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# On the mathematical modelling of cellular (discontinuous) precipitation

• Author Bio: E-mail address: O.Penrose@hw.ac.uk
• 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 $c_0$, the solute concentration in the metastable phase. It is hoped that our analytical treament 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 $c_0$ at which $v$ can be vanishingly small.

Mathematics Subject Classification: Primary:74N20 Dynamicsofphaseboundaries; Secondary:34B15Nonlinearboundaryvalueproblems, 74A50 Structuredsurfacesandinterfaces, coexistentphases, 74G15Numericalapproximationofsolutions, 74N25Transformationsinvolvingdiffusion.

 Citation: • • Figure 1.  Schematic cross-section of a cellular precipitation front, advancing upwards at velocity $v$. The as yet undisturbed metastable phase is labelled 0; the two (relatively) stable phases into which it separates after the front has passed are labelled $\alpha$ and $\beta$

Figure 2.  Notation for the coordinate axes and for angles and arc lengths

Figure 3.  The thermodynamic construction for phase equilibrium. The common tangent has slope $\mu_{TJ}$ and touches the curve at the points with abscissas $c_\alpha^{eq}$ and $c_\beta^{eq}.$

Figure 4.  Dihedral angles $\delta$, inclinations of arcs $\theta$ and surface tensions $\gamma$ at a triple junction

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