scholarly journals Coupled diffusion and phase transition: Phase fields, constraints, and the Cahn–Hilliard equation

Meccanica ◽  
2021 ◽  
Author(s):  
Fernando P. Duda ◽  
Adel F. Sarmiento ◽  
Eliot Fried
Keyword(s):  
Phase Field ◽  
Mass Fraction ◽  
Three Dimensional ◽  
The Body ◽  
Hilliard Equation ◽  
Coupled Diffusion ◽  
Phase Fields ◽  
Cahn Hilliard Equation ◽  
Distinguishing Features

AbstractWe develop a constrained theory for constituent migration in bodies with microstructure described by a scalar phase field. The distinguishing features of the theory stem from a systematic treatment and characterization of the reactions needed to maintain the internal constraint given by the coincidence of the mass fraction and the phase field. We also develop boundary conditions for situations in which the interface between the body and its environment is structureless and cannot support constituent transport. In addition to yielding a new derivation of the Cahn–Hilliard equation, the theory affords an interpretation of that equation as a limiting variant of an Allen–Cahn type diffusion system arising from the unconstrained theory obtained by considering the mass fraction and the phase field as independent quantities. We corroborate that interpretation with three-dimensional numerical simulations of a recently proposed benchmark problem.

scholarly journals Curve and Surface Smoothing Using a Modified Cahn-Hilliard Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Yongho Choi ◽  
Darea Jeong ◽  
Junseok Kim
Keyword(s):  
Piecewise Linear ◽  
Three Dimensional ◽  
Computational Results ◽  
Fourier Spectral Method ◽  
Splitting Scheme ◽  
Surface Smoothing ◽  
Three Dimensional Objects ◽  
Hilliard Equation ◽  
Stable Splitting ◽  
Cahn Hilliard Equation

We present a new method using the modified Cahn-Hilliard (CH) equation for smoothing piecewise linear shapes of two- and three-dimensional objects. The CH equation has good smoothing dynamics and it is coupled with a fidelity term which keeps the original given data; that is, it does not produce significant shrinkage. The modified CH equation is discretized using a linearly stable splitting scheme in time and the resulting scheme is solved by using a Fourier spectral method. We present computational results for both curve and surface smoothing problems. The computational results demonstrate that the proposed algorithm is fast and efficient.

scholarly journals Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains

2012 ◽  
Vol 468 (2147) ◽  
pp. 3705-3724 ◽  
Author(s):  
Markus Schmuck ◽  
Marc Pradas ◽  
Grigorios A. Pavliotis ◽  
Serafim Kalliadasis
Keyword(s):  
Porous Media ◽  
Free Energy ◽  
Phase Field ◽  
Single Channel ◽  
Confined Geometries ◽  
Science And Engineering ◽  
Hilliard Equation ◽  
Phase Field Models ◽  
Cahn Hilliard Equation ◽  
Double Well Potential

We derive a new, effective macroscopic Cahn–Hilliard equation whose homogeneous free energy is represented by fourth-order polynomials, which form the frequently applied double-well potential. This upscaling is done for perforated/strongly heterogeneous domains. To the best knowledge of the authors, this seems to be the first attempt of upscaling the Cahn–Hilliard equation in such domains. The new homogenized equation should have a broad range of applicability owing to the well-known versatility of phase-field models. The additionally introduced feature of systematically and reliably accounting for confined geometries by homogenization allows for new modelling and numerical perspectives in both science and engineering. Our results are applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls.

Investigation of Tumor Growth Based on Phase Field Model

2011 ◽  
Author(s):  
Zhi Zhu He ◽  
Jing Liu
Keyword(s):  
Cell Proliferation ◽  
Tumor Growth ◽  
Tumor Cell ◽  
Phase Field ◽  
Classical Model ◽  
Phase Field Method ◽  
Phase Method ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

This paper presents and investigates the tumor growth based on a phase model. The tumor core is necrotic and inhibitor chemical species are considered. The interface of tumor and health tissue is tracked using a phase field equation. The reformulation of a classical model, accounting for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion, is derived. The advantages of the finite difference methodology employed are generality and relative simplicity implication. We present simulations of the nonlinear evolution of growing tumors morphology and discuss the effects of tumor microenvironment. Mechanisms reflecting the tumor growth and development behavior was preliminarily interpreted. Recently numerous mathematical have been developed to investigate the growth dynamics of tumor [1–8]. One of most significant model developed by Wise [8] is based on Cahn-Hilliard equation, which is conservation phase field method. Allen-Chan nonconservation phase field has been developed to track the moving interface for multiphase simulation by Sun [9]. Allen-Chan equation is second order, while Cahn-Hilliard equation is fourth order in space. Thus, we introduce the Allen-Chan phase method [9–10] to simulate the tumor growth, which is very simple for numerical simulation The computation domain is illustrated in Fig. 1, where ΩH denotes host tissue, the tumor domains is comprised of viable tumor cell ΩV and dead tumor cell ΩD. The numerical results are presented at Fig. (2–4). One can find that the growth of tumor strongly depend on the nutrients and nonlinear unstable growth may lead to finger shaped pattern, which is in agreement with recent experimental observations [7] of in vivo tumor. In summary, a phase method has been developed to study diffusion and consumption of the nutrients and tumor cell proliferation, necrosis and migration, which discloses the evolution of complex shape of tumor.

scholarly journals Micromechanically informed reaction-induced damage and fracture in polymers due to oxidation

2020 ◽  
Author(s):  
Shabnam Konica ◽  
Trisha Sain
Keyword(s):  
Network Structure ◽  
Phase Field ◽  
Network Theory ◽  
Three Dimensional ◽  
Diffusion Reaction ◽  
Temperature Oxidation ◽  
Oxidative Aging ◽  
Coupled Diffusion ◽  
Damage And Fracture ◽  
Amorphous Network

High temperature oxidation in polymers is a complex phenomenon, driven by the coupled diffusion-reaction process, causing changes in the amorphous network structure and resulting in property degradation. Prolonged oxidation in polymers results in the formation of a coarse, oxide layer on the outer surface and induces spontaneous cracking inside the material. In this paper, we present a chemical reaction-driven evolving network theory coupled with phase-field fracture to describe the effect of oxidation in polymers across different length scales. The theory considers the coupling between oxygen diffusion, chemical reactions, large deformation of polymers, and phase-field fracture in a thermodynamically consistent way. Guided by the statistical mechanics, the network theory has been introduced to model the reaction induced chain scissions and crosslinking events causing significant changes in the three-dimensional network structure. Further, these microscale events have been considered as the reason behind macroscopic mechanical property degradation, namely oxidative embrittlement. Finally the network theory is coupled with a phase-field fracture model to capture the macroscale damage and fracture in the polymer under stress-coupled oxidation conditions. We derive the specific constitutive forms for all the physical-chemical processes based on the thermodynamic inequality conditions, and numerically implement the theory in finite elements by writing ABAQUS user-defined element (UEL) subroutine. To present the model's capability, numerical examples with standard fracture geometries have been studied. The simulation results have demonstrated the model's capability of predicting the effect of oxidative aging on the polymer's response.

scholarly journals Effect of the Particle Size Distribution on the Cahn-Hilliard Dynamics in a Cathode of Lithium-Ion Batteries

Batteries ◽  
2020 ◽  
Vol 6 (2) ◽  
pp. 29
Author(s):  
Pavel L’vov ◽  
Renat Sibatov
Keyword(s):  
Particle Size ◽  
Particle Size Distribution ◽  
Size Distribution ◽  
Phase Field ◽  
Single Phase ◽  
Field Model ◽  
Phase Field Model ◽  
Hilliard Equation ◽  
Wide Range ◽  
Cahn Hilliard Equation

The phase-field model based on the Cahn-Hilliard equation is employed to simulate lithium intercalation dynamics in a cathode with particles of distributed size. We start with a simplified phase-field model for a single submicron particle under galvanostatic condition. We observe two stages associated with single-phase and double-phase patterns typical for both charging and discharging processes. The single-phase stage takes approximately 10–15% of the process and plays an important role in the intercalation dynamics. We establish the laws for speed of front propagation and evolution of single-phase concentration valid for different sizes of electrode particles and a wide range of temperatures and C-rates. The universality of these laws allows us to formulate the boundary condition with time-dependent flux density for the Cahn-Hilliard equation and analyze the phase-field intercalation in a heterogeneous cathode characterized by the particle size distribution.

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