scholarly journals A mathematical model for phase separation: A generalized Cahn-Hilliard equation

2011 ◽  
Vol 34 (10) ◽  
pp. 1193-1201 ◽  
Author(s):  
A. Berti ◽  
I. Bochicchio
Keyword(s):  
Mathematical Model ◽  
Phase Separation ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

scholarly journals Phase Separation in the Advective Cahn–Hilliard Equation

2020 ◽  
Vol 30 (6) ◽  
pp. 2821-2845
Author(s):  
Yu Feng ◽  
Yuanyuan Feng ◽  
Gautam Iyer ◽  
Jean-Luc Thiffeault
Keyword(s):  
Phase Separation ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

PHASE SEPARATION DYNAMICS AND EXTERNAL FORCE FIELD

1988 ◽  
Vol 02 (06) ◽  
pp. 765-771 ◽  
Author(s):  
K. KITAHARA ◽  
Y. OONO ◽  
DAVID JASNOW
Keyword(s):  
Dynamical System ◽  
Phase Separation ◽  
Force Field ◽  
External Force ◽  
System Modeling ◽  
Bulk Phase ◽  
Hilliard Equation ◽  
External Force Field ◽  
Dynamical System Modeling ◽  
Cahn Hilliard Equation

If spinodal decomposition is modeled by the Cahn-Hilliard (-Cook) equation, the effect of a uniform external force such as gravitation does not appear in the bulk phase kinetics. In contrast, in the Kawasaki exchange modeling of the local dynamics of binary alloys, this effect directly modifies the bulk phase kinetics. We resolve this paradox through the cell-dynamical-system modeling of the Kawasaki exchange dynamics. Its continuum version has turned out to be a modified Cahn-Hilliard equation already proposed by Langer et al. about ten years ago. We demonstrate some examples in which the correction to the Cahn-Hilliard equation is significant.

Effect of Mo and Ni on Phase Separation in Fe-Cr=Mo-Ni Quaternary Alloys

2011 ◽  
Vol 409 ◽  
pp. 449-454
Author(s):  
Ling Ling Yang ◽  
Yoshiyuki Saito
Keyword(s):  
Numerical Simulation ◽  
Phase Separation ◽  
Numerical Model ◽  
Accurate Solution ◽  
Ternary Alloys ◽  
Quaternary Alloys ◽  
Hilliard Equation ◽  
Model Based ◽  
Newton Raphson ◽  
Cahn Hilliard Equation

Numerical simulation of phase separation in Fe-Cr-Mo and Fe-Cr-Ni ternary alloys and Fe-Cr-Mo-Ni quaternary alloys were performed with use of the Cahn-Hilliard equation for ternary and quaternary alloys. A new numerical model based on the Gauss-Seidel and Newton Raphson methods was utilized to obtain efficient and accurate solution.

scholarly journals Effective Cahn-Hilliard Equation for the Phase Separation of Active Brownian Particles

2014 ◽  
Vol 112 (21) ◽  
Author(s):  
Thomas Speck ◽  
Julian Bialké ◽  
Andreas M. Menzel ◽  
Hartmut Löwen
Keyword(s):  
Phase Separation ◽  
Hilliard Equation ◽  
Brownian Particles ◽  
Cahn Hilliard Equation

scholarly journals Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Luca Scarpa
Keyword(s):  
Phase Separation ◽  
Random Component ◽  
Optimal Controls ◽  
Convection Term ◽  
Velocity Control ◽  
Optimal Velocity ◽  
Magnetic Devices ◽  
Hilliard Equation ◽  
Backward Problem ◽  
Cahn Hilliard Equation

AbstractA Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid’s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

scholarly journals Numerical methods for a system of coupled Cahn-Hilliard equations

2021 ◽  
Vol 12 (1) ◽  
pp. 1-12
Author(s):  
Mattia Martini ◽  
Giacomo E. Sodini
Keyword(s):  
Phase Separation ◽  
Numerical Methods ◽  
Experimental Approach ◽  
Numerical Solutions ◽  
Characteristic Parameters ◽  
Hilliard Equation ◽  
Cahn Hilliard Equation

Abstract In this work, we consider a system of coupled Cahn-Hilliard equations describing the phase separation of a copolymer and a homopolymer blend. We propose some numerical methods to approximate the solution of the system which are based on suitable combinations of existing schemes for the single Cahn-Hilliard equation. As a verification for our experimental approach, we present some tests and a detailed description of the numerical solutions’ behaviour obtained by varying the values of the system’s characteristic parameters.

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