting multiscale processes coupled with emerging spatiotemporal patterns that lead to microenvironmental heterogeneity and further to infiltration tendency of tumors. Built upon decades of knowledge toward the biological processes involved in tumor progress, mathematical modeling and analysis have been employed to decipher key mechanisms underlying the sophisticated emerging phenomena in tumor growth observed experimentally and clinically. Among the efforts, we have recently developed a Cahn-Hilliard model of tumor growth at tissue scale, where the Cahn-Hilliard energy described cell aggregation and phase separation due to differentiated cell-cell adhesion. Various nontrivial emerging patterns resembling observed tumor morphology were identified over the parameter space of our model. However, the previous development and analysis of our model have been implicitly based on the assumption of a homogeneous microenvironmental background and unrestricted boundaries. Most clinically relevant tumors are constrained by particular organ tissue structures that may co-evolve with the progressing tumors and have profound impact on tumor-microenvironment interactions. Here we adopt a recently developed diffuse-domain approach, utilizing the Cahn-Hilliard equation framework we have previously established, to adapt partial-differential-equation models of tumor growth to a co-evolving tissue geometry. We will demonstrate this approach by modeling the growth of lymphoma within a lymph node and ductal carcinoma in situ within mammary ducts.

Branch interactions and long-term dynamics for the diblock copolymer model in one dimension

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki [26]. Their model yields a Cahn-Hilliard-like equation, where a nonlocal term is added to the standard Cahn-Hilliard energy. We study the long-term dynamics of this model on one-dimensional domains through a combination of bifurcation theoretic results and numerical simulations. Our results shed light on how the complicated bifurcation behavior of the diblock copolymer model is related to the better known bifurcation structure of the Cahn-Hilliard equation. In addition, we demonstrate that this knowledge can be used to predict the long-term dynamics of solutions originating close to the homogeneous equilibrium. In particular, we show that the periodicity of the long-term limit of such solutions can be predicted by tracking certain secondary bifurcation points in the bifurcation diagram, and that the long-term limit is in general not given by the global energy minimizer.

Count and Symmetry of Global and Local Minimizers of the Cahn-Hilliard Energy Over Cylindrical Domains

Abstract We address the problem of minimization of the Cahn-Hilliard energy functional under a mass constraint over two and three-dimensional cylindrical domains. Although existence is presented for a general case the focus is mainly on rectangles, parallelepipeds and circular cylinders. According to the symmetry of the domain the exact numbers of global and local minimizers are given as well as their geometric profile and interface locations; all are onedimensional increasing/decreasing and odd functions for domains with lateral symmetry in all axes and also for circular cylinders. The selection of global minimizers by the energy functional is made via the smallest interface area criterion. The approach utilizes Γ−convergence techniques to prove existence of an one-parameter family of local minimizers of the energy functional for any cylindrical domain. The exact numbers of global and local minimizers as well as their geometric profiles are accomplished via suitable applications of the unique continuation principle while exploring the domain geometry in each case and also the preservation of global minimizers through the process of Γ−convergence.

Curvature driven flow of bi-layer interfaces

We introduce the Functionalized Cahn–Hilliard (FCH) energy, a negative multiple of the Cahn–Hilliard energy balanced against the square of its own variational derivative, as a finite width regularization of the sharp-interface Canham–Helfrich energy. Mass-preserving gradient flows associated to the FCH energy are higher-order phase field models which develop not only single-layer, or front-type interfaces, but also bi-layer, or homoclinic interfaces with associated endcap and multi-junction structures. The single-layer interfaces manifest a fingering instability which grows into endcapped bi-layers. The meandering growth of the bi-layer interfaces and the subsequent merging lead to a multi-junction dominated network that bears a striking similarity to the phase separated domains of both perfluorosulfonic membranes and amphiphilic di-block co-polymer solutions. The bi-layers generated by the gradient flows of the FCH energy have an interfacial width which scales with ε≪1, however for fixed ε, there is a class of bi-layers parameterized by width and background state. Our primary result is the asymptotic derivation of the normal velocity of a closed bi-layer hypersurface in Rd (d≥2) coupled to the evolution for the surface width, curvature, and background state. We also show the convergence of the FCH energy to a scaled Canham–Helfrich type energy for both single and bi-layer interfaces, with the surface area coefficient of the limiting Canham–Helfrich energy coupling to the bi-layer width. Thus the bi-layer networks grow to maximize surface area while minimizing the square of curvature, up to the point that the increase in surface area stretches the bi-layers too thin.

Convergence of phase-field approximations to the Gibbs–Thomson law

We prove the convergence of phase-field approximations of the Gibbs–Thomson law. This establishes a relation between the first variation of the Van der Waals–Cahn–Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs–Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn–Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta–Kawasaki as a model for micro-phase separation in block-copolymers.

Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series)

9R3. Bifurcation Theory: An Introduction With Applications to PBEs (Applied Mathematical Sciences 156 Series). - Edited by H Kielhofer (Inst for Math, Univ of Augsburg, Universitatsstr 14, Raum 2011, Augsburg D-86135 Germany). Springer-Verlag, New York. 2004. 346 pp. ISBN 0-387-40401-5. $69.95.Reviewed by HW Haslach Jr (Dept of Mech Eng, Univ. of Maryland, College Park MD 20742-3035).This unified exposition of the single-parameter bifurcation response of an operator on infinite dimensional space is organized into three sections: an abstract development of local and then global bifurcation and applications to elliptic and parabolic partial differential equations. The local bifurcation theory, taking up about half the book, depends on the breakdown of the implicit function theorem and is based on the Lyapunov-Schmidt reduction for infinite dimensional spaces. The case that the Fre´chet derivative has a one-dimensional kernel includes the saddle-node and the various types of pitchfork bifurcations. A principle of exchange of stability is proved. A Hopf bifurcation theorem for periodic solutions is first proved for ordinary differential equations and then for retarded functional differential equations. A center manifold theorem is used for Hopf bifurcation of Hamiltonian, reversible, and conservative systems. A principle of exchange of stability for Hopf bifurcations is proved, and the stability of global continuation of solutions is examined. A principle of exchange of stability for period-doubling bifurcations is developed. The Newton polygon method is described for problems with a one-dimensional kernel in both the degenerate and nondegenerate cases. Floquet exponents are used to create a principle of exchange of stability for degenerate Hopf bifurcations. Some results are given for higher dimensional kernels of the Fre´chet derivative and multiparameter bifurcation. The global bifurcation theory, subject of the second section, is based on the local index of the operator F, obtained from the Leray-Schauder degree which takes the place of the Brouwer degree of finite-dimensional problems. The crossing number, which is related to a change in Morse index, is defined from the index of the operator. Then if DxF0,λ has an odd crossing number for a particular value of the bifurcation parameter λ, that value of the parameter gives a bifurcation point. A degree is defined for a class of Fredholm operators that includes some classes of elliptic operators and for potential operators, and global bifurcation theorems proved, often using Rabinowitz’s theorem. The applications forming the third section begin with results on elliptic operators of second order acting on scalar functions. An example using the degenerate Newton polygon is developed. The particular case of the Cahn-Hilliard energy that describes a binary alloy is solved by taking the mass as the bifurcation parameter and applying a symmetry constraint to obtain a one-dimensional kernel. Then local bifurcations of the free nonlinear vibrations described by the nonlinear wave equation with the period as bifurcation parameter are examined in the one-dimensional case and some extensions are given to higher dimensions. Results are also obtained for the wave equation on the unit sphere. A Hopf bifurcation is analyzed for a nonlinear parabolic problem in which the principle of exchange of stability holds. Finally global bifurcation and continuation results are given for quasi-linear elliptic problems and are applied to the Euler-Lagrange equation for the Cahn-Hilliard energy problem. This mathematics book requires substantial mathematical background. The book is directed to advanced readers rather than beginners so that some basic ideas are not defined. The author has separated the abstract description in the first two sections from the applications with the goal of clarity and allowing the results to be available for any application, but at the risk of making it harder to understand the significance of some of the ideas. In many applications to models of physical behavior, the hypotheses of the various theorems are quite difficult to verify. The book is very useful as a reference because it collects and organizes the bifurcation analysis of infinite-dimensional operators. It could also be used as a text in an advanced course on bifurcation theory with an emphasis on partial differential equations.

On a one-dimensional variational problem related to the Cahn–Hilliard energy in a bent strip-like domain

On a one-dimensional variational problem related to the Cahn–Hilliard energy in a bent strip-like domain

Minimizing Sequences Selected via Singular Perturbations, and their Pattern Formation

We study the Cahn-Hilliard energy E ɛ(u) over the unit square under the constraint of a constant mass m with (ɛ > 0) and without ɛ= 0) interfacial energy. Minimizers of E 0(u) have no preferred pattern and we select patterns via sequences of conditionally critical points of E ɛ(u) converging to minimizers as ɛ tends to zero. Those critical points are not minimizers if the singular limit has no minimal interface. We obtain them by a global bifurcation analysis of the Euler-Lagrange equations for E ɛ(u) where the mass m is the bifurcation parameter. We make use of the symmetry of the unit square, and the elliptic maximum principle, in turn, implies that the location of maxima and minima is fixed for all solutions on global branches. This property is used to guarantee the existence of a singular limit and to verify the Weierstrass-Erdmann corner condition which proves its minimizing property.

Forthcoming articles

We introduce the Functionalized Cahn–Hilliard (FCH) energy, a negative multiple of the Cahn–Hilliard energy balanced against the square of its own variational derivative, as a finite width regularization of the sharp-interface Canham–Helfrich energy. Mass-preserving gradient flows associated to the FCH energy are higher-order phase field models which develop not only single-layer, or front-type interfaces, but also bi-layer, or homoclinic interfaces with associated endcap and multi-junction structures. The single-layer interfaces manifest a fingering instability which grows into endcapped bi-layers. The meandering growth of the bi-layer interfaces and the subsequent merging lead to a multi-junction dominated network that bears a striking similarity to the phase separated domains of both perfluorosulfonic membranes and amphiphilic di-block co-polymer solutions. The bi-layers generated by the gradient flows of the FCH energy have an interfacial width which scales with ε≪1, however for fixed ε, there is a class of bi-layers parameterized by width and background state. Our primary result is the asymptotic derivation of the normal velocity of a closed bi-layer hypersurface in Rd (d≥2) coupled to the evolution for the surface width, curvature, and background state. We also show the convergence of the FCH energy to a scaled Canham–Helfrich type energy for both single and bi-layer interfaces, with the surface area coefficient of the limiting Canham–Helfrich energy coupling to the bi-layer width. Thus the bi-layer networks grow to maximize surface area while minimizing the square of curvature, up to the point that the increase in surface area stretches the bi-layers too thin.