Morphological instability in rapid directional solidification

Merchant, G. J.; Davis, Stephen H.

doi:10.1016/0956-7151(90)90282-l

Cited by 103 publications

(57 citation statements)

References 14 publications

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“…By considering both the velocity dependent solute partition model and the limitation of atomic kinetic attachment, the interface stability analysis has found the time-dependent oscillatory instability of planar interface with the velocities near the absolute stability limit velocity of a planar interface [77,78]. In this stability analysis, it also has been shown that the oscillatory instability is controlled by the non-equilibrium solidification effects, i.e.…”

Section: Chapter 1 Introductionmentioning

confidence: 92%

“…Merchant-Davis [78] modified the analysis of Coriell-Sekerka [77]. Using Aziz's solute trapping model for dilute solutions [2] and incorporating the kinetic undercooling at the interface, the Merchant-Davis's model allows velocity-dependent k V and T* in a thermodynamically consistent way.…”

Section: Interface Stability Analysis For Banded Structure Formationmentioning

confidence: 99%

“…The modified linear absolute stability analysis accounting for local non-equilibrium effect, such as solute trapping [77,78], may be related with the band structure formation. The result of the modified linear absolute stability analysis for planar interface showed that the velocity of absolute stability limit, V AS , is higher than that from the classical Mullins-Sekerka's stability analysis, and at velocities slightly less than V AS , there is an oscillatory instability with time for rather long wavelengths.…”

Section: Interface Stability Analysis For Banded Structure Formationmentioning

confidence: 99%

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Phase-field investigation on the non-equilibrium interface dynamics of rapid alloy solidification

Choi

2011

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Section: Chapter 1 Introductionmentioning

confidence: 92%

Section: Interface Stability Analysis For Banded Structure Formationmentioning

confidence: 99%

Section: Interface Stability Analysis For Banded Structure Formationmentioning

confidence: 99%

See 1 more Smart Citation

Phase-field investigation on the non-equilibrium interface dynamics of rapid alloy solidification

Choi

2011

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“…A major advantage of our approach is that it allows a transparent and, to a large extend, analytical evaluation. This includes a readjustment of the stability analysis by Merchant and Davis [11] who discovered an oscillatory instability, similar to that, discussed earlier by Coriell and Sekerka [12]. Also included is a clarifying analysis of the so far barely understood low-velocity sections of the cyclic trajectories, identified by Carrard et al [1], and by Karma and Sarkissian [9].…”

Section: Rudi Schmitzmentioning

confidence: 99%

Diffusion-Induced Oscillations of Extended Defects

2012

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From a simple model for the driven motion of a planar interface under the influence of a diffusion field we derive a damped nonlinear oscillator equation for the interface position. Inside an unstable regime, where the damping term is negative, we find limit-cycle solutions, describing an oscillatory propagation of the interface. In case of a growing solidification front this offers a transparent scenario for the formation of solute bands in binary alloys, and, taking into account the Mullins-Sekerka instability, of banded structures. The interaction of propagating extended defects with a diffusion field frequently leads to oscillations or jerky motions of the defects. A prime example of such an effect is the oscillation of a solidification front, induced by the diffusion of the solute component in a dilute binary alloy, which is growing in the setup of directional solidification. In a large number of metallic materials this leads to the formation of banded structures [1], reflecting a periodic array of layers with high and low solute concentrations where the former ones show a dendritic microstructure. The appearance of similar banding effects has recently been discussed [2] in rapid solidification of colloids.Layered structures are also generated by the oscillatory nucleation of a solid phase under the action of a diffusion field [3]. A related phenomenon is the oscillatory zoning, observed in solid solutions [4] and in natural minerals [5]. Another notable scenario is that of diffusion-controlled jerky motions of a driven grain boundary [6]. A similar behavior of dislocations in metallic alloys leads to the Portevin-Le Chatelier effect [7], denoting the appearance of jerky plastic deformations. We, finally, mention the oscillatory motion of a crack tip, which is coated by the nucleus of a new phase [8], replacing the attached cloud of a diffusion field.Theoretical discussions of such effects either are of a phenomenological type, like those in Ref. [7], and partly in Refs.[1] and [2], or they rely on a Fokker-Planck [6], or a diffusion equation with non-equilibrium boundary conditions [9]. In all approaches the source of oscillatory defect motions is identified as an unstable regime where a reduction of the driving force leads to an increase of the defect velocity. Additional information is provided by simulations, based on phase-field models for directional solidification [10] and for nucleation [3] processes.In the present Letter we introduce an extremely simple but powerful model for the diffusion-induced oscillatory motion of a planar interface, using the language adapted to the directional solidification of a dilute binary alloy. A major advantage of our approach is that it allows a transparent and, to a large extend, analytical evaluation. This includes a readjustment of the stability analysis by Merchant and Davis [11] who discovered an oscillatory instability, similar to that, discussed earlier by Coriell and Sekerka [12]. Also included is a clarifying analysis of the so far barely understood low-velo...

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“…This situation occurs in many applications and the coupled system of equations described above has been derived by many authors. We mention here some physical examples where two unstable modes can interact: double-layer convection [25], [24], [18]; crystal-growing experiments (where the convective and morphological modes can interact) [13], [26], [21]; gasless combustion [19]; sand ripple formation [31]. The coupled system of modulation equations has, for instance, been derived in [19], [17], [22].…”

Section: Introductionmentioning

confidence: 99%

Singularly Perturbed and Nonlocal Modulation Equations for Systems with Interacting Instability Mechanisms

Doelman

Rottschäfer

1997

Journal of Nonlinear Science

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Summary. Two related systems of coupled modulation equations are studied and compared in this paper. The modulation equations are derived for a certain class of basic systems which are subject to two distinct, interacting, destabilising mechanisms. We assume that, near criticality, the ratio of the widths of the unstable wavenumber-intervals of the two (weakly) unstable modes is small-as, for instance, can be the case in doublelayer convection. Based on these assumptions we first derive a singularly perturbed modulation equation and then a modulation equation with a nonlocal term. The reduction of the singularly perturbed system to the nonlocal system can be interpreted as a limit in which the width of the smallest unstable interval vanishes. We study and compare the behaviour of the stationary solutions of both systems. It is found that spatially periodic stationary solutions of the nonlocal system exist under the same conditions as spatially periodic stationary solutions of the singularly perturbed system. Moreover, these solutions can be interpreted as representing the same quasi-periodic patterns in the underlying basic system. Thus, the 'Landau reduction' to the nonlocal system has no significant influence on the stationary quasi-periodic patterns. However, a large variety of intricate heteroclinic and homoclinic connections is found for the singularly perturbed system. These orbits all correspond to so-called 'localised structures' in the underlying system: They connect simple periodic patterns at x → ±∞. None of these patterns can be described by the nonlocal system. So, one may conclude that the reduction to the nonlocal system destroys a rich and important set of patterns.

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Morphological instability in rapid directional solidification

Cited by 103 publications

References 14 publications

Phase-field investigation on the non-equilibrium interface dynamics of rapid alloy solidification

Phase-field investigation on the non-equilibrium interface dynamics of rapid alloy solidification

Diffusion-Induced Oscillations of Extended Defects

Singularly Perturbed and Nonlocal Modulation Equations for Systems with Interacting Instability Mechanisms

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