### Abstract

Cellular precipitation is a dynamic phase transition in solid solutions (such as alloys) where a metastable phase decomposes into two stable phases : an approximately planar (but corrugated) boundary advances into the metastable phase, leaving behind it interleaved plates (lamellas) of the two stable phases.

The forces acting on each interface (thermodynamic, elastic and surface tension) are modelled here using a first-order ODE and the diffusion of solute along the interface by a second-order ODE, with boundary conditions at the triple junctions where three interfaces meet. Careful attention is paid to the approximations and physical assumptions used in formulating the model.

These equations, previously studied by approximate (mostly numerical) methods, have the peculiarity that v; the velocity of advance of the interface, is not uniquely determined by the given physical data such as c0, the solute concentration in the metastable phase. It is hoped that our analytical treatment will help to improve the understanding of this.

We show how to solve the equations exactly in the limiting case where v = 0. For larger v, a successive approximation scheme is formulated. One result of the analysis is that there is just one value for c0 at which v can be vanishingly small.

The forces acting on each interface (thermodynamic, elastic and surface tension) are modelled here using a first-order ODE and the diffusion of solute along the interface by a second-order ODE, with boundary conditions at the triple junctions where three interfaces meet. Careful attention is paid to the approximations and physical assumptions used in formulating the model.

These equations, previously studied by approximate (mostly numerical) methods, have the peculiarity that v; the velocity of advance of the interface, is not uniquely determined by the given physical data such as c0, the solute concentration in the metastable phase. It is hoped that our analytical treatment will help to improve the understanding of this.

We show how to solve the equations exactly in the limiting case where v = 0. For larger v, a successive approximation scheme is formulated. One result of the analysis is that there is just one value for c0 at which v can be vanishingly small.

Original language | English |
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Pages (from-to) | 963–982 |

Number of pages | 20 |

Journal | Discrete and Continuous Dynamical Systems-Series A |

Volume | 37 |

Issue number | 2 |

Early online date | Nov 2016 |

DOIs | |

Publication status | Published - Feb 2017 |

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## Cite this

Penrose, O., & Cahn, J. W. (2017). On the mathematical modelling of cellular (discontinuous) precipitation.

*Discrete and Continuous Dynamical Systems-Series A*,*37*(2), 963–982. https://doi.org/10.3934/dcds.2017040