A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening

Allen, Samuel M.; Cahn, John W.

doi:10.1016/0001-6160(79)90196-2

Cited by 3,262 publications

(2,014 citation statements)

References 31 publications

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“…Moreover, recall the expressions (18) and (25) For the connection between Q − 1 and P 1 , recall that in the invariant plane {ε 1 = 0}, system (22a)-(22c) corresponds precisely to the original FKPP equation (1). Hence, the heteroclinic orbit connecting Q − 1 and P 1 is given by the corresponding FKPP orbit, denoted by − 1 above, after blow-up.…”

Section: Proposition 27 Let ϕ Be a Cut-off Function Which Satisfiesmentioning

confidence: 99%

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The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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The Fisher-Kolmogorov-Petrowskii-Piscounov (FKPP) equation with cut-off was introduced in (Brunet and Derrida 1997 Shift in the velocity of a front due to a cut-off Phys. Rev. E 56 2597-604) to model N-particle systems in which concentrations less than ε = 1/N are not attainable. It was conjectured that the cut-off function, which sets the reaction terms to zero if the concentration is below the small threshold ε, introduces a substantial shift in the propagation speed of the corresponding travelling waves. In this paper, we prove the conjecture of Brunet and Derrida, showing that the speed of propagation is given by c crit (ε) = 2 − π 2 /(ln ε) 2 + O((ln ε) −3 ), as ε → 0, for a large class of cut-off functions. Moreover, we extend this result to a more general family of scalar reaction-diffusion equations with cut-off. The main mathematical techniques used in our proof are the geometric singular perturbation theory and the blow-up method, which lead naturally to the identification of the reasons for the logarithmic dependence of c crit on ε as well as for the universality of the corresponding leading-order coefficient (π 2 ).

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Section: Proposition 27 Let ϕ Be a Cut-off Function Which Satisfiesmentioning

confidence: 99%

“…To analyse rigorously the transition through chart K 1 in the vicinity of the point P 1 (which is now located at the origin) for ε ∈ (0, ε 0 ) small, we have to describe the map 1 …”

Section: Transition Through Chart Kmentioning

confidence: 99%

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

2007

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“…The Allen-Cahn equation was originally introduced in [1] as a phenomenological model for anti-phase domain coarsening in a binary alloy. It has been subsequently applied to a wide range of other different problems such as the motion by mean curvature flows (cf., e.g., [12]) and the crystal growth (cf., e.g., [23]).…”

Section: Introductionmentioning

confidence: 99%

Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions

Calatroni

Colli

2013

Nonlinear Analysis: Theory, Methods & Applications

110

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Abstract. We prove well-posedness results for the solution to an initial and boundaryvalue problem for an Allen-Cahn type equation describing the phenomenon of phase transitions for a material contained in a bounded and regular domain. The dynamic boundary conditions for the order parameter have been recently proposed by some physicists to account for interactions with the walls. We show our results using suitable regularizations of the nonlinearities of the problem and performing some a priori estimates which allow us to pass to the limit thanks to compactness and monotonicity arguments.---------------------

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“…In case of significant damping (e.g., through friction) we have |ρφ| |νφ| so that (S21) reduces to the Allen-Cahn equation [10,11]. Of course, damping in the system can be arbitrarily complex, so the linear approximation chosen here is only a leading-order approximation which, however, worked excellently for our experimentally investigated 1D bistable networks [12,13].…”

Section: Discrete Network and Continuum Limitmentioning

confidence: 99%

“…In addition, if the curvature is small compared to the interface thickness (i.e., R c w), then [11] Qψ ( ) − P ,rr ≈ 0, reducing (S33) to˙ + P ς ,r = 0 (S34)…”

Section: Interface Motion In the Phase Field Modelmentioning

confidence: 99%

Atomimetic Mechanical Structures with Nonlinear Topological Domain Evolution Kinetics

Frazier

Kochmann

2017

Advanced Materials

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A mechanical metamaterial, a simple, periodic mechanical structure, is reported, which reproduces the nonlinear dynamic behavior of materials undergoing phase transitions and domain switching at the structural level. Tunable multistability is exploited to produce switching and transition phenomena whose kinetics are governed by the same Allen–Cahn law commonly used to describe material‐level, structural‐transition processes. The reported purely elastic mechanical system displays several key features commonly found in atomic‐ or mesoscale physics of solids. The rotating‐mass network shows qualitatively analogous features as, e.g., ferroic ceramics or phase‐transforming solids, and the discrete governing equation is shown to approach the phase field equation commonly used to simulate the above processes. This offers untapped opportunities for reproducing material‐level, dissipative and diffusive kinetic phenomena at the structural level, which, in turn, invites experimental realization and paves the road for new active, intelligent, or phase‐transforming mechanical metamaterials bringing small‐scale processes to the macroscopically observable scale.

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A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening

Cited by 3,262 publications

References 31 publications

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off

Global solution to the Allen–Cahn equation with singular potentials and dynamic boundary conditions

Atomimetic Mechanical Structures with Nonlinear Topological Domain Evolution Kinetics

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