Stochastic generalization for a hyperbolic model of spinodal decomposition

A model for diffusion and phase separation which takes into account exponential relaxation of the solute diffusion flux and its fluctuations is developed. The model describes a system undergoing phase separation governed by a partial differential equation of hyperbolic type. The analysis is done for the evolution of patterns in spinodal decomposition for the system supercooled below critical temperature. Analytical results show that relaxation processes of the solute diffusion flux lead to the selection of patterns with different wavenumbers. Considering spatial-temporal correlations of the flux fluctuations, we have found that the temporal correlations promote selecting large-period patterns, whereas the corresponding spatial correlations accelerate such processes.     

Non-equilibrium effects in spinodal decomposition of a binary system

A model for diffusion and phase separation which takes into account relaxation of the solute diffusion flux is developed. It is shown that the model predicts non-linearity in the amplification rate of decomposition, which is governed by the ratio between diffusion and correlation lengths. The predicted amplification rate is tested against experimental data on a binary phase-separated glass.     

Evolution of the structure factor in a hyperbolic model of spinodal decomposition

We consider the modification of the Cahn-Hilliard equation when a time delay process through a memory function is taken into account. The memory effects are seen to affect the dynamics of phase transition at short times. The process of fast spinodal decomposition associated with a conserved order parameter - concentration is studied numerically. Details of a semi-implicit numerical scheme used to simulate the kinetics of spinodal decomposition and evolution of the structure factor are discussed. Analysis of the modeled structure factor predicted by a hyperbolic model of spinodal decomposition is presented in comparison with the parabolic model of Cahn and Hilliard. It is shown that during initial periods of decomposition the structure factor exhibits wave behavior. Analytical treatments explain such behavior by existence of damped oscillations in structure factor at earliest stages of phase separation and at large values of the wave-number. These oscillations disappear gradually in time and the hyperbolic evolution approaches the pure dissipative parabolic evolution of spinodal decomposition.     

Model of spinodal decomposition of phases under hyperbolic diffusion

A model is proposed for the kinetics of spinodal decomposition at high diffusion rates described by a hyperbolic equation. The formation of new phases is described using a zero-radius nonlinear potential model. It is shown that a regular distribution in space of the phases during spinodal decomposition is initiated by suitable initial conditions for the concentration gradient with constant initial concentrations of the components. Fiz. Tverd. Tela (St. Petersburg) 41, 907–909 (May 1999)     

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.     

A soluble kinetic model for spinodal decomposition

We compare the two-dimensional voter model with approximate theories for spinodal decomposition. The cluster size distribution and the short-time dynamics of the voter model are studied by means of a Monte Carlo simulation. The time-dependent structure factor and the long-time scaling of the voter dynamics are known analytically.This paper is dedicated to Nico van Kampen on the occasion of his 67th birthday.     

Shock wave stability under the spinodal decomposition of binary mixtures

A diffusion equation for a binary mixture and spinodal decomposition in the case of phase separation are considered. It is shown that, if the binding force between polymer chain links is weak, the diffusion equation for a binary mixture allows for the reduction to the Burgers equation with “viscosity”; that is, the coexistence of rarefaction waves and shock density waves is a possibility. The effect of strong bonds between polymer chain links on the spinodal decomposition dynamics is studied. It is demonstrated that strong bonding may cause a multiflux wave system with alternate stability to arise when the viscosity varies.     

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.     

Experimental study of spinodal decomposition in a 1D conserved order parameter system

We study the zigzag instability coarsening of splay-bend walls formed in a nematic liquid crystal under external fields. The vertexes of zigzag can be considered as kinks in a one-dimensional order parameter system and the geometrical constraints associated with the necessary equal length sum of zig and zag segments impose a conserved quantity in this Cahn-Hilliard-type problem. In the late stage of coarsening, the characteristic length of the system L(t) shows a logarithmic increase in time and the dynamical scaling law holds. We then try to extract the nontrivial asymptotic scaling exponent lambda of the two-time correlation function, defined by lim( approximately [L(t)/L(t('))](-lambda). The scaling exponents with respective time references, t(')=32 and 64 s, after quench are found to be lambda approximately 2 which is larger than the value with respective time reference t(')=0, predicted by numerical simulation.     

The nature of isostructural spinodal decomposition in the Al-Zn system

The electronic structure and thermodynamic properties of disordered Al-Zn alloys are investigated in the coherent potential approximation by the KKR-ASA method. Analysis of the electronic density of states and the Fermi surfaces of disordered alloys reveals the presence of eight electronic topological transitions in the concentration interval from 0 to 70 at. % Zn. It is shown that the passage of the Fermi level through two minima of the density of states, which are due to the superposition of different types of electronic topological transitions, gives rise to singularities in the concentration dependence of the second derivative of the thermodynamic potential at points corresponding to the boundaries of the region of isostructural decomposition of the high-temperature solid solution, according to the phase diagram of the Al-Zn system. Fiz. Tverd. Tela (St. Petersburg) 39, 593–596 (April 1997)     

Experimental small-angle X-ray scattering investigation of initial stages of the spinodal decomposition in model sodium silicate glasses

The kinetics of the initial stages of the spinodal decomposition in model glasses of the Na₂O-SiO₂ system has been investigated in situ. It has been demonstrated that there is a quantitative agreement of the experimental results obtained in the framework of the Stephenson theory with the basic principles of modern theories and the data on direct determination of the viscosity, mobility, and diffusion. It has been found that the spatial-temporal evolution of the heterogeneous structure has a multistage character during spinodal decomposition. The characteristic size of the phase regions at each stage varies with time according to the power law. The sequence of stages and the values of exponents for the spinodal decomposition are as follows: 1/20, 1/4, 1/2, and 1/3.     

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.     

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.     

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.