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Journal ArticleDOI

Spinodal Decomposition for Multicomponent Cahn–Hilliard Systems

01 Feb 2000-Journal of Statistical Physics (Kluwer Academic Publishers-Plenum Publishers)-Vol. 98, Iss: 3, pp 871-896
TL;DR: In this paper, the authors considered the initial stage phase separation process in multicomponent Cahn-Hilliard systems through spinodal decomposition and established the existence of certain dominating subspaces determining the behavior of most solutions originating near a spatially homogeneous state.
Abstract: We consider the initial-stage phase separation process in multicomponent Cahn–Hilliard systems through spinodal decomposition. Relying on recent work of Maier-Paape and Wanner, we establish the existence of certain dominating subspaces determining the behavior of most solutions originating near a spatially homogeneous state. It turns out that, depending on the initial concentrations of the alloy components, several distinct phenomena can be observed. For ternary alloys we observe the following two phenomena: If the initial concentrations of the three components are almost equal, the dominating subspace consists of two copies of the finite-dimensional dominating subspace from the binary alloy case. For all other initial concentrations, only one copy of the binary dominating subspace determines the behavior. Thus, in the latter case we observe a strong mutual coupling of the concentrations in the alloy during the initial separation process.
Citations
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01 Jan 2016
TL;DR: In this paper, Zuc11b et al. this paper showed that 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15], 1/n [Per17] and 1/m [DFL17] were the most frequent p ≤ p ≥ ∞.
Abstract: (2 + 1) [XTpXpH12, CTH11]. + [Zuc11b]. 0 [Fed17]. 1 [BELP15, CAS11, Cor16, Fed17, GDL10, GBL16, Hau16, JV19, KT12, KM19c, Li19, MN14b, Nak17, Pal11, Pan14, RT14, RBS16b, RY12, SS18c, Sug10, dOP18]. 1 + 1 [Sak18, CP15b]. 1/2 [MD10]. 1/f [FDR12]. 1/n [Per17]. 1/|x− y| [MSV10, MSV13]. 13 [DFL17]. 1 ≤ p ≤ ∞ [Dud13]. 1/f [HPF15]. 2 [AB19, ADS19, BF12, BNT13, DSS15, EKD12, Her13, Ily12, Lan10, Li12, Li19, LZ11, Ny13, Ost16, PSS16, ST14, Sch13b, TJ15, WPB15, dWL10]. 2 + 1 [dWL14]. 2.5 [BC15a]. 2R [WLEC17]. 2× 2 [CLTT13]. 3 [BCF19, BLS17, ESPP14, Kar18, SH16, SWKS14, dCCS19]. 3/2 [DK10]. 38 [Cam13]. 4 [BBS14, Zha14]. 4× 4 [LN19a]. 5/2 [DK10, EKD12]. 6 [EC11]. 8 [Zha14]. 90◦ [YM11]. 3 [Afz12]. 1−x [EFO11]. 13 [CDCL18]. 2 [ML15, QR13, ST11c]. 4 [HBB10]. 6 [BCL10a, BCL10b, EFO11]. x [EFO11].

129 citations

Journal ArticleDOI
TL;DR: In this paper, the authors developed an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics, which is used in many areas of science, from particle physics to sociology.
Abstract: Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicomponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant interaction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics. Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion. We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell--jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation.

57 citations


Cites background from "Spinodal Decomposition for Multicom..."

  • ...The problem of the extension of the Cahn–Hilliard approach to multicomponent diffusion was discussed by various authors [76, 4]....

    [...]

Journal ArticleDOI
TL;DR: The scheme is based on a nonlinear splitting method and is solved by an efficient and accurate nonlinear multigrid method and allows the N-component Cahn–Hilliard system to be converted into a system of N−1 binary Cahn-Hilliard equations and significantly reduces the required computer memory and CPU time.
Abstract: We present a practically unconditionally gradient stable conservative nonlinear numerical scheme for the N-component Cahn?Hilliard system modeling the phase separation of an N-component mixture. The scheme is based on a nonlinear splitting method and is solved by an efficient and accurate nonlinear multigrid method. And the scheme allows us to convert the N-component Cahn?Hilliard system into a system of N-1 binary Cahn?Hilliard equations and significantly reduces the required computer memory and CPU time. We observe that our numerical solutions are consistent with the linear stability analysis results. We also demonstrate the efficiency of the proposed scheme with various numerical experiments.

56 citations


Cites background from "Spinodal Decomposition for Multicom..."

  • ...[27] explained the initial-stage phase separation process in multi-component CH systems through spinodal decomposition....

    [...]

Journal ArticleDOI
TL;DR: In this article, the authors developed an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics, and demonstrated how the mechanism-based approach can be in-cluded into the general thermodynamic framework, and prove the corresponding dissipation in equalities.
Abstract: Diffusion preserves the positivity of concentrations, therefore, multicomponent diffu- sion should be nonlinear if there exist non-diagonal terms. The vast variety of nonlinear multicom- ponent diffusion equations should be ordered and special tools are needed to provide the systematic construction of the nonlinear diffusion equations for multicomponent mixtures with significant in- teraction between components. We develop an approach to nonlinear multicomponent diffusion based on the idea of the reaction mechanism borrowed from chemical kinetics. Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion. We demonstrate, how the mechanism based approach to multicomponent diffusion can be in- cluded into the general thermodynamic framework, and prove the corresponding dissipation in- equalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell-jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation.

56 citations

Journal ArticleDOI
TL;DR: In this paper, a new generalization of the well-known Cahn-Hilliard two-phase model for the modeling of n-phase mixtures is proposed, which is derived using the consistency principle.
Abstract: In this paper, we propose a new generalization of the well-known Cahn–Hilliard two-phase model for the modeling of n-phase mixtures. The model is derived using the consistency principle: we require that our n-phase model exactly coincides with the classical two-phase model when only two phases are present in the system. We give conditions for the model to be well-posed. We also present numerical results (including simulations obtained when coupling the Cahn–Hilliard system with the Navier–Stokes so as to obtain a phase-field model for multiphase flows) to illustrate the capability of such modeling.

53 citations

References
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Journal ArticleDOI
TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
Abstract: It is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density is given by : NV∫V [f 0(c)+κ(▿c)2]dV, where NV is the number of molecules per unit volume, ▿c the composition or density gradient, f 0 the free energy per molecule of a homogeneous system, and κ a parameter which, in general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. This expression is used to determine the properties of a flat interface between two coexisting phases. In particular, we find that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc , and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2 . The predicted interfacial free energy and its temperature dependence are found to be in agreement with existing experimental data. The possibility of using optical measurements of the interface thickness to provide an additional check of our treatment is briefly discussed.

8,720 citations

Book
01 Jan 1947
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.

7,426 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present an algebraic extension of LINEAR TRANSFORMATIONS and QUADRATIC FORMS, and apply it to EIGEN-VARIATIONS.
Abstract: Partial table of contents: THE ALGEBRA OF LINEAR TRANSFORMATIONS AND QUADRATIC FORMS. Transformation to Principal Axes of Quadratic and Hermitian Forms. Minimum-Maximum Property of Eigenvalues. SERIES EXPANSION OF ARBITRARY FUNCTIONS. Orthogonal Systems of Functions. Measure of Independence and Dimension Number. Fourier Series. Legendre Polynomials. LINEAR INTEGRAL EQUATIONS. The Expansion Theorem and Its Applications. Neumann Series and the Reciprocal Kernel. The Fredholm Formulas. THE CALCULUS OF VARIATIONS. Direct Solutions. The Euler Equations. VIBRATION AND EIGENVALUE PROBLEMS. Systems of a Finite Number of Degrees of Freedom. The Vibrating String. The Vibrating Membrane. Green's Function (Influence Function) and Reduction of Differential Equations to Integral Equations. APPLICATION OF THE CALCULUS OF VARIATIONS TO EIGENVALUE PROBLEMS. Completeness and Expansion Theorems. Nodes of Eigenfunctions. SPECIAL FUNCTIONS DEFINED BY EIGENVALUE PROBLEMS. Bessel Functions. Asymptotic Expansions. Additional Bibliography. Index.

4,525 citations

Journal ArticleDOI
TL;DR: In this article, matched asymptotic expansions are used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions, when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature.
Abstract: The method of matched asymptotic expansions is used to describe solutions of the nonlinear Cahn-Hilliard equation for phase separation in N > 1 space dimensions The expansion is formally valid when the thickness of internal transition layers is small compared with the distance separating layers and with their radii of curvature On the dominant (slowest) timescale the interface velocity is determined by the mean curvature of the interface, by a non-local relation which is identical to that in a well-known quasi-static model of solidification, which exhibits a shape instability discovered by Mullins & Sekerka (J appl Phys 34, 323-329 (1963)) On a faster timescale, the Cahn-Hilliard equation regularizes a classic two-phase Stefan problem Similarity solutions of the two-phase Stefan problem should describe boundary layers Existence and uniqueness of such similarity solutions which admit metastable states is proved rigorously in an appendix

480 citations

Journal ArticleDOI
TL;DR: The thermodynamic treatment of non-uniform systems of Cahn and Hilliard is shown to be equivalent to the self-consistent thermodynamic formalism of Hart as discussed by the authors.
Abstract: The thermodynamic treatment of nonuniform systems of Cahn and Hilliard is shown to be equivalent to the self‐consistent thermodynamic formalism of Hart All parameters of the two treatments have been rigorously related and key equations have been shown to be equivalent

419 citations