Phase-Field Simulation of Process in Sintering Ceramics

2010 ◽  
Vol 154-155 ◽  
pp. 1674-1679 ◽  
Author(s):  
Liang Liang Liu ◽  
Feng Gao ◽  
Bo Li ◽  
Guo Xin Hu
Keyword(s):  
Phase Field ◽  
Landau Equation ◽  
Phase Field Model ◽  
Growth Exponent ◽  
Microstructural Development ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Ginzburg Landau ◽  
Cahn Hilliard Equation ◽  
Sintering Neck

A phase field model is used to describe the microstructural development during the ceramic sintering. The evolution of the density is governed by Cahn-Hilliard equation, while the long-range order (lro) parameter fields by the time-dependent Ginzburg-Landau equation. In the simulation, green microstructures that consist of circular particles with different particle-size distributions and green densities have been produced by the stochastic growth model. The porosity of 25.6% was considered. The formation and growth of sintering neck, the seal, spheroidization as well as disappearance of pores and growth of grains are observed during simulation. The simulation results show grain boundary diffusion and surface diffusion are the dominate mechanism at the initial sintering stage. The predicted growth exponent of sintering neck and grain is consistent with the existing theoretical analysis.

Phase Field Simulation of Ferroelectric Single Crystals

Aerospace ◽  
2005 ◽  
Author(s):  
T. Liu ◽  
C. S. Lynch
Keyword(s):  
Phase Field ◽  
Wall Motion ◽  
Field Model ◽  
Landau Equation ◽  
Phase Field Model ◽  
Domain Wall Motion ◽  
Ferroelectric Materials ◽  
Time Dependent ◽  
Ginzburg Landau ◽  
Domain Structures

Ferroelectric materials exhibit spontaneous polarization and domain structures below the Curie temperature. In this study a cubic to tetragonal phase transformation and the evolution of domain structures in ferroelectric crystals are simulated by using the time-dependent Ginzburg-Landau equation. The effects of electric boundary conditions on the formation of domain patterns and field induced polarization switching are discussed. The phase field model is used to simulate the formation of domain structures, domain wall motion and the macroscopic response of ferroelectric materials under external fields.

Domain Structures and Phase Diagram in 2D Ferroelectrics Under Applied Biaxial Strains - Phase Field Simulations and Thermodynamic Calculations

2005 ◽  
Vol 881 ◽  
Author(s):  
Jie Wang ◽  
Yulan Li ◽  
Long-Qing Chen ◽  
Tong-Yi Zhang
Keyword(s):  
Phase Diagram ◽  
Phase Field ◽  
Field Model ◽  
Landau Equation ◽  
Phase Field Model ◽  
Ginzburg Landau ◽  
Elastic Interactions ◽  
Domain Structures ◽  
Domain State ◽  
Different Temperatures

Absract:The microscopic domain structures in 2D ferroelectrics under applied biaxial strains are investigated using a phase field model based on the time-dependent Ginzburg-Landau equation that takes both long-range electric and -elastic interactions into account. The stable polarization patterns are simulated at different temperatures and applied inequiaxial strains. The results show that the ferroelectrics transfer from multi-domain state to single-domain state when temperature surpasses a critical value. On the other hand, the macroscopic equilibrium polarization states are also studied through a nonlinear thermodynamic theory. The corresponding transition from a1, a2 state (p1 ≠ 0, P2 ≠ 0) to a1 state (p1 ≠0, P2 = 0) or a2 state (p2 ≠ 0, P2 = 0) is also found from the “strain-straintemperature” phase diagram, which is constructed by minimizing Helmholtz free energy.

scholarly journals Phase-field model of precipitation processes with coherency loss

2021 ◽  
Vol 7 (1) ◽  
Author(s):  
Tian-Le Cheng ◽  
You-Hai Wen
Keyword(s):  
Phase Field ◽  
Field Model ◽  
Ostwald Ripening ◽  
Phase Field Model ◽  
Energy Criterion ◽  
Field Variable ◽  
Ginzburg Landau ◽  
Underlying Mechanisms ◽  
Flood Fill ◽  
Coherency Loss

AbstractA phase-field model is proposed to simulate coherency loss coupled with microstructure evolution. A special field variable is employed to describe the degree of coherency loss of each particle and its evolution is governed by a Ginzburg-Landau type kinetic equation. For the sake of computational efficiency, a flood-fill algorithm is introduced that can drastically reduce the required number of field variables, which allows the model to efficiently simulate a large number of particles sufficient for characterizing their statistical features during Ostwald ripening. The model can incorporate size dependence of coherency loss, metastability of coherent particles, and effectively incorporate the underlying mechanisms of coherency loss by introducing a so-called differential energy criterion. The model is applied to simulate coarsening of Al3Sc precipitates in aluminum alloy and comprehensively compared with experiments. Our results clearly show how the particle size distribution is changed during coherency loss and affects the coarsening rate.

scholarly journals Optimal control of stochastic phase-field models related to tumor growth

2020 ◽  
Vol 26 ◽  
pp. 104
Author(s):  
Carlo Orrieri ◽  
Elisabetta Rocca ◽  
Luca Scarpa
Keyword(s):  
Optimal Control ◽  
Tumor Growth ◽  
Phase Field ◽  
Phase Field Model ◽  
Growth Dynamics ◽  
Necessary Optimality Conditions ◽  
Reaction Diffusion ◽  
Adjoint System ◽  
Reaction Diffusion Equation ◽  
Cahn Hilliard Equation

We study a stochastic phase-field model for tumor growth dynamics coupling a stochastic Cahn-Hilliard equation for the tumor phase parameter with a stochastic reaction-diffusion equation governing the nutrient proportion. We prove strong well-posedness of the system in a general framework through monotonicity and stochastic compactness arguments. We introduce then suitable controls representing the concentration of cytotoxic drugs administered in medical treatment and we analyze a related optimal control problem. We derive existence of an optimal strategy and deduce first-order necessary optimality conditions by studying the corresponding linearized system and the backward adjoint system.

Phase Field Modeling of Ferroelectric Domain Wall Interactions With Charge Defects

2006 ◽  
Author(s):  
Chad M. Landis
Keyword(s):  
Phase Field ◽  
Domain Walls ◽  
Virtual Work ◽  
Landau Equation ◽  
Second Law Of Thermodynamics ◽  
Ferroelectric Material ◽  
Diffuse Interface ◽  
Element Formulation ◽  
Ginzburg Landau ◽  
Domain Structures

The overall objective of this work is to develop a theoretical model that can track the evolution of the domain structures in ferroelectric crystals, which are responsible for the non-linear electromechanical behavior of these materials. To this end, a continuum thermodynamics framework is devised, and the theory falls into the class of phase-field or diffuse-interface modeling approaches. Here a set of micro-forces and governing balance laws are postulated and applied within the second law of thermodynamics to identify the appropriate material constitutive relationships. The approach is shown to yield the commonly accepted Ginzburg-Landau equation for the evolution of the polarization order parameter. Within the theory a form for the free energy is postulated that can be applied to fit the general elastic, piezoelectric and dielectric properties of a ferroelectric material near its spontaneously polarized state. Thereafter, a principle of virtual work is specified for the theory and is implemented to devise a finite element formulation. The theory and numerical methods are used to investigate the interactions of 180° and 90° domain walls with an array of charge defects and to determine the electromechanical pinning strength of the array on the walls.

A phase-field study of domain dynamics in ferroelectric BCT-BZT system

MRS Advances ◽  
2016 ◽  
Vol 1 (40) ◽  
pp. 2783-2788 ◽  
Author(s):  
Soumya Bandyopadhyay ◽  
Tushar Jogi ◽  
Kumaraswamy Miriyala ◽  
Ranjith Ramadurai ◽  
Saswata Bhattacharyya
Keyword(s):  
Free Energy ◽  
Phase Field ◽  
Three Dimensional ◽  
Phase Field Model ◽  
Ginzburg Landau ◽  
Equimolar Composition ◽  
Electrical Boundary Conditions ◽  
Experimental Findings ◽  
Domain Dynamics ◽  
Three Dimensional Simulations

ABSTRACTWe present a thermodynamically consistent phase-field model describing the free energy of perovskite-based BCT-BZT solid solution containing an intermediate morphotropic phase boundaries. The Landau coefficients are derived as functions of composition of zirconium. The electrostrictive and elastic constants are appropriately chosen from experimental findings. The resulting Landau free energy is constructed to describe the stable polarization states as a function of composition. The evolution of the polarization order parameters at a particular composition is described by a set of time-dependent Ginzburg-Landau (TDGL) equations. Additionally, we solve Poisson’s equation and mechanical equilibrium equation to account for the ferroelectric/ferroelastic interactions. We have performed two dimensional and three-dimensional simulations with appropriate electrical boundary conditions to study the effect of external electric field on domain dynamics in BCT-BZT system at the equimolar composition.

Phase-field modeling of bimodal particle size distributions during continuous cooling

2003 ◽  
Vol 51 (4) ◽  
pp. 1123-1132 ◽  
Author(s):  
Y.H. Wen ◽  
J.P. Simmons ◽  
C. Shen ◽  
C. Woodward ◽  
Y. Wang
Keyword(s):  
Particle Size ◽  
Phase Field ◽  
Size Distributions ◽  
Particle Size Distributions ◽  
Phase Field Modeling ◽  
Continuous Cooling ◽  
Field Modeling

Liquid phase stratification induced by large temperature gradient during spinodal decomposition in Fe–Cr alloys: A phase-field study

2021 ◽  
pp. 2150374
Author(s):  
Lifei Du ◽  
Runbo Tian ◽  
Tiantian Shi ◽  
Youqi Cao
Keyword(s):  
Liquid Phase ◽  
Temperature Gradient ◽  
Spinodal Decomposition ◽  
Phase Field ◽  
Anisotropic Diffusion ◽  
Phase Field Model ◽  
Solute Diffusion ◽  
Large Temperature ◽  
Cahn Hilliard Equation ◽  
Kinetics Of

The spinodal decomposition in Fe-40at.%Cr binary alloy is numerically studied by implementing the phase-field model based on Cahn–Hilliard equation. Effects of different temperature gradients on the solute distributing characteristics during the spinodal decomposition are investigated. In the system with a temperature gradient, the phase decomposition happens gradually from low temperature to high temperature, and a metastable stratification is achieved with specified temperature distribution. The critical temperature and corresponding temperature gradient are specified for the obvious solute stratification in the binary Fe–Cr alloy. The kinetics of the solute diffusion during the spinodal decomposition is discussed to reveal the liquid phase stratification induced by the anisotropic diffusion with the nonuniform temperature field. Therefore, tailoring the heat treatment during the spinodal decomposition in Fe–Cr binary alloys might be an efficient way to obtain nanometer coherent microstructures with specified solute distribution.

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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