B. Roulet

Interface kinetics and solidification of alloys: A discussion of some phenomenological models

Abstract We discuss various phenomenological descriptions of uniform interface kinetics in binary mixtures. We show that the linearized description provided by out-of-equilibrium thermodynamics is unambiguous and internally coherent. We discuss how various simple microscopic pictures of atomic attachment are reflected in the values of the corresponding Onsager coefficients. We discuss the far-from-equilibrium models of Aziz and Jackson-Gilmer-Leamy and the relationship between solute trapping and interspecies atomic drag. We propose another type of model in which the interaction between the kinetics of the two species is of the catalytic type. We show that such an interaction may give rise to oscillatory modes of growth, i.e. to kinetically-induced striations of the solid alloy.

The Mullins-Sekerka instability in directional solidification of thin samples

Abstract When a solid-liquid front is in contact with a confining wall, the condition of mechanical equilibrium imposes the value of the contact angle θ, which induces a meniscus-shaped deformation of the front. We studied the influence of this capillary effect on the Mullins-Sekerka instability observed in directional solidification of thin transparent samples, in the limit of a very weakly curved meniscus (θ⋍ θ 2 ) . For this purpose, we first calculated the shape of the meniscus in the symmetric model, with the help of an expansion (up to third order) of the front equation in powers of the deformation amplitudes. We studied its dependence on thickness in the vicinity of the Mullins-Sekerka bifurcation. We then performed the analysis of linear stability of this meniscus-shaped front, and showed that both the position of the cellular velocity threshold and the period of the cells are thickness-dependent.

Non-equilibrium thermodynamics of the solidification problem

Abstract The theory of non-equilibrium thermodynamics elaborated by Bedeaux, Albano and Mazur for two immiscible fluids is extended to describe solidification. The description includes convection and the possibility of surface transport. The relevance of the latter effect in various physical situations is discussed. When it is negligible, reduced interface conservation laws and kinetic equations are obtained. The usefulness of this formalism is illustrated by the sample example of the effect of linear attachment kinetics on the cellular instability of binary mixtures.

On the stability of hexagonal interfacial patterns in directional solidification of binary mixtures

Abstract Extending the work of Wollkind, Sriranganathan and Oulton, we study the stability of periodic hexagonal cellular front patterns against small modulations of the periodic structure. For this purpose, we first write down the “amplitude equations” describing the slow space and time variations of the front deformation close to the Mullins-Sekerka bifurcation. We then study the stability of their stationary hexagonal solutions against phase diffusion. We find that, due to phase diffusion instabilities, the range of stability of these solutions is always smaller than their range of existence.

On fluctuations and relaxation in systems described by a one-dimensional Fokker-Planck equation with a time-dependent potential

We study the effect of a time variation of a bistable potential on Kramers relaxation and on Suzukis fluctuation enhancement, for a system described by a one-variable Fokker-Planck equation, in the limit of a small constant diffusion coefficient. We show that the two processes must be described with two different approximations: 1.(i) Kramers relaxation can be treated in a quasi-adiabatic scheme. We then show that, for a large range of potential modulation rates, the local populations in each well obey adiabatic balance equations, the solution of which we discuss in various limits.2.(ii) Suzukis description of the fluctuation enhancement can be extended to the dynamical case. We reformulate the technique set up by Ahlers et al. to describe the crossing of the bifurcation from a mono- to a bistable potential. We show that the extended Suzuki approximation is valid provided that the crossing velocity is larger than a limit which we estimate.

High-frequency power spectra of systems described by non-linear Langevin equations

We show that, for systems described by non-linear first- and second-order Langevin equations, the high-frequency correlations always have respectively, an ω−2 and ω−4 asymptotic behavior, irrespective of the shape and strength of the non-linearities, and whether detailed balance is obeyed in the stationary state or not.

Kramers time in weakly nonpotential systems

We consider a bistable Fokker-Planck system with a known stationary distribution and a small nonpotential part in the drift force. We perform a perturbation calculation of its Kramers time,ΤK, and compare it with the corresponding time,ΤK(0), for the potential system which has the same stationary distribution. We show thatΤK/ΤK(0) depends only on the properties of the drift force close to the “saddle-point.”

Effect of a small slow modulation on Pomeau-Manneville intermittencies☆

Abstract It is shown that a small slow modulation on a system exhibiting Pomeau-Manneville intermittencies results in a small downward shift of the intermittency threshold, but preserves the ( −1 2 ) power law for the duration of the periodic pulses in the vicinity of this threshold.

Dynamical structures in Directional Solidification of Mixtures

Solidification patterns in mixtures are discussed and it is shown that, although the understanding of the factors affecting dendritic growth is improving, more experiments on cellular patterns are needed. The use of transparent eutectic materials is emphasized and the need for more theoretical work on atomic attachment kinetics is stressed.