New computational method in the theory of spinodal decomposition

Langer, J. S.; Baron, M.; Miller, Harold D.

doi:10.1103/physreva.11.1417

Cited by 831 publications

(331 citation statements)

References 23 publications

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“…The authors of the paper [55] mentioned that the present computational scheme does appear to be accurate enough to justify its use in the study of realistic metallurgical systems. Instead of thermodynamic projector which was not known in 1975, they used the projection onto the first three non-trivial moments ( u 2 , u 3 , u 4 ).…”

Section: Langer-bar-on-miller Approximation For Spinodal Decompositionmentioning

confidence: 99%

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Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Gorban

Karlin

2004

Physica A: Statistical Mechanics and its Applications

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Three results are presented: First, we solve the problem of persistence of dissipation for reduction of kinetic models. Kinetic equations with thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic projector is proven: There exists only one projector which transforms any vector field equipped with the given Lyapunov function into a vector field with the same Lyapunov function for a given ansatz manifold which is not tangent to the Lyapunov function levels.Second, we use the thermodynamic projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production increases. (The theorem about entropy overproduction.)In example, we apply the thermodynamic projector to derivation the equations of reduced kinetics for the Fokker-Planck equation. A new class of closures is developed, the kinetic multipeak polyhedra. Distributions of this type are expected in kinetic models with multidimensional instability as universally, as the Gaussian distribution appears for stable systems. The number of possible relatively stable states of a nonequilibrium system grows as 2 m , and the number of macroscopic parameters is in order mn, where n is the dimension of configuration space, and m is the number of independent unstable directions in this space. The elaborated class of closures and equations pretends to describe the effects of "molecular individualism". This is the third result.

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Section: Langer-bar-on-miller Approximation For Spinodal Decompositionmentioning

confidence: 99%

“…The process of spinodal decomposition was described quantitatively in the paper [55]. This model consists of two coupled equations: for the single-point distribution function of fluctuations, and for the pair correlation function.…”

Section: Langer-bar-on-miller Approximation For Spinodal Decompositionmentioning

confidence: 99%

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Gorban

Karlin

2004

Physica A: Statistical Mechanics and its Applications

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“…Subsequently Cahn developed a more general linearized theory of spinodal instability pointing out the essential role played by nonlinear effects in determining the nature of the instability and then in limiting its growth [24].…”

Section: Introductionmentioning

confidence: 99%

“…For example an Al-rich Al-Zn alloy, when quenched rapidly from above 400 • C and then annealed at temperatures in the neighborhood of 100 • C, is known to decompose into Al-and Zn-rich regions via the spinodal mechanism [24].…”

Section: Introductionmentioning

confidence: 99%

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

Berti

Bochicchio

2011

Math. Meth. Appl. Sci.

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In this paper we present a mathematical model to describe the phenomenon of phase separation, which is modelled as space regions where an order parameter changes smoothly. The model proposed, including thermal and mixing effects, is deduced for an incompressible fluid, so the resulting differential system couples a generalized Cahn-Hilliard equation with the Navier-Stokes equation. Its consistency with the second law of thermodynamics in the classical Clausius-Duhem form is finally proved.

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“…Spinodal decomposition in alloys has been a phenomenon of much interest since the seminal presentation of Cahn and Hilliard of the nonlinear diffusion equation. 11,12 Analytical research has addressed the nonlinearity of the equation, [13][14][15] but today computers are capable of calculating the full diffusion equation. The kinetics of decomposition have been simulated, [16][17][18] with some recent work based on the Onsager equation.…”

Section: Introductionmentioning

confidence: 99%

Vibrational entropy of spinodal decomposition in FeCr

2005

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Inelastic neutron-scattering spectra were measured on stoichiometric Fe 0.50 Cr 0.50 prepared as a bodycentered-cubic ͑bcc͒ solid solution, and after increasing amounts of chemical unmixing on the bcc lattice induced by annealing the solid solution at 773 K. These spectra were reduced by a conventional procedure to a neutron-weighted vibrational density of states. Mössbauer spectrometry was used to characterize the extent of decomposition after annealing. A neutron-weight correction was performed, using results from the Mössbauer spectra and recent data on inelastic nuclear resonant scattering from 57 Fe-Cr. The vibrational entropy of decomposition was found to be −0.17± 0.01k B / atom, nearly equal to the change in configurational entropy after spinodal decomposition. Effects of vibrational entropy on the thermodynamics of unmixing are analyzed, showing a large effect on the free energy with the formation of Cr-rich zones, and a large effect on the critical temperature for spinodal decomposition for equiatomic Fe 0.50 Cr 0.50 .

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New computational method in the theory of spinodal decomposition

Cited by 831 publications

References 23 publications

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

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

Vibrational entropy of spinodal decomposition in FeCr

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