Morphological characterization of spinodal decomposition kinetics

The time evolution of the morphology of homogeneous phases during spinodal decomposition is described using a family of morphological measures known as Minkowski functionals. They provide the characteristic length scale L of patterns in a convenient, statistically robust, and computationally inexpensive way. They also allow one to study the scaling behavior of the content, shape, and connectivity of spatial structures and to define the crossover from the early stage decomposition to the late stage domain growth. We observe the scaling behavior with , , and depending on the viscosity of the fluid. When approaching the spinodal density , we recover the prediction for the early time spinodal decomposition. Received 3 March 1998     

On the kinetics of spinodal decomposition

We discuss the kinetics of phase separation based on the mechanism of spinodal decomposition. The starting point is Boltzmann's transport equation. A perturbation theory will be developed leading to a hierarchy of differential equations for the local density. The first member of this hierarchy is a nonlinear wave-type equation, which linearized version will be solved in order to discuss the behaviour of the amplification factor. Additionally, we compare our results with those of standard diffusion-type approximations and molecular dynamics calculations.Research supported in part by National Science Foundation, Grant ENG-7 515 882-A01     

On the theory of spinodal decomposition in solid and liquid binary mixtures

The kinetics of phase separation is discussed with emphasis on the transition between spinodal decomposition and nucleation. A reanalysis of the theory of Langer, Baron and Miller shows that it exhibits a spinodal line somewhat closer to the coexistence curve than the meanfield spinodal. There the same (as we think unphysical) critical singularities occur as in Cahn-Hilliard theory. The precise location of this spinodal line depends on the cell size of the coarse graining. For concentrations less than the spinodal one the structure factorS(k, t) converges then towards the structure factor of the metastable onephase state, implying an infinite lifetime of the latter.In order to include the effects of nucleation and growth we hence present an alternative treatment, extending our previous work on cluster dynamics. From a simple approximation for the radial concentration distribution function of clustersS(k, t) is computed numerically. Even at rather low concentrations the time evolution ofS(k, t) is then similar to what Langer et al. find at high concentrations, implying a very gradual transition from nucleation and growth to spinodal decomposition, at least for parameter values appropriate to the Ising model. This treatment, which is consistent with Lifshitz-Slyozov's coarsening law at late times, is extended to the early stages of phase separation in liquid mixtures.     

Aggregation kinetics for a one-dimensional zero-degree Kelvin model of spinodal decomposition

We study analytically the approach to equilibrium in a simple zero-temperature model for phase separation in a binary alloy, in which nearest neighbor interchange can occur only if the portion of AB bonds is thereby decreased. The approach to equilibrium is found analytically. Because of the existence of infinitely many possible stationary states, the asymptotic distribution of AB pairs depends on the details of the initial state and must be obtained by a recursion method.Chargé de recherches FNRS.     

Dynamical spinodal: The transition between nucleation and spinodal decomposition

The nonclassical regime in the two phase region between nucleation and spinodal decomposition of a binary model with medium range interaction has been investigated. The Monte Carlo results indicate a dynamical spinodal. At this dynamical spinodal a transient percolating structure occurs. However, the mean droplet size remains finite there.     

Theory of spinodal decomposition in alloys

A statistical theory of the thermally driven composition fluctuations in a binary alloy is developed for the purpose of studying the phenomenon of spinodal decomposition. The theory can be stated in the form of a Fokker-Planck equation, which reduces, upon taking a suitable moment, to the nonlinear generalized diffusion equation which has been the basis of recent work in this field. Using the full Fokker-Planck equation, it is possible to compute the lifetime of the stationary solutions of the diffusion equation, and thus to study the rate at which the structure of the alloy coarsens during aging.     

Asymptotic behaviour of spinodal decomposition

We report a numerical study of the asymptotic regime of spinodal decomposition. Specifically, we find (a) that the characteristic pattern size shows a power-law growth with exponent 0.33; and (b) that the effect of noise is asymptotically irrelevant.     

Non-linear dynamics of spinodal decomposition

We develop a new technique describing the non linear growth of interfaces. We apply this analytical approach to the one dimensional Cahn-Hilliard equation. The dynamics is captured through a solvability condition performed over a particular family of quasi-static solutions. The main result is that the dynamics along this particular class of solutions can be expressed in terms of a simple ordinary differential equation. The density profile of the stationary regime found at the end of the non-linear growth is also well characterized. Numerical simulations are compared in a satisfactory way with the analytical results through three different fitting methods and asymptotic dynamics are well recovered, even far from the region where the approximations hold. Received 16 October 2001 / Received in final form 15 March 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: josseran@lmm.jussieu.fr RID="b" ID="b"UMR CNRS 7607     

Evidence for spinodal decomposition in nuclear multifragmentation

Multifragmentation of a "fused system" was observed for central collisions between 32 MeV/nucleon 129Xe and (nat)Sn. Most of the resulting charged products were well identified due to the high performances of the INDRA 4pi array. Experimental higher-order charge correlations for fragments show a weak but nonambiguous enhancement of events with nearly equal-sized fragments. Supported by dynamical calculations in which spinodal decomposition is simulated, this observed enhancement is interpreted as a "fossil" signal of spinodal instabilities in finite nuclear systems.     

Oscillatory solutions to the system of Allen-Cahn and Cahn-Hilliard equations: A spinodal decomposition model

The boundary-value problem for the system of Cahn-Novick-Cohen equations is analyzed. This problem is a quasi-continuum model of the corresponding lattice model for an Fe-Al alloy and simultaneously describes the processes of phase separation (spinodal decomposition) and atomic ordering in sublattices. It is demonstrated that the evolution of the system can occur according to the following three scenarios. (i) Against the background of a disordered state v = 0, spatially nonuniform distributions of the concentration u with respect to a stationary distribution u = u _m that is dependent on the mean mass m are developed at long times t → ∞. (ii) Against the background of a stationary distribution of the concentration u = u _m , spatially nonuniform distributions of the order parameter are developed at t → ∞. (iii) The first and second scenarios can proceed simultaneously for a specific set of parameters (for example, with the dimensionless temperature ? = T/T _c , where T _c is the critical temperature). The results of the calculations performed with so-called asymmetric boundary conditions of wetting in a constant magnetic field for a thin quasi-one-dimensional film consisting of a binary mixture are compared with data obtained from numerical and real experiments.     

Kinetics of spinodal decomposition in Fe-Si alloys

The spinodal decomposition in the two-phase region of the phase diagram (B₂ + DO₃) of Fe-Si alloys is studied on the basis of the generalized diffusion equation. The configurational free energy is calculated in the Bragg-Williams-Gorsky approximation. The effect of the coherency strain energy is estimated. In the main part of the paper the Langer's method is applied in the investigation of the coarsening rate in Fe-Si system with a good agreement with experimental data in a broad range of annealing times. The limitations of such approach are discussed with respect to the Khachaturyan's theory of the periodic distributions of decomposition products.The author is grateful to Dr. F.Kroupa for his guidance and encouragement during the course of this work, to Dr. S.Libovický and Dr. A.Gemperle for discussions of their latest experimental results.     

The spinodal decomposition of Cu-Ti alloys

The wave length of periodic modulated microstructure in an aged Cu-Ti alloy has been measured. Simultaneously the microstructure changes and development were studied in dependence on the ageing time at 573 K.     

A particle model for spinodal decomposition

We study a one-dimensional lattice gas where particles jump stochastically obeying an exclusion rule and having a small drift toward regions of higher concentration. We prove convergence in the continuum limit to a nonlinear parabolic equation whenever the initial density profile satisfies suitable conditions which depend on the strengtha of the drift. There is a critical valuea _c ofa. Fora<a _c, the density values are unrestricted, while foraa _c, they should all be to the right or to the left of a given interval (a). The diffusion coefficient of the limiting equation can be continued analytically to (a), and, in the interior of (a), it has negative values which should correspond to particle aggregation phenomena. We also show that the dynamics can be obtained as a limit of a Kawasaki evolution associated to a Kac potential. The coefficienta plays the role of the inverse temperature. The critical value ofa coincides with the critical inverse temperature in the van der Waals limit and (a) with the spinodal region. It is finally seen that in a scaling intermediate between the microscopic and the hydrodynamic, the system evolves according to an integrodifferential equation. The instanton solutions of this equation, as studied by Dal Passo and De Mottoni, are then related to the phase transition region in the thermodynamic phase diagram; analogies with the Cahn-Hilliard equations are also discussed.This paper is dedicated to Jerry Percus with great affection on the occasion of his 65th birthday.     

Asymmetric landscapes of early spinodal decomposition

We address the effect of an asymmetric concentration-dependent mobility on the early stages of spinodal decomposition in polymer blends and solutions, and characterise it quantitatively. This is particularly important when one of the components has a slower dynamics than the other, e.g., because it is closer to its glass transition, or is weakly cross-linked. Composition mode coupling is proposed as the underlying physical mechanism, and then investigated by means of numerical simulations of the Cahn-Hilliard equation in one (1d) and two (2d) dimensions. In general, this coupling broadens the peak in the structure factor: in 1d an asymmetric concentration profile is obtained, with sharpened interfaces, whereas in 2d the formation is favoured of sharp peaks of the phase rich in the more mobile component. It is shown how the changing morphology of this phase-separating system can be described with the aid of Minkowski functionals. Received 28 August 2000 and Received in final form 18 July 2001